%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: camera.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips camera %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.06.01:1055 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (camera.dvi) @start /Fa 42 127 df45 D<70F8F8F8700505798414>I< 0006000E000E001C001C003800380070007000E000E001C001C003800380038007000700 0E000E001C001C003800380070007000E000E000C0000F1D7E9914>I<0FE03FF8783C70 1CE00EE00EE00EE00E701C1EF003801FF03838701CE00EE00EE00EE00EF01E701C38381F F007C00F177E9614>56 D<07C01FE038307038601CE01CE00CE00EE00E601E701E383E1F EE07CE000E001C001C001C7038703070E03FC01F000F177E9614>I<70F8F8F870000000 00000070F8F8F8700510798F14>I<01F18007FF800E1F801C0780380780700380700380 700380E00000E00000E00000E00000E00000E00000E00000700380700380700380380700 1C07000E0E0007FC0001F00011177F9614>67 DII<03C60FFE1C3E181E 381E700E700E600EE000E000E000E000E07FE07FE00E600E700E700E381E181E1C3E0FFE 03CE10177F9614>71 D73 D77 D<1FF07FFC783C701CE00EE00EE00EE00EE00EE00EE00EE00EE00EE00EE00E E00EE00EE00EE00E701C783C7FFC1FF00F177E9614>79 D82 D<0FCC1FFC307C603CE01CE01CE01CE00070007E003FE00FF001F8001C001E000E600EE0 0EE00EF01CF838FFF0C7E00F177E9614>I<7FFF80FFFF80E1C380E1C380E1C380E1C380 01C00001C00001C00001C00001C00001C00001C00001C00001C00001C00001C00001C000 01C00001C00001C0000FF8000FF80011177F9614>I87 D<1FC0007FF000707800 201800001C00001C0007FC001FFC003C1C00701C00E01C00E01C00E01C00707C003FFF80 0F8F8011107E8F14>97 DI<03F80FFE1C0E380470006000E000E000E000E00060 007000380E1C1E0FFC03F00F107E8F14>I<007E00007E00000E00000E00000E00000E00 000E0007CE000FFE001C3E00301E00700E00E00E00E00E00E00E00E00E00E00E00E00E00 700E00301E00383E001FEFC007CFC012177F9614>I<07E00FF01C38301C700CE00EE00E FFFEFFFEE00060007000380E1C1E0FFC03F00F107E8F14>I<00FC01FE038E0704070007 0007007FFEFFFE070007000700070007000700070007000700070007000700FFF8FFF80F 177F9614>I<07CF001FFF80383B80301800701C00701C00701C003018003838003FF000 37C0007000007000003FF8001FFC003FFE00700F00E00380E00380E00380E00380700700 3C1E001FFC0007F00011197F8F14>II<06000F000F0006000000000000000000 FF00FF00070007000700070007000700070007000700070007000700FFF8FFF80D187C97 14>I<006000F000F0006000000000000000001FF01FF000700070007000700070007000 700070007000700070007000700070007000700070007040E0E0C07F803F000C207E9714 >IIIII<07C01FF03C78701C701CE0 0EE00EE00EE00EE00EE00E701C783C3C781FF007C00F107E8F14>II114 D<0FD83FF86038C038C038F0007F803FF0 07F8001C6006E006F006F81CFFF8CFE00F107E8F14>I<06000E000E000E000E007FFCFF FC0E000E000E000E000E000E000E000E000E0E0E0E0E0E0E1C07F801F00F157F9414>I< FC3F00FC3F001C07001C07001C07001C07001C07001C07001C07001C07001C07001C0700 1C07001C1F000FFFE003E7E01310808F14>III121 D<3FFF7FFF700E701C7038007000E001C0038007000E001C0738077007FFFFFFFF10107F 8F14>I<1C103F38E7E041C00D047D9614>126 D E /Fb 64 123 df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c 1 51 df<7FFFFCFFFFFEC00006C00006C00006C00006C00006C00006C0 0006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C0 0006FFFFFEFFFFFE17177D991F>50 D E /Fd 5 112 df<18F818181818181818181818 FF080D7D8C0E>49 D<3E00418080C0C0C000C000C0018003000400084030407F80FF800A 0D7E8C0E>I<3DE04220C300C30042007C00C0007F007F80C0C0C0C0C0C061803F000B0E 7E880E>103 D108 D<1E0061804080C0C0C0C0C0C0408061801E000A097E880E>111 D E /Fe 7 89 df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f 4 113 df<004000C001800180018003000300060006000C000C000C00 1800180030003000600060006000C000C0000A157E8F0F>61 D<00C00080000000000000 0000070019801180030003000300030006000600060006008C00F0000A137F8E0C>106 D<3800180018001800300030C03360344078007E0063006320C340C1800B0E7E8D10>I< 39C02E302C30183018301830186038C03780300030006000F8000C0D7F880F>112 D E /Fg 6 107 df0 D<040004000400C460F5E03F800E00 3F80F5E0C4600400040004000B0D7E8D11>3 D<040E0E1C1C1C38383070706060C0C007 0F7F8F0A>48 D<03FC0FFC1C003000600060006000C000C000FFFCFFFCC000C000600060 00600030001C000FFC03FC0E147D9016>50 D<03F0020FF00610700E20700C40700C8060 1C00601800E01800E03800C03007FFF00FFFF00180700180600380600300E00300E00700 E10600E20600FC0C00F818157F931C>72 D106 D E /Fh 1 49 df<181818303030606060 C0C0050B7E8B09>48 D E /Fi 48 123 df<003FC00001F0300003C0380007C07C000F80 7C000F807C000F8038000F8000000F8000000F8000000F800000FFFFFC00FFFFFC000F80 7C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F80 7C000F807C000F807C000F807C000F807C007FE1FF807FE1FF80191D809C1B>12 D<0020004001800380030006000E001C001C003C0038003800780078007800F800F000F0 00F000F000F000F000F000F000F000F800780078007800380038003C001C001C000E0006 00030003800180004000200B297C9E13>40 D<800040003000380018000C000E00070007 0007800380038003C003C003C003E001E001E001E001E001E001E001E001E001E003E003 C003C003C0038003800780070007000E000C00180038003000400080000B297D9E13>I< FFF8FFF8FFF8FFF80D04808A10>45 D<78FCFCFCFC7806067D850D>I<00600001E0000F E000FFE000F3E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003 E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0007FFF807F FF80111B7D9A18>49 D<07F8001FFE00383F80780FC0FC07C0FC07E0FC03E0FC03E07803 E00007E00007C00007C0000F80001F00001E0000380000700000E0000180600300600600 600800E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131B7E9A18>I<07F8001FFE003C1F003C 0F807C07C07E07C07C07C03807C0000F80000F80001E00003C0003F800001E00000F8000 07C00007C00007E03007E07807E0FC07E0FC07E0FC07C0780F80781F001FFE0007F80013 1B7E9A18>I<000180000380000780000F80001F80003F80006F8000CF80008F80018F80 030F80060F800C0F80180F80300F80600F80C00F80FFFFF8FFFFF8000F80000F80000F80 000F80000F80000F8000FFF800FFF8151B7F9A18>I<78FCFCFCFC7800000000000078FC FCFCFC7806127D910D>58 D<00038000000380000007C0000007C0000007C000000FE000 000FE000001FF000001BF000003BF8000031F8000031F8000060FC000060FC0000E0FE00 00C07E0000C07E0001803F0001FFFF0003FFFF8003001F8007001FC006000FC006000FC0 0C0007E00C0007E0FF803FFEFF803FFE1F1C7E9B24>65 D<001FE02000FFF8E003F80FE0 07C003E00F8001E01F0000E03E0000E03E0000607E0000607C000060FC000000FC000000 FC000000FC000000FC000000FC000000FC000000FC0000007C0000607E0000603E000060 3E0000C01F0000C00F80018007C0030003F80E0000FFFC00001FE0001B1C7D9B22>67 DIII72 D I76 D78 D<003FE00001F07C0003C01E000F800F801F0007C01E0003C03E0003E07E0003F07C0001 F07C0001F0FC0001F8FC0001F8FC0001F8FC0001F8FC0001F8FC0001F8FC0001F8FC0001 F87C0001F07E0003F07E0003F03E0003E03F0007E01F0007C00F800F8003C01E0001F07C 00003FE0001D1C7D9B24>II<003FE00001F07C0003C01E000F 800F801F0007C01F0007C03E0003E07E0003F07C0001F07C0001F0FC0001F8FC0001F8FC 0001F8FC0001F8FC0001F8FC0001F8FC0001F8FC0001F87C0001F07C0001F07E0003F03E 0003E03E0703E01F08C7C00F906F8003D03E0001F87C00003FF8080000180800001C1800 001FF800001FF800000FF000000FF0000007E0000003C01D247D9B24>II<07F820 1FFEE03C07E07801E07000E0F000E0F00060F00060F80000FE0000FFE0007FFE003FFF00 3FFF800FFFC007FFE0007FE00003F00001F00000F0C000F0C000F0C000E0E000E0F001C0 FC03C0EFFF0083FC00141C7D9B1B>I<7FFFFFE07FFFFFE0781F81E0701F80E0601F8060 E01F8070C01F8030C01F8030C01F8030C01F8030001F8000001F8000001F8000001F8000 001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000 001F8000001F8000001F800007FFFE0007FFFE001C1C7E9B21>I<0FF8001C1E003E0F80 3E07803E07C01C07C00007C0007FC007E7C01F07C03C07C07C07C0F807C0F807C0F807C0 780BC03E13F80FE1F815127F9117>97 DI<03FC000E0E001C1F003C1F00781F00780E00F80000F80000F80000F80000F80000 F800007800007801803C01801C03000E0E0003F80011127E9115>I<000FF0000FF00001 F00001F00001F00001F00001F00001F00001F00001F00001F001F9F00F07F01C03F03C01 F07801F07801F0F801F0F801F0F801F0F801F0F801F0F801F07801F07801F03C01F01C03 F00F0FFE03F9FE171D7E9C1B>I<03FC000F07001C03803C01C07801C07801E0F801E0F8 01E0FFFFE0F80000F80000F800007800007800603C00601E00C00F038001FC0013127F91 16>I<007F0001E38003C7C00787C00F87C00F83800F80000F80000F80000F80000F8000 FFF800FFF8000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8000 0F80000F80000F80000F80007FF0007FF000121D809C0F>I<03F0F00E1F383C0F383807 307807807807807807807807803807003C0F001E1C0033F0003000003000003000003FFF 003FFFC01FFFE01FFFF07801F8F00078F00078F000787000707800F01E03C007FF00151B 7F9118>II<1E003F003F003F003F 001E000000000000000000000000007F007F001F001F001F001F001F001F001F001F001F 001F001F001F001F001F00FFC0FFC00A1E7F9D0E>I107 DIII<01FC000F07801C01C03C01E07800F07800F0F8 00F8F800F8F800F8F800F8F800F8F800F87800F07800F03C01E01E03C00F078001FC0015 127F9118>II114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007CC01C C01CE01CE018F830CFC00E127E9113>I<0300030003000300070007000F000F003FFCFF FC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F 9913>III120 DI<7FFF80781F00703F00603E00607C 0060FC0060F80001F00003F00007E00007C1800F81801F81801F03803E03007E07007C0F 00FFFF0011127F9115>I E /Fj 11 112 df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k 41 122 df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l 25 121 df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m 18 113 df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n 43 121 df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o 83 125 df<007E1F0001C1B1800303E3C00703C3C00E03C1800E01C0000E01C0000E01C0000E01 C0000E01C0000E01C000FFFFFC000E01C0000E01C0000E01C0000E01C0000E01C0000E01 C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01 C0000E01C0007F87FC001A1D809C18>11 D<007E0001C1800301800703C00E03C00E0180 0E00000E00000E00000E00000E0000FFFFC00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C07F87F8151D80 9C17>I<007FC001C1C00303C00703C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C0FFFFC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C00E01C00E01C00E01C07FCFF8151D809C17>I<003F07E00001C09C1800 0380F018000701F03C000E01E03C000E00E018000E00E000000E00E000000E00E000000E 00E000000E00E00000FFFFFFFC000E00E01C000E00E01C000E00E01C000E00E01C000E00 E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E0 1C000E00E01C000E00E01C000E00E01C000E00E01C007FC7FCFF80211D809C23>I<0E0E 1E3870E0800707779C15>19 D<00000100003F830000E0E20003803C0007001C000E000E 001C0017003800338038002380780043C0700041C0700081C0F00101E0F00101E0F00201 E0F00401E0F00401E0F00801E0F01001E0F01001E0702001C0704001C0784003C0388003 80398003801D0007000E000E0007001C000780380008E0E000183F8000100000001B207E 9D20>31 D<60F0F0F0F0F0F0F060606060606060606060606060000000000060F0F06004 1E7C9D0C>33 D<6060F0F0F8F86868080808080808101010102020404080800D0C7F9C15 >I<0F0000C0188000C030600380703807006027FB00E0100600E0100C00E0100C00E010 1800E0101800E0103000E0106000602060007020C00030418000188180000F0303C00006 062000060C10000C1C08001818080018380400303804006038040060380400C0380400C0 3804018038040300180803001C0806000C100C000620040003C01E217E9E23>37 D<60F0F8680808081010204080050C7C9C0C>39 D<00800100020006000C000C00180018 003000300030006000600060006000E000E000E000E000E000E000E000E000E000E000E0 00E0006000600060006000300030003000180018000C000C000600020001000080092A7C 9E10>I<8000400020003000180018000C000C0006000600060003000300030003000380 038003800380038003800380038003800380038003800300030003000300060006000600 0C000C00180018003000200040008000092A7E9E10>I<0300030003000300C30CE31C73 381FE0078007801FE07338E31CC30C03000300030003000E127D9E15>I<000600000006 000000060000000600000006000000060000000600000006000000060000000600000006 00000006000000060000FFFFFFE0FFFFFFE0000600000006000000060000000600000006 000000060000000600000006000000060000000600000006000000060000000600001B1C 7E9720>I<60F0F0701010101020204080040C7C830C>II<60F0 F06004047C830C>I<00010003000600060006000C000C000C0018001800180030003000 300060006000C000C000C0018001800180030003000300060006000C000C000C00180018 001800300030003000600060006000C000C00010297E9E15>I<03C00C301818300C300C 700E60066006E007E007E007E007E007E007E007E007E007E007E007E007E00760066006 700E300C300C18180C3007E0101D7E9B15>I<010007003F00C700070007000700070007 000700070007000700070007000700070007000700070007000700070007000700070007 00FFF80D1C7C9B15>I<07C01830201C400C400EF00FF80FF807F8077007000F000E000E 001C001C00380070006000C00180030006010C01180110023FFE7FFEFFFE101C7E9B15> I<07E01830201C201C781E780E781E381E001C001C00180030006007C00030001C001C00 0E000F000F700FF80FF80FF80FF00E401C201C183007C0101D7E9B15>I<000C00000C00 001C00003C00003C00005C0000DC00009C00011C00031C00021C00041C000C1C00081C00 101C00301C00201C00401C00C01C00FFFFC0001C00001C00001C00001C00001C00001C00 001C0001FFC0121C7F9B15>I<300C3FF83FF03FC020002000200020002000200023E02C 303018301C200E000E000F000F000F600FF00FF00FF00F800E401E401C2038187007C010 1D7E9B15>I<00F0030C04040C0E181E301E300C700070006000E3E0E430E818F00CF00E E006E007E007E007E007E007600760077006300E300C18180C3003E0101D7E9B15>I<40 00007FFF807FFF007FFF0040020080040080040080080000100000100000200000600000 400000C00000C00001C00001800001800003800003800003800003800007800007800007 8000078000078000078000030000111D7E9B15>I<03E00C301008200C20066006600670 0670067C0C3E183FB01FE007F007F818FC307E601E600FC007C003C003C003C003600260 04300C1C1007E0101D7E9B15>I<03C00C301818300C700C600EE006E006E007E007E007 E007E0076007700F300F18170C2707C700060006000E300C780C78187018203010C00F80 101D7E9B15>I<60F0F0600000000000000000000060F0F06004127C910C>I<60F0F06000 00000000000000000060F0F0701010101020204080041A7C910C>I<7FFFFFC0FFFFFFE0 0000000000000000000000000000000000000000000000000000000000000000FFFFFFE0 7FFFFFC01B0C7E8F20>61 D<000600000006000000060000000F0000000F0000000F0000 00178000001780000037C0000023C0000023C0000043E0000041E0000041E0000080F000 0080F0000080F000010078000100780001FFF80002003C0002003C0002003C0004001E00 04001E000C001F000C000F001E001F00FF00FFF01C1D7F9C1F>65 DI<001F808000E0618001801980070007800E0003801C000380 1C00018038000180780000807800008070000080F0000000F0000000F0000000F0000000 F0000000F0000000F0000000F0000000700000807800008078000080380000801C000100 1C0001000E000200070004000180080000E03000001FC000191E7E9C1E>IIII<001F808000E0618001801980070007800E0003801C0003801C000180 38000180780000807800008070000080F0000000F0000000F0000000F0000000F0000000 F0000000F000FFF0F0000F80700007807800078078000780380007801C0007801C000780 0E00078007000B800180118000E06080001F80001C1E7E9C21>III75 DIII<003F800000E0E0000380380007001C000E000E001C0007003C000780380003807800 03C0780003C0700001C0F00001E0F00001E0F00001E0F00001E0F00001E0F00001E0F000 01E0F00001E0700001C0780003C0780003C0380003803C0007801C0007000E000E000700 1C000380380000E0E000003F80001B1E7E9C20>II<003F800000E0E0000380380007001C000E000E001C0007003C00078038000380 780003C0780003C0700001C0F00001E0F00001E0F00001E0F00001E0F00001E0F00001E0 F00001E0F00001E0700001C0780003C0780003C0380003803C0E07801C1107000E208E00 07205C0003A0780000F0E020003FE0200000602000003060000038E000003FC000003FC0 00001F8000000F001B257E9C20>II<07E0801C1980 300580300380600180E00180E00080E00080E00080F00000F800007C00007FC0003FF800 1FFE0007FF0000FF80000F800007C00003C00001C08001C08001C08001C0C00180C00180 E00300D00200CC0C0083F800121E7E9C17>I<7FFFFFC0700F01C0600F00C0400F004040 0F0040C00F0020800F0020800F0020800F0020000F0000000F0000000F0000000F000000 0F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F000000 0F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E>IIII91 D<08081010202040404040808080808080B0B0F8F8787830300D0C7A 9C15>II<1FC000307000783800781C00301C00001C00001C 0001FC000F1C00381C00701C00601C00E01C40E01C40E01C40603C40304E801F87001212 7E9115>97 DI<03F80C0C181E301E 700C6000E000E000E000E000E000E00060007002300218040C1803E00F127F9112>I<00 1F8000038000038000038000038000038000038000038000038000038000038003F3800E 0B80180780300380700380600380E00380E00380E00380E00380E00380E0038060038070 03803003801807800E1B8003E3F0141D7F9C17>I<07E00C301818300C700E6006E006FF FEE000E000E000E00060007002300218040C1803E00F127F9112>I<00F8018C071E061E 0E0C0E000E000E000E000E000E00FFE00E000E000E000E000E000E000E000E000E000E00 0E000E000E000E000E000E007FE00F1D809C0D>I<00038007C4C01C78C0383880301800 701C00701C00701C00701C003018003838001C700027C0002000002000003000003FF800 1FFF001FFF802003806001C0C000C0C000C0C000C06001803003001C0E0007F800121C7F 9215>II<18003C003C0018000000 000000000000000000000000FC001C001C001C001C001C001C001C001C001C001C001C00 1C001C001C001C001C00FF80091D7F9C0C>I<00C001E001E000C0000000000000000000 000000000007E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000 E000E000E000E000E000E060E0F0C0F1C061803E000B25839C0D>IIIII<03F0000E1C00180600300300700380 600180E001C0E001C0E001C0E001C0E001C0E001C06001807003803003001806000E1C00 03F00012127F9115>II<03E0800E198018058038078070 0380700380E00380E00380E00380E00380E00380E003807003807003803807801807800E 1B8003E380000380000380000380000380000380000380000380001FF0141A7F9116>I< FCE01D381E781E781C301C001C001C001C001C001C001C001C001C001C001C001C00FFC0 0D127F9110>I<1F9020704030C010C010E010F8007F803FE00FF000F880388018C018C0 18E010D0608FC00D127F9110>I<04000400040004000C000C001C003C00FFE01C001C00 1C001C001C001C001C001C001C001C101C101C101C101C100C100E2003C00C1A7F9910> IIIIII<7FFC70386038407040F040E0 41C003C0038007000F040E041C043C0C380870087038FFF80E127F9112>I124 D E /Fp 30 118 df<70F8FCFCFC7C04040808102040060D7D850C> 44 D<78FCFCFCFC7806067D850C>46 D<01800780FF80FF800780078007800780078007 80078007800780078007800780078007800780078007800780FFFCFFFC0E187C9716>49 D<0FF0003FFC00787E00FC1F00FC1F80FC0F80FC0F80780F80001F80001F00001E00003C 0000780000700000E0000180000301800601800C01801003803FFF007FFF00FFFF00FFFF 0011187E9716>I<3006003FFC003FF8003FF0003FE0003F800030000030000030000037 F000381C00201E00000F00000F00000F80700F80F80F80F80F80F80F80F80F00601F0038 3E001FF80007E00011187E9716>53 D<07E0001FF8003C1C00781E00780F00F80F00F80F 00F80F80F80F80F80F80F80F80781F80381F801C2F8007CF80000F80000F00380F007C1F 007C1E00783C003078001FF0000FC00011187E9716>57 D66 D<001FE02000FFFCE003F80FE007C003E01F8001E01F0000E03E0000E07E0000607C0000 60FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC0000007C0000 607E0000603E0000601F0000C01F8000C007C0038003F80F0000FFFC00001FF0001B1A7E 9920>III76 D80 D<07F0401FFDC03C0FC07803C07001C0F001C0F000C0F000C0F80000FF00007FF8003FFF 001FFF800FFFC001FFE0000FE00003F00001F0C000F0C000F0C000F0E000E0F001E0FC03 C0EFFF8083FE00141A7E9919>83 D86 D<1FF000383C007C1E007C0F007C0F00380F 00000F0000FF000FCF003E0F007C0F00F80F00F80F00F80F00F817007C27E01FC3E01311 7F9015>97 DI<03FC000F0E001C1F003C1F00781F00780E 00F80000F80000F80000F80000F800007800007800003C01801C03000F060003FC001111 7F9014>I<0007E00007E00001E00001E00001E00001E00001E00001E00001E003F9E00F 07E01C03E03C01E07801E07801E0F801E0F801E0F801E0F801E0F801E07801E07801E03C 01E01C03E00F0DF803F9F8151A7F9919>I<03F0000E1C001C0E003C0700780700780780 F80780F80780FFFF80F80000F800007800007800003C01801C03000E060003FC0011117F 9014>I<00FE0003C700078F800F0F800F0F800F07000F00000F00000F0000FFF000FFF0 000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00 000F00003FE0003FE000111A80990E>I<07F1C01C1EE0380EE0780FC0780F00780F0078 0F00780F00380E001C1C0037F0002000003000003000003FFE001FFF801FFFC07003C0E0 00E0E000E0E000E0E000E07001C03C078007FC0013197F9016>II<3C7E7E7E7E3C00000000FCFC3C3C3C3C3C3C3C3C3C3C3C3C3CFFFF081B7E9A0D> I108 D110 D<03F8000E0E003C07803803 807803C07803C0F803E0F803E0F803E0F803E0F803E0F803E07803C07C07C03C07800E0E 0003F80013117F9016>I114 D<1FD820784018C018C018F000FF807FE03FF0 07F8003CC00CC00CE00CE008F830CFE00E117F9011>I<06000600060006000E000E001E 003FF0FFF01E001E001E001E001E001E001E001E001E181E181E181E181E180F3003E00D 187F9711>II E /Fq 32 122 df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r 52 122 df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s 3 123 df<0C000C00CCC0EDC07F800C007F80EDC0CCC00C 000C000A0B7D8B10>3 D<1818181818FFFF18181818181818181818181818181808167D 900E>121 D<1818181818FF18181818180018181818FFFF1818181808167D900E>I E /Ft 1 108 df<1F80000380000380000380000700000700000700000700000E00000E 00000E07000E19801C23801C41801C80001D00003E00003FC00038E00038700070710070 7100707100707200E03200601C00111A7F9914>107 D E /Fu 82 128 df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v 35 121 df<00180000780001F800FFF800FFF80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 007FFFE07FFFE013207C9F1C>49 D<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8 FE03F8FE01F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E00 00380000700000E01801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0 FFFFF015207D9F1C>I<00FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03 F81F03F00003F00003E00007C0001F8001FE0001FF000007C00001F00001F80000FC0000 FC3C00FE7E00FEFF00FEFF00FEFF00FEFF00FC7E01FC7801F81E07F00FFFC001FE001720 7E9F1C>I<0000E00001E00003E00003E00007E0000FE0001FE0001FE00037E00077E000 E7E001C7E00187E00307E00707E00E07E00C07E01807E03807E07007E0E007E0FFFFFFFF FFFF0007E00007E00007E00007E00007E00007E00007E000FFFF00FFFF18207E9F1C>I< 3000203E01E03FFFC03FFF803FFF003FFE003FF80033C000300000300000300000300000 31FC0037FF003E0FC03807E03003E00003F00003F00003F80003F83803F87C03F8FE03F8 FE03F8FC03F0FC03F07007E03007C03C1F800FFF0003F80015207D9F1C>I<001F8000FF E003E07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FCFF E0FD80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C00 7C3E007C1E00F80F00F00783E003FFC000FF0017207E9F1C>I<00007000000000700000 0000F800000000F800000000F800000001FC00000001FC00000003FE00000003FE000000 03FE000000067F000000067F0000000C7F8000000C3F8000000C3F800000181FC0000018 1FC00000301FE00000300FE00000700FF000006007F000006007F00000C007F80000FFFF F80001FFFFFC00018001FC00018001FC00030001FE00030000FE00070000FF000600007F 000600007F00FFE007FFF8FFE007FFF825227EA12A>65 D<0003FE0080001FFF818000FF 01E38001F8003F8003E0001F8007C0000F800F800007801F800007803F000003803F0000 03807F000001807E000001807E00000180FE00000000FE00000000FE00000000FE000000 00FE00000000FE00000000FE00000000FE000000007E000000007E000001807F00000180 3F000001803F000003801F800003000F8000030007C000060003F0000C0001F800380000 FF00F000001FFFC0000003FE000021227DA128>67 D69 D73 D80 D82 D<7FFFFFFF807FFFFFFF807E03F80F 807803F807807003F803806003F80180E003F801C0E003F801C0C003F800C0C003F800C0 C003F800C0C003F800C00003F800000003F800000003F800000003F800000003F8000000 03F800000003F800000003F800000003F800000003F800000003F800000003F800000003 F800000003F800000003F800000003F800000003F800000003F800000003F800000003F8 000001FFFFF00001FFFFF00022227EA127>84 D87 D<0FFC003FFF807E07C07E03E07E01E07E01F03C01F000 01F00001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC01F0FC01F0FC01F07E02F07E 0CF81FF87F07E03F18167E951B>97 DI<00FF8007FFE00F83F01F03F03E03F07E03 F07C01E07C0000FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00 181F00300FC06007FFC000FF0015167E9519>I<0001FE000001FE0000003E0000003E00 00003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E00 01FC3E0007FFBE000F81FE001F007E003E003E007E003E007C003E00FC003E00FC003E00 FC003E00FC003E00FC003E00FC003E00FC003E00FC003E007C003E007C003E003E007E00 1F00FE000F83BE0007FF3FC001FC3FC01A237EA21F>I<00FE0007FF800F87C01E01E03E 01F07C00F07C00F8FC00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E 00003E00181F00300FC07003FFC000FF0015167E951A>I<001FC0007FE000F1F001E3F0 03E3F007C3F007C1E007C00007C00007C00007C00007C00007C000FFFE00FFFE0007C000 07C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C000 07C00007C00007C00007C00007C0003FFC003FFC00142380A211>I<01FE0F0007FFBF80 0F87C7801F03E7801E01E0003E01F0003E01F0003E01F0003E01F0003E01F0001E01E000 1F03E0000F87C0000FFF800009FE000018000000180000001C0000001FFFE0000FFFF800 07FFFE001FFFFF003C003F0078000F80F0000780F0000780F0000780F000078078000F00 3C001E001F007C000FFFF80001FFC00019217F951C>II<1C003E007F007F007F00 3E001C000000000000000000000000000000FF00FF001F001F001F001F001F001F001F00 1F001F001F001F001F001F001F001F001F001F001F00FFE0FFE00B247EA310>I 107 DIII<00FE0007FFC00F83E01E 00F03E00F87C007C7C007C7C007CFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C 007C7C007C3E00F81F01F00F83E007FFC000FE0017167E951C>II114 D<0FF3003FFF00781F00600700E00300 E00300F00300FC00007FE0007FF8003FFE000FFF0001FF00000F80C00780C00380E00380 E00380F00700FC0E00EFFC00C7F00011167E9516>I<0180000180000180000180000380 000380000780000780000F80003F8000FFFF00FFFF000F80000F80000F80000F80000F80 000F80000F80000F80000F80000F80000F80000F81800F81800F81800F81800F81800F83 0007C30003FE0000F80011207F9F16>III120 D E /Fw 20 122 df<183E7E7F3F1F070E0E1CFCF8 60080D77851A>44 D<3078FCFC7830060676851A>46 D<003E0001FF8003FFC007C1E00F 00E01E0F703C3FF0387FF07070F870E07870E078E1C038E1C038E1C038E1C038E1C038E1 C038E1C038E1C03870E07070E0707070E0387FE03C3FC01E0F000F003807C0F803FFF001 FFE0003F00151E7E9D1A>64 D<1FF0003FFC007FFE00780F00300700000380000380007F 8007FF801FFF803F8380780380700380E00380E00380E00380700780780F803FFFFC1FFD FC07F0FC16157D941A>97 D<00FF8003FFC00FFFE01F01E03C00C0780000700000700000 E00000E00000E00000E00000E000007000007000007800703C00701F01F00FFFE003FFC0 00FE0014157D941A>99 D<000FC0001FC0000FC00001C00001C00001C00001C00001C000 01C001F1C007FDC00FFFC01E0FC03C07C07803C07001C0E001C0E001C0E001C0E001C0E0 01C0E001C0E001C07003C07003C03807C03E0FC01FFFF807FDFC01F1F8161E7E9D1A>I< 01F80007FF000FFF801E07C03C01C07800E07000E0E00070E00070FFFFF0FFFFF0FFFFF0 E000007000007000007800703C00701F01F00FFFE003FF8000FE0014157D941A>I<0007 E0001FF0003FF800787800F03000E00000E00000E00000E0007FFFF0FFFFF0FFFFF000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E0003FFF807FFFC03FFF80151E7F9D1A>I<01F87C07FFFE0FFFFE1E078C1C 03803801C03801C03801C03801C03801C01C03801E07801FFF001FFE0039F80038000038 00001C00001FFF801FFFE03FFFF878007C70001CE0000EE0000EE0000EE0000E70001C78 003C3E00F81FFFF007FFC001FF0017217F941A>I<00C00001E00001E00000C000000000 0000000000000000000000000000007FE0007FE0007FE00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0007FFF80 FFFFC07FFF80121F7C9E1A>105 D<000C001E001E000C00000000000000000000000007 FE0FFE07FE000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E000E000E000E000E000E001C601CF038FFF87FF01FC00F2A7E9E1A>II<7E3E00FEFF807FFFC00FC1C00F 80E00F00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E 00E00E00E07FC3FCFFE7FE7FC3FC17157F941A>110 D<01F00007FC001FFF003E0F803C 07807803C07001C0E000E0E000E0E000E0E000E0E000E0E000E0F001E07001C07803C03C 07803E0F801FFF0007FC0001F00013157D941A>I<7F81F8FF8FFC7F9FFE03FE1E03F80C 03E00003E00003C000038000038000038000038000038000038000038000038000038000 0380007FFF00FFFF007FFF0017157F941A>114 D<07FB801FFF807FFF80780780E00380 E00380E003807800007FC0001FFC0007FE00003F800007806001C0E001C0E001C0F003C0 FC0780FFFF00EFFE00E3F80012157C941A>I<0180000380000380000380000380000380 000380007FFFE0FFFFE0FFFFE00380000380000380000380000380000380000380000380 0003800003800003807003807003807003807001C1E001FFE000FF80003F00141C7F9B1A >I<7E07E0FE0FE07E07E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E00E 00E00E00E00E00E00E00E00E00E00E01E00F03E007FFFC03FFFE00FCFC17157F941A>I< 7F83FCFFC7FE7F83FC0E00E00E00E00E00E00701C00701C00701C0038380038380038380 01C70001C70001C70000EE0000EE0000EE00007C00007C0000380017157F941A>I<7F83 FCFFC7FE7F83FC0E00E00E00E00700E00701C00701C00381C003838003C38001C38001C7 0000E70000E70000E600006600006E00003C00003C00003C000038000038000038000070 0000700030F00078E00071E0007FC0003F80001E000017207F941A>121 D E /Fx 2 104 df<0003C0001E0000380000700000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 01C0000380000F0000F800000F000003800001C00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 007000003800001E000003C012317DA419>102 DI E /Fy 27 122 df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z 3 123 df<020002000200C218F2783AE00F800F803AE0F278C2180200020002000D0E7E8E12>3 D<06000600060006000600060006000600FFF0FFF0060006000600060006000600060006 00060006000600060006000600060006000600060006000C1D7E9611>121 D<060006000600060006000600FFF0FFF006000600060006000600060000000600060006 00060006000600FFF0FFF00600060006000600060006000C1D7E9611>I E /FA 20 122 df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df68 D72 D<001FF8018000FFFF038003FFFFC78007F007EF800F8000FF801F00007F803E 00001F803E00000F807C00000F807C00000780FC00000780FC00000780FC00000380FE00 000380FE00000380FF00000000FFC00000007FF00000007FFF8000003FFFF800003FFFFF 80001FFFFFF0000FFFFFF80007FFFFFC0003FFFFFF0000FFFFFF00003FFFFF800001FFFF C000001FFFE0000001FFE00000003FE00000001FF00000000FF000000007F060000007F0 E0000003F0E0000003F0E0000003F0E0000003E0F0000003E0F0000003E0F8000007C0FC 000007C0FF00000F80FFC0001F00FBFC00FE00F1FFFFF800E03FFFF000C003FF80002431 7CB02D>83 D<00FFF0000003FFFF00000F803F80000FC00FE0001FE007F0001FE007F000 1FE003F8000FC003FC00078003FC00000003FC00000003FC00000003FC00000003FC0000 00FFFC00001FFFFC0000FFE3FC0003FC03FC000FF003FC001FC003FC003FC003FC007F80 03FC007F8003FC00FF0003FC00FF0003FC00FF0003FC00FF0007FC00FF0007FC007F800D FC003FC01DFE001FE078FFF007FFE07FF000FF803FF024207E9F27>97 D<000FFF00007FFFC001FC01F003F003F007E007F80FE007F81FC007F83FC003F03FC001 E07F8000007F8000007F800000FF800000FF800000FF800000FF800000FF800000FF8000 00FF800000FF8000007F8000007F8000007F8000003FC0001C3FC0001C1FC000380FE000 3807E0007003F001E001FC07C0007FFF00000FF8001E207D9F24>99 D<000FFC00007FFF8001FC0FC003F003E007E001F00FE001F81FC000FC3FC000FE3FC000 FE7F80007E7F80007F7F80007FFF80007FFF80007FFFFFFFFFFFFFFFFFFF800000FF8000 00FF800000FF8000007F8000007F8000007F8000003FC000071FC000071FC0000E0FE000 0E07F0001C03F8007800FE03E0003FFFC00007FE0020207E9F25>101 D<001FF007C000FFFE3FE001F83F79F007E00FC3F00FE00FE1F00FC007E0E01FC007F000 1FC007F0003FC007F8003FC007F8003FC007F8003FC007F8003FC007F8001FC007F0001F C007F0000FC007E0000FE00FE00007E00FC00003F83F000006FFFE00000E1FF000000E00 0000001E000000001E000000001F000000001F800000001FFFFF80000FFFFFF0000FFFFF FC0007FFFFFE0003FFFFFF0003FFFFFF800FFFFFFFC03F00007FC07E00001FE07C00000F E0FC000007E0FC000007E0FC000007E0FC000007E07E00000FC03E00000F803F00001F80 0FC0007E0007F803FC0001FFFFF000001FFF0000242F7E9F28>103 D<01F8000000FFF8000000FFF8000000FFF80000000FF800000007F800000007F8000000 07F800000007F800000007F800000007F800000007F800000007F800000007F800000007 F800000007F800000007F800000007F800000007F807F80007F83FFE0007F8783F0007F8 C03F8007F9801FC007FB001FC007FE001FE007FC001FE007FC001FE007FC001FE007F800 1FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001F E007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE0 07F8001FE007F8001FE007F8001FE007F8001FE0FFFFC3FFFFFFFFC3FFFFFFFFC3FFFF28 327DB12D>I<03C00007E0000FF0001FF8001FF8001FF8001FF8000FF00007E00003C000 00000000000000000000000000000000000000000000000000000001F8007FF8007FF800 7FF80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F800FFFF80FFFF80FFFF8011337DB217>I<01F800FFF800FFF800FFF8000FF8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F800FFFFC0FFFFC0FFFFC012327DB117>108 D<03F007F8001FE000FFF03FFE00FFF800FFF0783F01E0FC00FFF0C03F8300FE000FF180 1FC6007F0007F3001FCC007F0007F6001FF8007F8007FC001FF0007F8007FC001FF0007F 8007FC001FF0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F800 1FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F 8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F800 1FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F 8007F8001FE0007F8007F8001FE0007F80FFFFC3FFFF0FFFFCFFFFC3FFFF0FFFFCFFFFC3 FFFF0FFFFC3E207D9F43>I<03F007F800FFF03FFE00FFF0783F00FFF0C03F800FF1801F C007F3001FC007F6001FE007FC001FE007FC001FE007FC001FE007F8001FE007F8001FE0 07F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007 F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8 001FE007F8001FE007F8001FE0FFFFC3FFFFFFFFC3FFFFFFFFC3FFFF28207D9F2D>I<00 07FC0000007FFFC00001FC07F00003F001F80007E000FC000FC0007E001FC0007F003FC0 007F803F80003F807F80003FC07F80003FC07F80003FC0FF80003FE0FF80003FE0FF8000 3FE0FF80003FE0FF80003FE0FF80003FE0FF80003FE0FF80003FE07F80003FC07F80003F C07F80003FC03FC0007F803FC0007F801FC0007F000FE000FE0007E000FC0003F803F800 01FE0FF000007FFFC0000007FC000023207E9F28>I<03F03F00FFF07FC0FFF1C3E0FFF1 87E00FF30FF007F60FF007F60FF007FC07E007FC03C007FC000007FC000007F8000007F8 000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8 000007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FFFF E000FFFFE0001C207E9F21>114 D<01FF860007FFFE001F00FE003C003E0078001E0078 000E00F8000E00F8000E00F8000E00FC000000FF800000FFFC00007FFFC0003FFFF0003F FFF8001FFFFC0007FFFE0001FFFF00003FFF000000FF8000003F8060001F80E0000F80E0 000F80F0000F80F0000F00F8000F00FC001E00FE001C00FF807800F3FFF000C07F800019 207D9F20>I<001C0000001C0000001C0000001C0000001C0000003C0000003C0000003C 0000007C0000007C000000FC000001FC000003FC000007FC00001FFFFE00FFFFFE00FFFF FE0003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC038003FC 038003FC038003FC038003FC038003FC038003FC038001FC038001FC070000FE0700007F 0E00003FFC000007F000192E7FAD1F>I118 D121 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 96 50 a FB(Similarit)n(y)25 b(Searc)n(h)h(in)g(High)h (Dimensions)e(via)h(Hashing)221 238 y FA(Aristides)18 b(Gionis)648 220 y Fz(\003)834 238 y FA(Piotr)g(Ind)o(yk)1139 220 y Fz(y)1324 238 y FA(Rajeev)f(Motw)l(ani)1745 220 y Fz(z)627 293 y Fy(Departmen)o(t)d(of)j(Computer)e(Science)774 349 y(Stanford)i(Univ)o(ersit)o(y)771 404 y(Stanford,)g(CA)f(94305)509 459 y Fx(f)p Fw(gionis,ind)o(yk,)o(ra)o(jee)o(v)p Fx(g)p Fw(@)o(cs.)o(sta)o(nfo)o(rd)o(.ed)o(u)350 792 y Fv(Abstract)45 894 y Fu(The)23 b(nearest-)h(or)f(near-neigh)o(b)q(or)j(query)e (problems)45 942 y(arise)17 b(in)g(a)f(large)h(v)n(ariet)o(y)g(of)f (database)i(applications,)45 989 y(usually)e(in)f(the)f(con)o(text)g (of)g(similarit)o(y)i(searc)o(hing.)21 b(Of)45 1036 y(late,)c(there)f (has)h(b)q(een)g(increasing)h(in)o(terest)f(in)g(build-)45 1084 y(ing)h(searc)o(h/index)h(structures)e(for)g(p)q(erforming)h (simi-)45 1131 y(larit)o(y)13 b(searc)o(h)f(o)o(v)o(er)g(high-dimensi)q (onal)k(data,)c(e.g.,)f(im-)45 1178 y(age)18 b(databases,)h(do)q(cumen) o(t)f(collections,)j(time-series)45 1226 y(databases,)12 b(and)f(genome)g(databases.)18 b(Unfortunately)m(,)45 1273 y(all)c(kno)o(wn)f(tec)o(hniques)i(for)d(solving)j(this)f(problem) g(fall)45 1320 y(prey)k(to)g(the)g(\\curse)h(of)f(dimensionali)q(t)o(y) m(.")35 b(That)18 b(is,)45 1368 y(the)12 b(data)f(structures)i(scale)f (p)q(o)q(orly)h(with)f(data)g(dimen-)45 1415 y(sionalit)o(y;)27 b(in)22 b(fact,)g(if)f(the)g(n)o(um)o(b)q(er)h(of)e(dimensions)45 1462 y(exceeds)e(10)f(to)f(20,)i(searc)o(hing)g(in)g Ft(k)q Fu(-d)f(trees)g(and)g(re-)45 1510 y(lated)d(structures)g(in)o(v) o(olv)o(es)h(the)e(insp)q(ection)i(of)e(a)g(large)45 1557 y(fraction)i(of)e(the)h(database,)g(thereb)o(y)h(doing)g(no)f(b)q (etter)45 1604 y(than)j(brute-force)g(linear)i(searc)o(h.)28 b(It)16 b(has)h(b)q(een)h(sug-)45 1652 y(gested)13 b(that)f(since)h (the)f(selection)i(of)e(features)g(and)h(the)45 1699 y(c)o(hoice)g(of)f(a)h(distance)g(metric)g(in)g(t)o(ypical)h (applicatio)q(ns)45 1746 y(is)19 b(rather)g(heuristic,)i(determining)g (an)e(appro)o(ximate)45 1793 y(nearest)g(neigh)o(b)q(or)i(should)f (su\016ce)f(for)f(most)h(practi-)45 1841 y(cal)d(purp)q(oses.)26 b(In)15 b(this)h(pap)q(er,)h(w)o(e)d(examine)j(a)f(no)o(v)o(el)45 1888 y(sc)o(heme)k(for)f(appro)o(ximate)i(similarit)o(y)h(searc)o(h)e (based)45 1935 y(on)15 b(hashing.)23 b(The)14 b(basic)i(idea)f(is)g(to) g(hash)g(the)f(p)q(oin)o(ts)p -30 1983 390 2 v 34 2010 a Fs(\003)52 2021 y Fr(Supp)q(orted)i(b)o(y)i(NA)l(VY)j(N00014-96-1-1)o (221)15 b(gran)o(t)i(and)h(NSF)-30 2059 y(Gran)o(t)10 b(I)q(IS-9811904)o(.)34 2087 y Fs(y)44 2099 y Fr(Supp)q(orted)e(b)o(y)h (Stanford)f(Graduate)g(F)m(ello)o(wship)h(and)h(NSF)g(NYI)-30 2136 y(Aw)o(ard)i(CCR-9357849.)34 2165 y Fs(z)43 2177 y Fr(Supp)q(orted)7 b(b)o(y)i(AR)o(O)h(MURI)h(Gran)o(t)d(D)o (AAH04-96-1-0007,)f(NSF)-30 2214 y(Gran)o(t)12 b(I)q(IS-9811904)o(,)f (and)i(NSF)g(Y)m(oung)g(In)o(v)o(estigato)o(r)e(Aw)o(ard)i(CCR-)-30 2252 y(9357849,)f(with)i(matc)o(hing)d(funds)h(from)h(IBM,)h (Mitsubishi,)f(Sc)o(hlum-)-30 2289 y(b)q(erger)d(F)m(oundation,)e (Shell)j(F)m(oundation,)d(and)j(Xero)o(x)g(Corp)q(oration.)-30 2342 y Fq(Permission)16 b(to)g(c)n(opy)h(without)f(fe)n(e)h(al)r(l)f (or)g(p)n(art)g(of)g(this)g(material)g(is)-30 2379 y(gr)n(ante)n(d)d (pr)n(ovide)n(d)h(that)d(the)h(c)n(opies)h(ar)n(e)f(not)g(made)h(or)f (distribute)n(d)g(for)-30 2417 y(dir)n(e)n(ct)17 b(c)n(ommer)n(cial)i (advantage,)f(the)f(VLDB)g(c)n(opyright)g(notic)n(e)h(and)-30 2454 y(the)f(title)f(of)h(the)g(public)n(ation)h(and)f(its)e(date)i (app)n(e)n(ar,)i(and)e(notic)n(e)g(is)-30 2492 y(given)11 b(that)g(c)n(opying)g(is)f(by)h(p)n(ermission)f(of)g(the)h(V)m(ery)h(L) n(ar)n(ge)f(Data)g(Base)-30 2529 y(Endowment.)20 b(T)m(o)13 b(c)n(opy)i(otherwise,)f(or)g(to)f(r)n(epublish,)i(r)n(e)n(quir)n(es)g (a)f(fe)n(e)-30 2567 y(and/or)g(sp)n(e)n(cial)g(p)n(ermission)f(fr)n (om)g(the)g(Endowment.)-30 2619 y Fp(Pro)q(ceedings)i(of)f(the)h(25th)f (VLDB)h(Conference,)-30 2663 y(Edin)o(burgh,)d(Scotland,)i(1999.)1095 696 y Fu(from)g(the)g(database)h(so)f(as)g(to)g(ensure)g(that)g(the)g (prob-)1095 743 y(abilit)o(y)h(of)d(collision)k(is)d(m)o(uc)o(h)g (higher)h(for)e(ob)r(jects)h(that)1095 790 y(are)f(close)i(to)e(eac)o (h)h(other)f(than)h(for)f(those)h(that)g(are)f(far)1095 838 y(apart.)k(W)m(e)10 b(pro)o(vide)h(exp)q(erimen)o(tal)h(evidence)g (that)d(our)1095 885 y(metho)q(d)14 b(giv)o(es)h(signi\014can)o(t)h (impro)o(v)o(emen)o(t)f(in)g(running)1095 932 y(time)20 b(o)o(v)o(er)f(other)h(metho)q(ds)g(for)f(searc)o(hing)i(in)f(high-) 1095 979 y(dimensional)d(spaces)d(based)g(on)f(hierarc)o(hical)k(tree)c (de-)1095 1027 y(comp)q(osition.)32 b(Exp)q(erimen)o(tal)20 b(results)e(also)h(indicate)1095 1074 y(that)d(our)h(sc)o(heme)f (scales)h(w)o(ell)g(ev)o(en)f(for)g(a)g(relativ)o(ely)1095 1121 y(large)e(n)o(um)o(b)q(er)g(of)f(dimensions)i(\(more)e(than)h (50\).)1020 1220 y Fv(1)56 b(In)n(tro)r(duction)1020 1290 y Fo(A)11 b(similarit)o(y)d(searc)o(h)k(problem)e(in)o(v)o(olv)o (es)g(a)h(collection)g(of)f(ob-)1020 1338 y(jects)16 b(\(e.g.,)e(do)q(cumen)o(ts,)g(images\))f(that)i(are)g(c)o (haracterized)1020 1385 y(b)o(y)21 b(a)g(collection)f(of)h(relev)n(an)o (t)g(features)h(and)f(represen)o(ted)1020 1432 y(as)14 b(p)q(oin)o(ts)h(in)e(a)h(high-dimensional)e(attribute)i(space;)h(giv)o (en)1020 1480 y(queries)h(in)f(the)h(form)d(of)i(p)q(oin)o(ts)g(in)f (this)h(space,)i(w)o(e)e(are)g(re-)1020 1527 y(quired)e(to)g(\014nd)h (the)g(nearest)g(\(most)e(similar\))f(ob)r(ject)j(to)f(the)1020 1574 y(query)m(.)33 b(The)20 b(particularly)e(in)o(teresting)h(and)g(w) o(ell-studied)1020 1622 y(case)d(is)f(the)g Fn(d)p Fo(-dimensional)d (Euclidean)j(space.)22 b(The)15 b(prob-)1020 1669 y(lem)9 b(is)h(of)g(ma)r(jor)f(imp)q(ortance)h(to)g(a)h(v)n(ariet)o(y)f(of)f (applications;)1020 1716 y(some)15 b(examples)f(are:)22 b(data)16 b(compression)f([20)o(];)g(databases)1020 1764 y(and)10 b(data)g(mining)e([21)o(];)j(information)d(retriev)n(al)i([11) o(,)g(16)o(,)g(38];)1020 1811 y(image)16 b(and)i(video)g(databases)h ([15)o(,)e(17,)g(37,)h(42)o(];)h(mac)o(hine)1020 1858 y(learning)13 b([7)o(];)f(pattern)i(recognition)f([9,)f(13];)g(and,)h (statistics)1020 1906 y(and)k(data)f(analysis)g([12)o(,)g(27].)26 b(T)o(ypically)m(,)15 b(the)j(features)g(of)1020 1953 y(the)d(ob)r(jects)g(of)e(in)o(terest)j(are)e(represen)o(ted)j(as)d(p)q (oin)o(ts)g(in)g Fm(<)1976 1938 y Fl(d)1020 2000 y Fo(and)g(a)g (distance)g(metric)g(is)g(used)h(to)f(measure)g(similarit)o(y)d(of)1020 2048 y(ob)r(jects.)18 b(The)10 b(basic)g(problem)e(then)i(is)f(to)h(p)q (erform)f(indexing)1020 2095 y(or)14 b(similarit)o(y)c(searc)o(hing)k (for)f(query)i(ob)r(jects.)k(The)14 b(n)o(um)o(b)q(er)1020 2142 y(of)h(features)i(\(i.e.,)e(the)i(dimensionalit)o(y\))c(ranges)j (an)o(ywhere)1020 2190 y(from)f(tens)i(to)g(thousands.)26 b(F)m(or)16 b(example,)g(in)g(m)o(ultim)o(edia)1020 2237 y(applications)j(suc)o(h)j(as)e(IBM's)h(QBIC)g(\(Query)g(b)o(y)f(Image) 1020 2284 y(Con)o(ten)o(t\),)11 b(the)h(n)o(um)o(b)q(er)e(of)g (features)i(could)f(b)q(e)h(sev)o(eral)f(h)o(un-)1020 2332 y(dreds)19 b([15)o(,)e(17)o(].)28 b(In)18 b(information)c(retriev) n(al)k(for)f(text)h(do)q(c-)1020 2379 y(umen)o(ts,)23 b(v)o(ector-space)g(represen)o(tations)h(in)o(v)o(olv)o(e)c(sev)o(eral) 1020 2426 y(thousands)c(of)e(dimensions,)g(and)h(it)g(is)g(considered)h (to)f(b)q(e)h(a)1020 2473 y(dramatic)10 b(impro)o(v)o(emen)o(t)f(that)j (dimension-reduction)e(tec)o(h-)1020 2521 y(niques,)i(suc)o(h)h(as)f (the)g(Karh)o(unen-Lo)o(\023)-20 b(ev)o(e)13 b(transform)e([26)o(,)g (30)o(])1020 2568 y(\(also)16 b(kno)o(wn)h(as)f(principal)g(comp)q (onen)o(ts)h(analysis)f([22)o(])g(or)1020 2615 y(laten)o(t)f(seman)o (tic)f(indexing)g([11)o(]\),)g(can)h(reduce)i(the)e(dimen-)1020 2663 y(sionalit)o(y)d(to)i(a)g(mere)f(few)h(h)o(undreds!)p eop %%Page: 2 2 2 1 bop 20 -33 a Fo(The)16 b(lo)o(w-dimensional)d(case)18 b(\(sa)o(y)m(,)d(for)h Fn(d)g Fo(equal)g(to)g(2)g(or)-30 14 y(3\))d(is)g(w)o(ell-solv)o(ed)f([14)o(],)g(so)h(the)h(main)d(issue) j(is)f(that)g(of)f(deal-)-30 61 y(ing)k(with)h(a)g(large)g(n)o(um)o(b)q (er)f(of)g(dimensions,)h(the)g(so-called)-30 109 y(\\curse)j(of)d (dimensionalit)o(y)m(.")28 b(Despite)19 b(decades)h(of)e(in)o(ten-)-30 156 y(siv)o(e)f(e\013ort,)h(the)f(curren)o(t)h(solutions)e(are)h(not)g (en)o(tirely)g(sat-)-30 203 y(isfactory;)h(in)e(fact,)i(for)e(large)h (enough)g Fn(d)p Fo(,)g(in)f(theory)i(or)e(in)-30 251 y(practice,)e(they)f(pro)o(vide)f(little)g(impro)o(v)o(emen)o(t)e(o)o (v)o(er)j(a)f(linear)-30 298 y(algorithm)f(whic)o(h)i(compares)h(a)f (query)h(to)g(eac)o(h)g(p)q(oin)o(t)f(from)-30 345 y(the)h(database.)k (In)c(particular,)f(it)g(w)o(as)h(sho)o(wn)f(in)g([45)o(])g(that,)-30 393 y(b)q(oth)k(empirically)e(and)i(theoretically)m(,)h Fk(al)r(l)e Fo(curren)o(t)j(index-)-30 440 y(ing)d(tec)o(hniques)i (\(based)g(on)e(space)i(partitioning\))d(degrade)-30 487 y(to)g(linear)f(searc)o(h)i(for)f(su\016cien)o(tly)g(high)f (dimensions.)20 b(This)-30 535 y(situation)9 b(p)q(oses)i(a)e(serious)h (obstacle)g(to)g(the)g(future)g(dev)o(elop-)-30 582 y(men)o(t)i(of)h (large)g(scale)h(similarit)o(y)c(searc)o(h)15 b(systems.)j(Imagine)-30 629 y(for)i(example)g(a)g(searc)o(h)i(engine)f(whic)o(h)f(enables)h (con)o(ten)o(t-)-30 677 y(based)15 b(image)d(retriev)n(al)i(on)g(the)h (W)m(orld-Wide)e(W)m(eb.)18 b(If)c(the)-30 724 y(system)h(w)o(as)f(to)h (index)g(a)g(signi\014can)o(t)f(fraction)g(of)h(the)g(w)o(eb,)-30 771 y(the)k(n)o(um)o(b)q(er)f(of)g(images)f(to)i(index)f(w)o(ould)g(b)q (e)h(at)g(least)g(of)-30 819 y(the)13 b(order)g(tens)g(\(if)e(not)h(h)o (undreds\))i(of)d(million.)j(Clearly)m(,)d(no)-30 866 y(indexing)k(metho)q(d)g(exhibiting)f(linear)h(\(or)h(close)g(to)f (linear\))-30 913 y(dep)q(endence)f(on)d(the)g(data)g(size)h(could)f (manage)e(suc)o(h)j(a)f(h)o(uge)-30 961 y(data)j(set.)20 1031 y(The)21 b(premise)f(of)g(this)g(pap)q(er)h(is)g(that)f(in)g(man)o (y)f(cases)-30 1078 y(it)e(is)h(not)g(necessary)i(to)d(insist)h(on)g (the)g Fk(exact)h Fo(answ)o(er;)h(in-)-30 1125 y(stead,)14 b(determining)e(an)h Fk(appr)n(oximate)h Fo(answ)o(er)g(should)f(suf-) -30 1173 y(\014ce)18 b(\(refer)g(to)e(Section)i(2)e(for)h(a)f(formal)e (de\014nition\).)27 b(This)-30 1220 y(observ)n(ation)16 b(underlies)i(a)e(large)h(b)q(o)q(dy)g(of)f(recen)o(t)i(researc)o(h)-30 1267 y(in)11 b(databases,)g(including)g(using)f(random)g(sampling)e (for)j(his-)-30 1315 y(togram)g(estimation)g([8)o(])h(and)g(median)f (appro)o(ximation)f([33)o(],)-30 1362 y(using)17 b(w)o(a)o(v)o(elets)g (for)g(selectivit)o(y)h(estimation)d([34)o(])i(and)g(ap-)-30 1409 y(pro)o(ximate)12 b(SVD)i([25)o(].)k(W)m(e)13 b(observ)o(e)i(that) f(there)h(are)g(man)o(y)-30 1457 y(applications)h(of)g(nearest)j(neigh) o(b)q(or)d(searc)o(h)j(where)f(an)e(ap-)-30 1504 y(pro)o(ximate)j(answ) o(er)i(is)g(go)q(o)q(d)f(enough.)38 b(F)m(or)21 b(example,)f(it)-30 1551 y(often)12 b(happ)q(ens)g(\(e.g.,)g(see)h([23)o(]\))e(that)h(the)g (relev)n(an)o(t)g(answ)o(ers)-30 1599 y(are)21 b Fk(much)g Fo(closer)g(to)f(the)h(query)g(p)q(oin)o(t)f(than)h(the)g(irrele-)-30 1646 y(v)n(an)o(t)f(ones;)j(in)d(fact,)h(this)g(is)f(a)g(desirable)g (prop)q(ert)o(y)i(of)d(a)-30 1693 y(go)q(o)q(d)d(similarit)o(y)e (measure.)27 b(In)17 b(suc)o(h)h(cases,)g(the)g(appro)o(xi-)-30 1741 y(mate)10 b(algorithm)f(\(with)i(a)g(suitable)g(appro)o(ximation)d (factor\))-30 1788 y(will)14 b(return)j(the)f(same)e(result)i(as)g(an)f (exact)h(algorithm.)k(In)-30 1835 y(other)g(situations,)g(an)f(appro)o (ximate)e(algorithm)g(pro)o(vides)-30 1883 y(the)g(user)g(with)f(a)g (time-qualit)o(y)e(tradeo\013)j(|)e(the)i(user)h(can)-30 1930 y(decide)j(whether)h(to)f(sp)q(end)g(more)f(time)f(w)o(aiting)g (for)h(the)-30 1977 y(exact)e(answ)o(er,)g(or)f(to)g(b)q(e)h (satis\014ed)f(with)g(a)g(m)o(uc)o(h)f(quic)o(k)o(er)-30 2025 y(appro)o(ximation)11 b(\(e.g.,)i(see)i([5)o(]\).)20 2095 y(The)h(ab)q(o)o(v)o(e)g(argumen)o(ts)f(rely)g(on)h(the)g (assumption)f(that)-30 2142 y(appro)o(ximate)c(similarit)o(y)g(searc)o (h)j(can)f(b)q(e)h(p)q(erformed)f(m)o(uc)o(h)-30 2190 y(faster)i(than)f(the)g(exact)h(one.)j(In)c(this)g(pap)q(er)h(w)o(e)f (sho)o(w)g(that)-30 2237 y(this)19 b(is)g(indeed)h(the)g(case.)35 b(Sp)q(eci\014cally)m(,)19 b(w)o(e)g(in)o(tro)q(duce)h(a)-30 2284 y(new)d(indexing)e(metho)q(d)h(for)f(appro)o(ximate)g(nearest)i (neigh-)-30 2332 y(b)q(or)12 b(with)f(a)g(truly)h(sublinear)f(dep)q (endence)j(on)e(the)g(data)f(size)-30 2379 y(ev)o(en)h(for)g (high-dimensional)c(data.)17 b(Instead)c(of)e(using)g(space)-30 2426 y(partitioning,)h(it)i(relies)g(on)g(a)g(new)g(metho)q(d)f(called) h Fk(lo)n(c)n(ality-)-30 2473 y(sensitive)j(hashing)h(\(LSH\))p Fo(.)e(The)g(k)o(ey)h(idea)f(is)g(to)g(hash)g(the)-30 2521 y(p)q(oin)o(ts)9 b(using)g(sev)o(eral)h(hash)g(functions)f(so)h (as)f(to)g(ensure)i(that,)-30 2568 y(for)k(eac)o(h)i(function,)e(the)h (probabilit)o(y)e(of)h(collision)g(is)g(m)o(uc)o(h)-30 2615 y(higher)h(for)g(ob)r(jects)h(whic)o(h)f(are)g(close)h(to)f(eac)o (h)g(other)h(than)-30 2663 y(for)h(those)g(whic)o(h)g(are)h(far)e (apart.)30 b(Then,)19 b(one)g(can)f(deter-)1020 -33 y(mine)d(near)i (neigh)o(b)q(ors)g(b)o(y)g(hashing)f(the)h(query)g(p)q(oin)o(t)f(and) 1020 14 y(retrieving)h(elemen)o(ts)g(stored)g(in)f(buc)o(k)o(ets)i(con) o(taining)e(that)1020 61 y(p)q(oin)o(t.)31 b(W)m(e)18 b(pro)o(vide)h(suc)o(h)g(lo)q(calit)o(y-sensitiv)o(e)e(hash)i(func-) 1020 109 y(tions)13 b(that)f(are)i(simple)d(and)i(easy)g(to)g(implem)o (en)o(t;)d(they)k(can)1020 156 y(also)d(b)q(e)h(naturally)e(extended)j (to)e(the)h Fk(dynamic)g Fo(setting,)f(i.e.,)1020 203 y(when)h(insertion)g(and)g(deletion)f(op)q(erations)h(also)f(need)i(to) e(b)q(e)1020 251 y(supp)q(orted.)23 b(Although)15 b(in)f(this)h(pap)q (er)h(w)o(e)g(are)f(fo)q(cused)h(on)1020 298 y(Euclidean)f(spaces,)g (di\013eren)o(t)h(LSH)f(functions)f(can)h(b)q(e)h(also)1020 345 y(used)h(for)e(other)i(similarit)o(y)c(measures,)j(suc)o(h)h(as)f (dot)g(pro)q(d-)1020 393 y(uct)e([5].)1070 441 y(Lo)q(calit)o (y-Sensitiv)o(e)9 b(Hashing)h(w)o(as)g(in)o(tro)q(duced)g(b)o(y)g (Indyk)1020 488 y(and)19 b(Mot)o(w)o(ani)e([24)o(])i(for)f(the)h(purp)q (oses)i(of)d(devising)g(main)1020 536 y(memory)7 b(algorithms)g(for)j (nearest)h(neigh)o(b)q(or)e(searc)o(h;)j(in)d(par-)1020 583 y(ticular,)16 b(it)f(enabled)i(us)f(to)g(ac)o(hiev)o(e)g(w)o (orst-case)h Fn(O)q Fo(\()p Fn(dn)1916 568 y Fj(1)p Fl(=\017)1965 583 y Fo(\)-)1020 630 y(time)e(for)g(appro)o(ximate)f(nearest)k(neigh)o (b)q(or)e(query)g(o)o(v)o(er)g(an)1020 678 y Fn(n)p Fo(-p)q(oin)o(t)c (database.)18 b(In)13 b(this)f(pap)q(er)i(w)o(e)f(impro)o(v)o(e)d(that) j(tec)o(h-)1020 725 y(nique)g(and)g(ac)o(hiev)o(e)h(a)f(signi\014can)o (tly)f(impro)o(v)o(ed)f(query)j(time)1020 772 y(of)j Fn(O)q Fo(\()p Fn(dn)1167 757 y Fj(1)p Fl(=)p Fj(\(1+)p Fl(\017)p Fj(\))1284 772 y Fo(\).)29 b(This)18 b(yields)f(an)h(appro)o (ximate)e(nearest)1020 820 y(neigh)o(b)q(or)f(algorithm)e(running)i(in) h(sublinear)f(time)f(for)h(an)o(y)1020 867 y Fn(\017)f(>)h Fo(0.)23 b(F)m(urthermore,)15 b(w)o(e)h(generalize)g(the)g(algorithm)d (and)1020 914 y(its)h(analysis)f(to)h(the)g(case)h(of)e Fk(external)i(memory)p Fo(.)1070 963 y(W)m(e)k(supp)q(ort)h(our)f (theoretical)g(argumen)o(ts)g(b)o(y)g(empiri-)1020 1010 y(cal)14 b(evidence.)23 b(W)m(e)14 b(p)q(erformed)g(exp)q(erimen)o(ts)h (on)g(t)o(w)o(o)f(data)1020 1057 y(sets.)42 b(The)22 b(\014rst)h(con)o(tains)e(20,000)f(histograms)g(of)h(color)1020 1105 y(images,)f(where)i(eac)o(h)f(histogram)d(w)o(as)i(represen)o(ted) k(as)c(a)1020 1152 y(p)q(oin)o(t)13 b(in)g Fn(d)p Fo(-dimensional)e (space,)j(for)g Fn(d)f Fo(up)h(to)f(64.)18 b(The)c(sec-)1020 1199 y(ond)i(con)o(tains)f(around)h(270,000)e(p)q(oin)o(ts)h(represen)o (ting)j(tex-)1020 1247 y(ture)c(information)9 b(of)j(blo)q(c)o(ks)h(of) f(large)h(aerial)e(photographs.)1020 1294 y(All)22 b(our)g(tables)h(w)o (ere)g(stored)g(on)f(disk.)44 b(W)m(e)22 b(compared)1020 1341 y(the)i(p)q(erformance)f(of)g(our)h(algorithm)c(with)k(the)g(p)q (erfor-)1020 1389 y(mance)16 b(of)f(the)i(Sphere/Rectangle-tree)i (\(SR-tree\))e([28)o(],)f(a)1020 1436 y(recen)o(t)23 b(data)e(structure)i(whic)o(h)f(w)o(as)f(sho)o(wn)g(to)g(b)q(e)h(com-) 1020 1483 y(parable)e(to)h(or)f(signi\014can)o(tly)f(more)h(e\016cien)o (t)h(than)f(other)1020 1531 y(tree-decomp)q(osition-based)i(indexing)f (metho)q(ds)g(for)g(spa-)1020 1578 y(tial)11 b(data.)18 b(The)12 b(exp)q(erimen)o(ts)h(sho)o(w)f(that)h(our)f(algorithm)e(is) 1020 1625 y(signi\014can)o(tly)15 b(faster)i(than)f(the)h(earlier)f (metho)q(ds,)g(in)g(some)1020 1673 y(cases)23 b(ev)o(en)f(b)o(y)f(sev)o (eral)g(orders)i(of)e(magnitude.)38 b(It)22 b(also)1020 1720 y(scales)11 b(w)o(ell)e(as)g(the)i(data)e(size)i(and)e (dimensionalit)o(y)e(increase.)1020 1767 y(Th)o(us,)15 b(it)f(enables)h(a)g(new)g(approac)o(h)f(to)h(high-p)q(erformance)1020 1815 y(similarit)o(y)f(searc)o(h)k(|)e(fast)h(retriev)n(al)g(of)f (appro)o(ximate)f(an-)1020 1862 y(sw)o(er,)f(p)q(ossibly)g(follo)o(w)o (ed)e(b)o(y)i(a)g(slo)o(w)o(er)g(but)g(more)f(accurate)1020 1909 y(computation)18 b(in)h(the)i(few)e(cases)i(where)g(the)g(user)f (is)g(not)1020 1957 y(satis\014ed)15 b(with)e(the)i(appro)o(ximate)d (answ)o(er.)1070 2005 y(The)18 b(rest)h(of)e(this)h(pap)q(er)h(is)f (organized)g(as)g(follo)o(ws.)28 b(In)1020 2052 y(Section)18 b(2)f(w)o(e)h(in)o(tro)q(duce)g(the)g(notation)f(and)g(giv)o(e)g (formal)1020 2100 y(de\014nitions)e(of)f(the)h(similarit)o(y)d(searc)o (h)k(problems.)j(Then)c(in)1020 2147 y(Section)21 b(3)g(w)o(e)g (describ)q(e)h(lo)q(calit)o(y-sensitiv)o(e)e(hashing)g(and)1020 2194 y(sho)o(w)d(ho)o(w)g(to)f(apply)h(it)f(to)h(nearest)i(neigh)o(b)q (or)d(searc)o(h.)29 b(In)1020 2242 y(Section)20 b(4)f(w)o(e)h(rep)q (ort)g(the)g(results)h(of)e(exp)q(erimen)o(ts)g(with)1020 2289 y(LSH.)c(The)i(related)f(w)o(ork)g(is)g(describ)q(ed)h(in)f (Section)g(5.)24 b(Fi-)1020 2336 y(nally)m(,)10 b(in)h(Section)h(6)f(w) o(e)h(presen)o(t)h(conclusions)e(and)h(ideas)f(for)1020 2384 y(future)j(researc)o(h.)1020 2490 y Fv(2)56 b(Preliminaries)1020 2563 y Fo(W)m(e)12 b(use)i Fn(l)1174 2548 y Fl(d)1173 2573 y(p)1206 2563 y Fo(to)f(denote)g(the)g(Euclidean)g(space)h Fm(<)1789 2548 y Fl(d)1821 2563 y Fo(under)f(the)1020 2610 y Fn(l)1032 2616 y Fl(p)1064 2610 y Fo(norm,)e(i.e.,)g(when)h(the) h(length)g(of)e(a)h(v)o(ector)h(\()p Fn(x)1803 2616 y Fj(1)1822 2610 y Fn(;)7 b(:)g(:)g(:)e(x)1920 2616 y Fl(d)1939 2610 y Fo(\))12 b(is)1020 2663 y(de\014ned)k(as)e(\()p Fm(j)p Fn(x)1267 2669 y Fj(1)1285 2663 y Fm(j)1297 2648 y Fl(p)1326 2663 y Fo(+)c Fn(:)d(:)g(:)h Fo(+)i Fm(j)p Fn(x)1504 2669 y Fl(d)1523 2663 y Fm(j)1535 2648 y Fl(p)1553 2663 y Fo(\))1569 2648 y Fj(1)p Fl(=p)1622 2663 y Fo(.)20 b(F)m(urther,)c Fn(d)1840 2669 y Fl(p)1858 2663 y Fo(\()p Fn(p;)7 b(q)q Fo(\))13 b(=)p eop %%Page: 3 3 3 2 bop -30 -33 a Fm(jj)p Fn(p)s Fm(\000)s Fn(q)q Fm(jj)97 -27 y Fl(p)125 -33 y Fo(denotes)12 b(the)g(distance)f(b)q(et)o(w)o(een) h(the)g(p)q(oin)o(ts)e Fn(p)h Fo(and)-30 14 y Fn(q)19 b Fo(in)f Fn(l)74 -1 y Fl(d)73 24 y(p)94 14 y Fo(.)31 b(W)m(e)18 b(use)i Fn(H)328 -1 y Fl(d)365 14 y Fo(to)e(denote)i(the)f Fk(Hamming)g(metric)-30 61 y(sp)n(ac)n(e)c Fo(of)g(dimension)e Fn(d)p Fo(,)i(i.e.,)f(the)i(space)g(of)e(binary)h(v)o(ectors)-30 109 y(of)f(length)h Fn(d)f Fo(under)i(the)g(standard)f(Hammi)o(ng)d (metric.)20 b(W)m(e)-30 156 y(use)g Fn(d)69 162 y Fl(H)101 156 y Fo(\()p Fn(p;)7 b(q)q Fo(\))19 b(denote)h(the)h Fk(Hamming)f(distanc)n(e)p Fo(,)h(i.e.,)e(the)-30 203 y(n)o(um)o(b)q(er)13 b(of)g(bits)h(on)g(whic)o(h)g Fn(p)g Fo(and)f Fn(q)i Fo(di\013er.)20 251 y(The)h(nearest)i(neigh)o(b)q(or)e (searc)o(h)i(problem)c(is)j(de\014ned)g(as)-30 298 y(follo)o(ws:)-30 385 y Fi(De\014nitio)o(n)c(1)j(\(Nearest)f(Neigh)o(b)q(or)e(Searc)o(h)i (\(NNS\)\))-30 432 y Fk(Given)i(a)g(set)f Fn(P)22 b Fk(of)17 b Fn(n)f Fk(obje)n(cts)h(r)n(epr)n(esente)n(d)f(as)g(p)n(oints)h(in)g (a)-30 480 y(norme)n(d)f(sp)n(ac)n(e)g Fn(l)242 464 y Fl(d)241 490 y(p)262 480 y Fk(,)g(pr)n(epr)n(o)n(c)n(ess)f Fn(P)21 b Fk(so)16 b(as)g(to)g(e\016ciently)g(an-)-30 527 y(swer)10 b(queries)h(by)g(\014nding)h(the)f(p)n(oint)g(in)g Fn(P)16 b Fk(closest)11 b(to)g(a)g(query)-30 574 y(p)n(oint)k Fn(q)q Fk(.)20 661 y Fo(The)26 b(de\014nition)g(generalizes)h (naturally)e(to)g(the)i(case)-30 708 y(where)13 b(w)o(e)f(w)o(an)o(t)g (to)g(return)h Fn(K)h(>)e Fo(1)g(p)q(oin)o(ts.)17 b(Sp)q(eci\014cally)m (,)11 b(in)-30 756 y(the)i Fn(K)s Fk(-Ne)n(ar)n(est)f(Neighb)n(ors)h (Se)n(ar)n(ch)h(\()p Fn(K)s Fk(-NNS\))p Fo(,)e(w)o(e)g(wish)g(to)-30 803 y(return)h(the)f Fn(K)j Fo(p)q(oin)o(ts)d(in)f(the)h(database)g (that)g(are)g(closest)h(to)-30 850 y(the)h(query)f(p)q(oin)o(t.)k(The)d (appro)o(ximate)d(v)o(ersion)i(of)f(the)i(NNS)-30 898 y(problem)f(is)g(de\014ned)i(as)f(follo)o(ws:)-30 985 y Fi(De\014nitio)o(n)f(2)j(\()p Fn(\017)p Fi(-Nearest)e(Neigh)o(b)q(or) g(Searc)o(h)h(\()p Fn(\017)p Fi(-NNS\)\))-30 1032 y Fk(Given)g(a)h(set) f Fn(P)20 b Fk(of)15 b(p)n(oints)g(in)g(a)h(norme)n(d)f(sp)n(ac)n(e)g Fn(l)769 1017 y Fl(d)768 1042 y(p)789 1032 y Fk(,)g(pr)n(epr)n(o-)-30 1079 y(c)n(ess)e Fn(P)19 b Fk(so)13 b(as)g(to)g(e\016ciently)g(r)n (eturn)f(a)h(p)n(oint)h Fn(p)d Fm(2)g Fn(P)19 b Fk(for)12 b(any)-30 1126 y(given)j(query)f(p)n(oint)h Fn(q)q Fk(,)f(such)h(that)f Fn(d)p Fo(\()p Fn(q)q(;)7 b(p)p Fo(\))k Fm(\024)h Fo(\(1)c(+)g Fn(\017)p Fo(\))p Fn(d)p Fo(\()p Fn(q)q(;)f(P)f Fo(\))p Fk(,)-30 1174 y(wher)n(e)11 b Fn(d)p Fo(\()p Fn(q)q(;)c(P)f Fo(\))k Fk(is)h(the)g(distanc)n(e)h(of)f Fn(q)h Fk(to)g(the)f(its)g (closest)g(p)n(oint)-30 1221 y(in)k Fn(P)6 b Fk(.)20 1308 y Fo(Again,)14 b(this)h(de\014nition)f(generalizes)i(naturally)e (to)h(\014nd-)-30 1355 y(ing)e Fn(K)i(>)d Fo(1)i(appro)o(ximate)e (nearest)j(neigh)o(b)q(ors.)k(In)14 b(the)g Fk(Ap-)-30 1403 y(pr)n(oximate)h Fn(K)s Fk(-NNS)g Fo(problem,)e(w)o(e)h(wish)h(to) f(\014nd)g Fn(K)k Fo(p)q(oin)o(ts)-30 1450 y Fn(p)-9 1456 y Fj(1)10 1450 y Fn(;)7 b(:)g(:)g(:)t(;)g(p)123 1456 y Fl(K)165 1450 y Fo(suc)o(h)12 b(that)e(the)i(distance)f(of)f Fn(p)634 1456 y Fl(i)658 1450 y Fo(to)h(the)g(query)g Fn(q)h Fo(is)-30 1497 y(at)g(most)f(\(1)6 b(+)g Fn(\017)p Fo(\))12 b(times)f(the)i(distance)g(from)d(the)j Fn(i)p Fo(th)g(nearest)-30 1545 y(p)q(oin)o(t)g(to)h Fn(q)q Fo(.)-30 1645 y Fv(3)56 b(The)18 b(Algorithm)-30 1715 y Fo(In)13 b(this)g(section)h(w)o(e)f(presen)o(t)h(e\016cien)o(t)g (solutions)e(to)h(the)g(ap-)-30 1763 y(pro)o(ximate)f(v)o(ersions)h(of) g(the)h(NNS)f(problem.)k(Without)c(sig-)-30 1810 y(ni\014can)o(t)j (loss)g(of)g(generalit)o(y)m(,)g(w)o(e)g(will)f(mak)o(e)g(the)i(follo)o (wing)-30 1857 y(t)o(w)o(o)c(assumptions)g(ab)q(out)h(the)h(data:)0 1936 y(1.)20 b(the)d(distance)g(is)f(de\014ned)h(b)o(y)f(the)h Fn(l)626 1942 y Fj(1)661 1936 y Fo(norm)e(\(see)i(com-)53 1984 y(men)o(ts)c(b)q(elo)o(w\),)0 2063 y(2.)20 b(all)9 b(co)q(ordinates)i(of)f(p)q(oin)o(ts)h(in)f Fn(P)16 b Fo(are)10 b(p)q(ositiv)o(e)h(in)o(tegers.)20 2142 y(The)i(\014rst)i (assumption)c(is)i(not)h(v)o(ery)f(restrictiv)o(e,)h(as)g(usu-)-30 2190 y(ally)g(there)j(is)f(no)g(clear)g(adv)n(an)o(tage)f(in,)g(or)h (ev)o(en)g(di\013erence)-30 2237 y(b)q(et)o(w)o(een,)e(using)f Fn(l)261 2243 y Fj(2)293 2237 y Fo(or)g Fn(l)355 2243 y Fj(1)388 2237 y Fo(norm)e(for)i(similarit)o(y)d(searc)o(h.)19 b(F)m(or)-30 2284 y(example,)e(the)i(exp)q(erimen)o(ts)f(done)g(for)f (the)i(W)m(ebseek)f([43)o(])-30 2332 y(pro)r(ject)12 b(\(see)g([40)o(],)f(c)o(hapter)h(4\))e(sho)o(w)h(that)g(comparing)e (color)-30 2379 y(histograms)16 b(using)i Fn(l)309 2385 y Fj(1)345 2379 y Fo(and)g Fn(l)442 2385 y Fj(2)478 2379 y Fo(norms)f(yields)g(v)o(ery)h(similar)-30 2426 y(results)h(\()p Fn(l)134 2432 y Fj(1)171 2426 y Fo(is)e(marginally)e(b)q(etter\).)31 b(Both)18 b(our)g(data)g(sets)-30 2473 y(\(see)h(Section)f(4\))g(ha)o (v)o(e)g(a)f(similar)f(prop)q(ert)o(y)m(.)30 b(Sp)q(eci\014cally)m(,) -30 2521 y(w)o(e)22 b(observ)o(ed)h(that)f(a)g(nearest)h(neigh)o(b)q (or)f(of)f(an)h(a)o(v)o(erage)-30 2568 y(query)16 b(p)q(oin)o(t)e (computed)h(under)h(the)f Fn(l)592 2574 y Fj(1)626 2568 y Fo(norm)f(w)o(as)h(also)f(an)-30 2615 y Fn(\017)p Fo(-appro)o(ximate) c(neigh)o(b)q(or)h(under)i(the)f Fn(l)604 2621 y Fj(2)635 2615 y Fo(norm)f(with)g(an)h(a)o(v-)-30 2663 y(erage)17 b(v)n(alue)g(of)f Fn(\017)g Fo(less)i(than)f(3\045)f(\(this)h(observ)n (ation)f(holds)1020 -33 y(for)f(b)q(oth)g(data)f(sets\).)23 b(Moreo)o(v)o(er,)16 b(in)e(most)g(cases)i(\(i.e.,)e(for)1020 14 y(67\045)f(of)g(the)h(queries)h(in)e(the)i(\014rst)f(set)h(and)e (73\045)g(in)h(the)g(sec-)1020 61 y(ond)h(set\))h(the)g(nearest)h (neigh)o(b)q(ors)e(under)h Fn(l)1721 67 y Fj(1)1756 61 y Fo(and)f Fn(l)1850 67 y Fj(2)1884 61 y Fo(norms)1020 109 y(w)o(ere)20 b Fk(exactly)f Fo(the)g(same.)32 b(This)19 b(observ)n(ation)f(is)h(in)o(terest-)1020 156 y(ing)f(in)h(its)g(o)o (wn)g(righ)o(t,)h(and)f(can)g(b)q(e)h(partially)d(explained)1020 203 y(via)12 b(the)h(theorem)f(b)o(y)h(Figiel)e(et)j(al)d(\(see)j([19)o (])f(and)f(references)1020 251 y(therein\).)19 b(They)12 b(sho)o(w)o(ed)h(analytically)d(that)i(b)o(y)g(simply)e(ap-)1020 298 y(plying)18 b(scaling)h(and)h(random)e(rotation)g(to)i(the)g(space) g Fn(l)1964 304 y Fj(2)1983 298 y Fo(,)1020 345 y(w)o(e)e(can)g(mak)o (e)e(the)j(distances)g(induced)f(b)o(y)g(the)h Fn(l)1843 351 y Fj(1)1879 345 y Fo(and)f Fn(l)1976 351 y Fj(2)1020 393 y Fo(norms)d(almost)e(equal)i(up)h(to)f(an)h(arbitrarily)e(small)f (factor.)1020 440 y(It)j(seems)f(plausible)g(that)h(real)f(data)g(is)h (already)f(randomly)1020 487 y(rotated,)21 b(th)o(us)e(the)h (di\013erence)h(b)q(et)o(w)o(een)g Fn(l)1726 493 y Fj(1)1764 487 y Fo(and)e Fn(l)1862 493 y Fj(2)1900 487 y Fo(norm)1020 535 y(is)f(v)o(ery)g(small.)27 b(Moreo)o(v)o(er,)19 b(for)f(the)g(data) g(sets)h(for)e(whic)o(h)1020 582 y(this)h(prop)q(ert)o(y)h(do)q(es)f (not)g(hold,)g(w)o(e)g(are)g(guaran)o(teed)h(that)1020 629 y(after)d(p)q(erforming)f(scaling)h(and)g(random)e(rotation)i(our)g (al-)1020 677 y(gorithms)d(can)i(b)q(e)g(used)g(for)f(the)h Fn(l)1571 683 y Fj(2)1605 677 y Fo(norm)e(with)h(arbitrarily)1020 724 y(small)e(loss)h(of)h(precision.)1070 772 y(As)27 b(far)g(as)g(the)g(second)h(assumption)e(is)g(concerned,)1020 819 y(clearly)15 b(all)g(co)q(ordinates)h(can)f(b)q(e)h(made)f(p)q (ositiv)o(e)g(b)o(y)g(prop-)1020 866 y(erly)j(translating)g(the)h (origin)e(of)h Fm(<)1601 851 y Fl(d)1620 866 y Fo(.)31 b(W)m(e)18 b(can)h(then)g(con-)1020 914 y(v)o(ert)h(all)d(co)q (ordinates)j(to)f(in)o(tegers)h(b)o(y)e(m)o(ultiplying)e(them)1020 961 y(b)o(y)e(a)f(suitably)h(large)g(n)o(um)o(b)q(er)f(and)h(rounding)f (to)h(the)h(near-)1020 1008 y(est)h(in)o(teger.)23 b(It)16 b(can)g(b)q(e)g(easily)f(v)o(eri\014ed)h(that)f(b)o(y)g(c)o(ho)q(osing) 1020 1056 y(prop)q(er)f(parameters,)e(the)i(error)f(induced)h(b)o(y)e (rounding)h(can)1020 1103 y(b)q(e)g(made)f(arbitrarily)f(small.)k (Notice)e(that)g(after)g(this)f(op)q(er-)1020 1150 y(ation)h(the)i (minim)n(um)9 b(in)o(terp)q(oin)o(t)14 b(distance)h(is)e(1.)1020 1245 y Fi(3.1)48 b(Lo)q(calit)o(y-Sensi)o(ti)o(v)o(e)12 b(Hashing)1020 1316 y Fo(In)22 b(this)f(section)i(w)o(e)e(presen)o(t)j (lo)q(calit)o(y-sensitiv)o(e)c(hashing)1020 1363 y(\(LSH\).)h(This)f (tec)o(hnique)i(w)o(as)e(originally)f(in)o(tro)q(duced)i(b)o(y)1020 1411 y(Indyk)c(and)f(Mot)o(w)o(ani)g([24)o(])h(for)f(the)i(purp)q(oses) g(of)e(devising)1020 1458 y(main)8 b(memory)g(algorithms)h(for)h(the)h Fn(\017)p Fo(-NNS)f(problem.)16 b(Here)1020 1505 y(w)o(e)h(giv)o(e)f (an)g(impro)o(v)o(ed)e(v)o(ersion)j(of)f(their)h(algorithm.)23 b(The)1020 1553 y(new)15 b(algorithm)d(is)j(in)f(man)o(y)f(resp)q(ects) k(more)d(natural)g(than)1020 1600 y(the)e(earlier)h(one:)k(it)11 b(do)q(es)i(not)f(require)h(the)f(hash)g(buc)o(k)o(ets)h(to)1020 1647 y(store)i(only)f(one)g(p)q(oin)o(t;)f(it)h(has)h(b)q(etter)g (running)f(time)f(guar-)1020 1695 y(an)o(tees;)k(and,)f(the)g(analysis) f(is)h(generalized)g(to)g(the)g(case)h(of)1020 1742 y(secondary)e (memory)m(.)1070 1790 y(Let)h Fn(C)i Fo(b)q(e)e(the)g(largest)f(co)q (ordinate)h(in)f(all)f(p)q(oin)o(ts)h(in)g Fn(P)6 b Fo(.)1020 1837 y(Then,)13 b(as)h(p)q(er)g([29)o(],)e(w)o(e)i(can)g(em)o(b)q(ed)e Fn(P)19 b Fo(in)o(to)13 b(the)h(Hammi)o(ng)1020 1891 y(cub)q(e)19 b Fn(H)1160 1876 y Fl(d)1177 1863 y Fh(0)1208 1891 y Fo(with)e Fn(d)1328 1876 y Fg(0)1357 1891 y Fo(=)i Fn(C)s(d)p Fo(,)e(b)o(y)g(transforming)f(eac)o(h)i(p)q(oin)o(t)1020 1938 y Fn(p)11 b Fo(=)h(\()p Fn(x)1136 1944 y Fj(1)1155 1938 y Fn(;)7 b(:)g(:)g(:)e(x)1253 1944 y Fl(d)1272 1938 y Fo(\))14 b(in)o(to)f(a)g(binary)h(v)o(ector)1189 2028 y Fn(v)q Fo(\()p Fn(p)p Fo(\))f(=)f(Unary)1433 2038 y Fl(C)1461 2028 y Fo(\()p Fn(x)1501 2034 y Fj(1)1519 2028 y Fo(\))7 b Fn(:)g(:)g(:)f Fo(Unary)1711 2038 y Fl(C)1739 2028 y Fo(\()p Fn(x)1779 2034 y Fl(d)1798 2028 y Fo(\))p Fn(;)1020 2117 y Fo(where)15 b(Unary)1253 2127 y Fl(C)1281 2117 y Fo(\()p Fn(x)p Fo(\))e(denotes)i(the)g(unary)e(represen)o (tation)j(of)1020 2165 y Fn(x)p Fo(,)11 b(i.e.,)f(is)h(a)g(sequence)j (of)c Fn(x)h Fo(ones)h(follo)o(w)o(ed)e(b)o(y)h Fn(C)6 b Fm(\000)t Fn(x)12 b Fo(zero)q(es.)1020 2247 y Fi(F)l(act)k(1)21 b Fk(F)m(or)14 b(any)h(p)n(air)e(of)i(p)n(oints)f Fn(p;)7 b(q)15 b Fk(with)f(c)n(o)n(or)n(dinates)g(in)1020 2294 y(the)h(set)g Fm(f)p Fo(1)7 b Fn(:)g(:)g(:)t(C)s Fm(g)p Fk(,)1282 2384 y Fn(d)1304 2390 y Fj(1)1322 2384 y Fo(\()p Fn(p;)g(q)q Fo(\))k(=)h Fn(d)1491 2390 y Fl(H)1522 2384 y Fo(\()p Fn(v)q Fo(\()p Fn(p)p Fo(\))p Fn(;)7 b(v)q Fo(\()p Fn(q)q Fo(\)\))p Fn(:)1020 2473 y Fo(That)21 b(is,)h(the)g(em)o(b)q(edding)e(preserv)o(es)k(the)d(distances)h(b)q (e-)1020 2521 y(t)o(w)o(een)g(the)g(p)q(oin)o(ts.)41 b(Therefore,)25 b(in)c(the)h(sequel)g(w)o(e)g(can)1020 2568 y(concen)o(trate)c(on)e(solving)g Fn(\017)p Fo(-NNS)g(in)g(the)h (Hamming)c(space)1020 2615 y Fn(H)1058 2600 y Fl(d)1075 2588 y Fh(0)1088 2615 y Fo(.)25 b(Ho)o(w)o(ev)o(er,)16 b(w)o(e)h(emphasize)e(that)h(w)o(e)h Fk(do)g(not)f Fo(need)h(to)1020 2663 y(actually)12 b Fk(c)n(onvert)i Fo(the)g(data)f(to)g(the)h(unary)f (represen)o(tation,)p eop %%Page: 4 4 4 3 bop -30 -33 a Fo(whic)o(h)14 b(could)h(b)q(e)g(exp)q(ensiv)o(e)g (when)g Fn(C)j Fo(is)c(large;)g(in)g(fact,)g(all)-30 14 y(our)j(algorithms)e(can)i(b)q(e)h(made)e(to)h(run)h(in)e(time)g Fk(indep)n(en-)-30 61 y(dent)e Fo(on)g Fn(C)s Fo(.)k(Rather,)c(the)g (unary)g(represen)o(tation)i(pro)o(vides)-30 109 y(us)f(with)e(a)h(con) o(v)o(enien)o(t)h(framew)o(ork)d(for)i(description)h(of)e(the)-30 156 y(algorithms)k(whic)o(h)h(w)o(ould)g(b)q(e)i(more)e(complicated)f (other-)-30 203 y(wise.)20 255 y(W)m(e)10 b(de\014ne)h(the)f(hash)h (functions)f(as)g(follo)o(ws.)15 b(F)m(or)10 b(an)g(in)o(te-)-30 302 y(ger)j Fn(l)g Fo(to)g(b)q(e)g(sp)q(eci\014ed)h(later,)e(c)o(ho)q (ose)h Fn(l)h Fo(subsets)g Fn(I)757 308 y Fj(1)776 302 y Fn(;)7 b(:)g(:)g(:)e(;)i(I)887 308 y Fl(l)912 302 y Fo(of)-30 349 y Fm(f)p Fo(1)p Fn(;)g(:)g(:)g(:)t(;)g(d)126 334 y Fg(0)137 349 y Fm(g)p Fo(.)16 b(Let)11 b Fn(p)278 356 y Fg(j)p Fl(I)316 349 y Fo(denote)g(the)f(pro)r(jection)h(of)e(v)o (ector)i Fn(p)e Fo(on)-30 397 y(the)15 b(co)q(ordinate)f(set)i Fn(I)s Fo(,)e(i.e.,)f(w)o(e)h(compute)g Fn(p)689 404 y Fg(j)p Fl(I)732 397 y Fo(b)o(y)g(selecting)-30 444 y(the)19 b(co)q(ordinate)g(p)q(ositions)g(as)f(p)q(er)i Fn(I)i Fo(and)d(concatenating)-30 491 y(the)e(bits)f(in)g(those)h(p)q (ositions.)24 b(Denote)17 b Fn(g)651 497 y Fl(j)668 491 y Fo(\()p Fn(p)p Fo(\))f(=)f Fn(p)805 498 y Fg(j)p Fl(I)830 502 y Ff(j)847 491 y Fo(.)25 b(F)m(or)-30 539 y(the)16 b(prepro)q(cessing,)i(w)o(e)e(store)h(eac)o(h)f Fn(p)e Fm(2)h Fn(P)21 b Fo(in)15 b(the)h(buc)o(k)o(et)-30 586 y Fn(g)-10 592 y Fl(j)7 586 y Fo(\()p Fn(p)p Fo(\),)e(for)f Fn(j)h Fo(=)e(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(l)q Fo(.)18 b(As)c(the)h(total)e(n)o(um)o(b)q(er)g(of)g(buc)o(k)o(ets)-30 633 y(ma)o(y)k(b)q(e)i(large,)g(w)o(e)f(compress)h(the)g(buc)o(k)o(ets) h(b)o(y)e(resorting)-30 681 y(to)c(standard)g(hashing.)j(Th)o(us,)d(w)o (e)g(use)h(t)o(w)o(o)e(lev)o(els)h(of)f(hash-)-30 728 y(ing:)k(the)c(LSH)g(function)g(maps)e(a)i(p)q(oin)o(t)f Fn(p)h Fo(to)f(buc)o(k)o(et)i Fn(g)863 734 y Fl(j)880 728 y Fo(\()p Fn(p)p Fo(\),)-30 775 y(and)20 b(a)g(standard)g(hash)g (function)g(maps)f(the)h(con)o(ten)o(ts)i(of)-30 823 y(these)16 b(buc)o(k)o(ets)g(in)o(to)e(a)h(hash)g(table)g(of)f(size)h Fn(M)5 b Fo(.)21 b(The)15 b(maxi-)-30 870 y(mal)e(buc)o(k)o(et)i(size)g (of)f(the)i(latter)f(hash)f(table)h(is)g(denoted)g(b)o(y)-30 917 y Fn(B)r Fo(.)k(F)m(or)13 b(the)i(algorithm's)c(analysis,)i(w)o(e)h (will)e(assume)i(hash-)-30 965 y(ing)h(with)h(c)o(haining,)f(i.e.,)f (when)j(the)f(n)o(um)o(b)q(er)f(of)h(p)q(oin)o(ts)f(in)-30 1012 y(a)h(buc)o(k)o(et)i(exceeds)g Fn(B)r Fo(,)g(a)e(new)h(buc)o(k)o (et)g(\(also)f(of)g(size)i Fn(B)r Fo(\))f(is)-30 1059 y(allo)q(cated)11 b(and)h(link)o(ed)f(to)g(and)h(from)e(the)i(old)f (buc)o(k)o(et.)18 b(Ho)o(w-)-30 1107 y(ev)o(er,)j(our)e(implemen)o (tatio)o(n)d(do)q(es)k(not)f(emplo)o(y)e(c)o(haining,)-30 1154 y(but)12 b(relies)g(on)g(a)f(simpler)f(approac)o(h:)17 b(if)11 b(a)g(buc)o(k)o(et)i(in)e(a)g(giv)o(en)-30 1201 y(index)j(is)g(full,)e(a)i(new)h(p)q(oin)o(t)e(cannot)i(b)q(e)f(added)h (to)f(it,)f(since)-30 1249 y(it)i(will)f(b)q(e)j(added)f(to)f(some)g (other)h(index)g(with)f(high)g(prob-)-30 1296 y(abilit)o(y)m(.)h(This)e (sa)o(v)o(es)g(us)g(the)h(o)o(v)o(erhead)f(of)f(main)o(taining)d(the) -30 1343 y(link)j(structure.)20 1395 y(The)20 b(n)o(um)o(b)q(er)e Fn(n)h Fo(of)g(p)q(oin)o(ts,)h(the)g(size)g Fn(M)k Fo(of)19 b(the)h(hash)-30 1442 y(table,)14 b(and)g(the)h(maxim)n(um)10 b(buc)o(k)o(et)15 b(size)g Fn(B)i Fo(are)d(related)h(b)o(y)-30 1489 y(the)f(follo)o(wing)e(equation:)367 1597 y Fn(M)k Fo(=)c Fn(\013)503 1569 y(n)p 499 1588 34 2 v 499 1626 a(B)537 1597 y(;)-30 1708 y Fo(where)j Fn(\013)e Fo(is)g(the)h(memory)d (utilization)h(parameter,)g(i.e.,)g(the)-30 1756 y(ratio)g(of)g(the)h (memory)d(allo)q(cated)i(for)g(the)h(index)g(to)f(the)h(size)-30 1803 y(of)g(the)i(data)e(set.)20 1854 y(T)m(o)32 b(pro)q(cess)j(a)d (query)i Fn(q)q Fo(,)j(w)o(e)c(searc)o(h)h(all)e(indices)-30 1902 y Fn(g)-10 1908 y Fj(1)8 1902 y Fo(\()p Fn(q)q Fo(\))p Fn(;)7 b(:)g(:)g(:)f(;)h(g)174 1908 y Fl(l)186 1902 y Fo(\()p Fn(q)q Fo(\))16 b(un)o(til)f(w)o(e)h(either)h(encoun)o(ter)g (at)f(least)g Fn(c)11 b Fm(\001)f Fn(l)-30 1949 y Fo(p)q(oin)o(ts)f (\(for)h Fn(c)f Fo(sp)q(eci\014ed)j(later\))d(or)h(use)g(all)f Fn(l)i Fo(indices.)17 b(Clearly)m(,)-30 1996 y(the)f(n)o(um)o(b)q(er)f (of)f(disk)i(accesses)h(is)f(alw)o(a)o(ys)e(upp)q(er)i(b)q(ounded)-30 2044 y(b)o(y)k(the)g(n)o(um)o(b)q(er)g(of)f(indices,)j(whic)o(h)d(is)h (equal)g(to)g Fn(l)q Fo(.)36 b(Let)-30 2091 y Fn(p)-9 2097 y Fj(1)10 2091 y Fn(;)7 b(:)g(:)g(:)t(;)g(p)123 2097 y Fl(t)158 2091 y Fo(b)q(e)21 b(the)g(p)q(oin)o(ts)g(encoun)o (tered)h(in)e(the)i(pro)q(cess.)-30 2138 y(F)m(or)14 b(Appro)o(ximate)g Fn(K)s Fo(-NNS,)g(w)o(e)h(output)g(the)h Fn(K)i Fo(p)q(oin)o(ts)c Fn(p)931 2144 y Fl(i)-30 2185 y Fo(closest)i(to)e Fn(q)q Fo(;)h(in)f(general,)h(w)o(e)f(ma)o(y)f (return)j(few)o(er)f(p)q(oin)o(ts)g(if)-30 2233 y(the)f(n)o(um)o(b)q (er)g(of)f(p)q(oin)o(ts)h(encoun)o(tered)i(is)d(less)i(than)f Fn(K)s Fo(.)20 2284 y(It)j(remains)g(to)g(sp)q(ecify)h(the)g(c)o(hoice) g(of)f(the)h(subsets)i Fn(I)916 2290 y Fl(j)933 2284 y Fo(.)-30 2332 y(F)m(or)e(eac)o(h)h Fn(j)j Fm(2)d(f)p Fo(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(l)q Fm(g)p Fo(,)19 b(the)g(set)h Fn(I)596 2338 y Fl(j)632 2332 y Fo(consists)f(of)f Fn(k)i Fo(ele-)-30 2379 y(men)o(ts)c(from)e Fm(f)p Fo(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(d)350 2364 y Fg(0)361 2379 y Fm(g)16 b Fo(sampled)f(uniformly)e(at)k(random)-30 2426 y(with)c(replacemen)o (t.)k(The)d(optimal)c(v)n(alue)i(of)h Fn(k)g Fo(is)g(c)o(hosen)h(to)-30 2473 y(maxim)o(ize)h(the)i(probabilit)o(y)e(that)i(a)g(p)q(oin)o(t)f Fn(p)g Fo(\\close")h(to)g Fn(q)-30 2521 y Fo(will)e(fall)g(in)o(to)g (the)i(same)f(buc)o(k)o(et)h(as)g Fn(q)q Fo(,)f(and)g(also)g(to)g (mini-)-30 2568 y(mize)c(the)h(probabilit)o(y)e(that)i(a)f(p)q(oin)o(t) g Fn(p)597 2553 y Fg(0)622 2568 y Fo(\\far)g(a)o(w)o(a)o(y")f(from)g Fn(q)-30 2615 y Fo(will)f(fall)h(in)o(to)g(the)h(same)f(buc)o(k)o(et.) 18 b(The)12 b(c)o(hoice)h(of)e(the)h(v)n(alues)-30 2663 y(of)h Fn(l)i Fo(and)f Fn(k)h Fo(is)f(deferred)h(to)f(the)g(next)h (section.)p 1020 -73 1000 2 v 1019 495 2 568 v 1045 7 a Fi(Algorithm)c Fo(Prepro)q(cessing)1045 55 y Fi(Input)h Fo(A)i(set)h(of)e(p)q(oin)o(ts)h Fn(P)6 b Fo(,)1089 102 y Fn(l)15 b Fo(\(n)o(um)o(b)q(er)e(of)g(hash)h(tables\),)1045 149 y Fi(Output)d Fo(Hash)j(tables)h Fm(T)1460 155 y Fl(i)1473 149 y Fo(,)e Fn(i)f Fo(=)g(1)p Fn(;)7 b(:)g(:)g(:)e(;)i(l) 1045 197 y Fi(F)l(oreac)o(h)13 b Fn(i)f Fo(=)f(1)p Fn(;)c(:)g(:)g(:)e (;)i(l)1089 244 y Fo(Initialize)13 b(hash)h(table)f Fm(T)1484 250 y Fl(i)1512 244 y Fo(b)o(y)g(generating)1089 291 y(a)g(random)g(hash)h(function)f Fn(g)1555 297 y Fl(i)1569 291 y Fo(\()p Fm(\001)p Fo(\))1045 339 y Fi(F)l(oreac)o(h)g Fn(i)f Fo(=)f(1)p Fn(;)c(:)g(:)g(:)e(;)i(l)1089 386 y Fi(F)l(oreac)o(h)13 b Fn(j)h Fo(=)d(1)p Fn(;)c(:)g(:)g(:)e(;)i(n)1133 433 y Fo(Store)14 b(p)q(oin)o(t)f Fn(p)1370 439 y Fl(j)1402 433 y Fo(on)g(buc)o(k)o(et)i Fn(g)1612 439 y Fl(i)1626 433 y Fo(\()p Fn(p)1663 439 y Fl(j)1680 433 y Fo(\))f(of)f(hash)h (table)g Fm(T)1981 439 y Fl(i)p 2019 495 V 1020 496 1000 2 v 1020 571 a Fo(Figure)j(1:)23 b(Prepro)q(cessing)c(algorithm)14 b(for)j(p)q(oin)o(ts)f(already)1020 618 y(em)o(b)q(edded)e(in)f(the)i (Hamming)10 b(cub)q(e.)p 1020 685 1012 2 v 1019 1300 2 616 v 1045 766 a Fi(Algorithm)h Fo(Appro)o(ximate)h(Nearest)j(Neigh)o (b)q(or)f(Query)1045 813 y Fi(Input)e Fo(A)i(query)g(p)q(oin)o(t)g Fn(q)q Fo(,)1089 860 y Fn(K)j Fo(\(n)o(um)o(b)q(er)c(of)g(appr.)19 b(nearest)c(neigh)o(b)q(ors\))1045 908 y Fi(Access)f Fo(T)m(o)f(hash)h(tables)g Fm(T)1498 914 y Fl(i)1512 908 y Fo(,)f Fn(i)f Fo(=)g(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(l)1089 955 y Fo(generated)15 b(b)o(y)f(the)g(prepro)q(cessing)i(algorithm)1045 1002 y Fi(Output)11 b Fn(K)17 b Fo(\(or)d(less\))h(appr.)j(nearest)e (neigh)o(b)q(ors)1045 1050 y Fn(S)e Fm( )d Fo(\037)1045 1097 y Fi(F)l(oreac)o(h)i Fn(i)f Fo(=)f(1)p Fn(;)c(:)g(:)g(:)e(;)i(l) 1089 1144 y(S)14 b Fm( )d Fn(S)h Fm([)d(f)p Fo(p)q(oin)o(ts)14 b(found)f(in)h Fn(g)1586 1150 y Fl(i)1599 1144 y Fo(\()p Fn(q)q Fo(\))g(buc)o(k)o(et)h(of)e(table)h Fm(T)1972 1150 y Fl(i)1985 1144 y Fm(g)1045 1192 y Fo(Return)g(the)h Fn(K)i Fo(nearest)e(neigh)o(b)q(ors)f(of)g Fn(q)g Fo(found)g(in)f(set)i Fn(S)1045 1239 y Fo(/*)e(Can)h(b)q(e)g(found)g(b)o(y)g(main)d(memory)h (linear)h(searc)o(h)i(*/)p 2030 1300 V 1020 1302 1012 2 v 1020 1377 a(Figure)20 b(2:)29 b(Appro)o(ximate)18 b(Nearest)j(Neigh)o(b)q(or)e(query)h(an-)1020 1424 y(sw)o(ering)14 b(algorithm.)1070 1515 y(Although)e(w)o(e)i(are)f(mainly)e(in)o (terested)k(in)d(the)i(I/O)f(com-)1020 1562 y(plexit)o(y)18 b(of)h(our)g(sc)o(heme,)h(it)e(is)h(w)o(orth)g(p)q(oin)o(ting)f(out)h (that)1020 1610 y(the)e(hash)g(functions)f(can)h(b)q(e)g(e\016cien)o (tly)g(computed)f(if)f(the)1020 1657 y(data)c(set)i(is)f(obtained)f(b)o (y)h(mapping)d Fn(l)1620 1642 y Fl(d)1619 1667 y Fj(1)1651 1657 y Fo(in)o(to)i Fn(d)1755 1642 y Fg(0)1766 1657 y Fo(-dimensional)1020 1704 y(Hamming)e(space.)19 b(Let)13 b Fn(p)g Fo(b)q(e)g(an)o(y)g(p)q(oin)o(t)f(from)f(the)j(data)e(set)1020 1752 y(and)k(let)g Fn(p)1186 1737 y Fg(0)1213 1752 y Fo(denote)h(its)f(image)e(after)i(the)g(mapping.)22 b(Let)16 b Fn(I)1020 1799 y Fo(b)q(e)i(the)h(set)f(of)f(co)q(ordinates)i(and)e (recall)h(that)g(w)o(e)f(need)i(to)1020 1846 y(compute)14 b Fn(p)1210 1831 y Fg(0)1210 1860 y(j)p Fl(I)1238 1846 y Fo(.)19 b(F)m(or)13 b Fn(i)g Fo(=)f(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(d)p Fo(,)13 b(let)h Fn(I)1652 1853 y Fg(j)p Fl(i)1690 1846 y Fo(denote,)g(in)g(sorted)1020 1900 y(order,)20 b(the)g(co)q (ordinates)f(in)f Fn(I)23 b Fo(whic)o(h)18 b(corresp)q(ond)j(to)d(the) 1020 1947 y Fn(i)p Fo(th)c(co)q(ordinate)h(of)e Fn(p)p Fo(.)18 b(Observ)o(e,)d(that)f(pro)r(jecting)h Fn(p)1870 1932 y Fg(0)1895 1947 y Fo(on)f Fn(I)1971 1954 y Fg(j)p Fl(i)1020 1995 y Fo(results)k(in)f(a)g(sequence)j(of)c(bits)i(whic)o(h) f(is)g(monotone,)f(i.e.,)1020 2042 y(consists)j(of)d(a)i(n)o(um)o(b)q (er,)f(sa)o(y)g Fn(o)1528 2048 y Fl(i)1542 2042 y Fo(,)h(of)f(ones)h (follo)o(w)o(ed)e(b)o(y)h(ze-)1020 2089 y(ros.)j(Therefore,)c(in)e (order)h(to)g(represen)o(t)h Fn(p)1716 2074 y Fg(0)1716 2101 y Fl(I)1750 2089 y Fo(it)e(is)g(su\016cien)o(t)1020 2136 y(to)19 b(compute)f Fn(o)1269 2142 y Fl(i)1301 2136 y Fo(for)h Fn(i)h Fo(=)f(1)p Fn(;)7 b(:)g(:)g(:)e(;)i(d)p Fo(.)32 b(Ho)o(w)o(ev)o(er,)19 b(the)h(latter)1020 2184 y(task)e(is)g(equiv)n(alen)o(t)f(to)h(\014nding)f(the)i(n)o(um)o(b)q (er)e(of)g(elemen)o(ts)1020 2231 y(in)d(the)i(sorted)f(arra)o(y)g Fn(I)1395 2238 y Fg(j)p Fl(i)1433 2231 y Fo(whic)o(h)g(are)g(smaller)e (than)i(a)f(giv)o(en)1020 2278 y(v)n(alue,)h(i.e.,)g(the)h Fn(i)p Fo(th)h(co)q(ordinate)f(of)f Fn(p)p Fo(.)24 b(This)15 b(can)h(b)q(e)h(done)1020 2326 y(via)h(binary)f(searc)o(h)j(in)e(log)6 b Fn(C)21 b Fo(time,)d(or)g(ev)o(en)h(in)f(constan)o(t)1020 2373 y(time)c(using)h(a)g(precomputed)g(arra)o(y)g(of)f Fn(C)k Fo(bits.)k(Th)o(us,)15 b(the)1020 2420 y(total)k(time)f(needed)j (to)e(compute)g(the)h(function)f(is)g(either)1020 2468 y Fn(O)q Fo(\()p Fn(d)7 b Fo(log)f Fn(C)s Fo(\))17 b(or)f Fn(O)q Fo(\()p Fn(d)p Fo(\),)h(dep)q(ending)g(on)g(resources)i(used.)28 b(In)1020 2515 y(our)14 b(exp)q(erimen)o(tal)f(section,)h(the)h(v)n (alue)e(of)h Fn(C)i Fo(can)f(b)q(e)f(made)1020 2562 y(v)o(ery)h(small,) d(and)j(therefore)h(w)o(e)f(will)f(resort)i(to)e(the)i(second)1020 2610 y(metho)q(d.)1070 2663 y(F)m(or)10 b(quic)o(k)h(reference)i(w)o(e) f(summarize)c(the)k(prepro)q(cessing)p eop %%Page: 5 5 5 4 bop -30 -33 a Fo(and)14 b(query)g(answ)o(ering)g(algorithms)e(in)h (Figures)h(1)g(and)g(2.)-30 60 y Fi(3.2)48 b(Analysis)14 b(of)h(Lo)q(calit)o(y-Sensi)o(ti)o(v)o(e)e(Hashing)-30 130 y Fo(The)g(principle)g(b)q(ehind)f(our)h(metho)q(d)f(is)g(that)h (the)g(probabil-)-30 177 y(it)o(y)h(of)g(collision)f(of)h(t)o(w)o(o)g (p)q(oin)o(ts)g Fn(p)h Fo(and)f Fn(q)h Fo(is)g(closely)f(related)-30 224 y(to)h(the)g(distance)h(b)q(et)o(w)o(een)g(them.)k(Sp)q (eci\014cally)m(,)14 b(the)h(larger)-30 272 y(the)c(distance,)g(the)g (smaller)e(the)i(collision)e(probabilit)o(y)m(.)15 b(This)-30 319 y(in)o(tuition)e(is)h(formalized)f(as)h(follo)o(ws)f([24)o(].)19 b(Let)c Fn(D)q Fo(\()p Fm(\001)p Fn(;)7 b Fm(\001)p Fo(\))13 b(b)q(e)i(a)-30 366 y(distance)20 b(function)g(of)f(elemen)o(ts)g(from) f(a)h(set)i Fn(S)r Fo(,)g(and)e(for)-30 414 y(an)o(y)c Fn(p)f Fm(2)f Fn(S)19 b Fo(let)c Fm(B)q Fo(\()p Fn(p;)7 b(r)q Fo(\))15 b(denote)i(the)e(set)i(of)e(elemen)o(ts)g(from)-30 461 y Fn(S)i Fo(within)c(the)h(distance)h Fn(r)g Fo(from)d Fn(p)p Fo(.)-30 548 y Fi(De\014nitio)o(n)h(3)21 b Fk(A)d(family)h Fm(H)g Fk(of)g(functions)g(fr)n(om)f Fn(S)k Fk(to)d Fn(U)-30 596 y Fk(is)g(c)n(al)r(le)n(d)g Fo(\()p Fn(r)173 602 y Fj(1)192 596 y Fn(;)7 b(r)230 602 y Fj(2)247 596 y Fn(;)g(p)287 602 y Fj(1)305 596 y Fn(;)g(p)345 602 y Fj(2)363 596 y Fo(\)-sensitiv)o(e)20 b Fk(for)f Fn(D)q Fo(\()p Fm(\001)p Fn(;)7 b Fm(\001)p Fo(\))19 b Fk(if)g(for)f(any)-30 643 y Fn(q)q(;)7 b(p)k Fm(2)g Fn(S)12 692 y Fm(\017)20 b Fk(if)14 b Fn(p)e Fm(2)f(B)q Fo(\()p Fn(q)q(;)c(r)267 698 y Fj(1)285 692 y Fo(\))15 b Fk(then)h Fo(Pr)453 698 y Fg(H)484 692 y Fo([)p Fn(h)p Fo(\()p Fn(q)q Fo(\))11 b(=)h Fn(h)p Fo(\()p Fn(p)p Fo(\)])f Fm(\025)h Fn(p)792 698 y Fj(1)810 692 y Fk(,)12 772 y Fm(\017)20 b Fk(if)14 b Fn(p)i(=)-25 b Fm(2)11 b(B)q Fo(\()p Fn(q)q(;)c(r)267 778 y Fj(2)285 772 y Fo(\))15 b Fk(then)h Fo(Pr)453 778 y Fg(H)484 772 y Fo([)p Fn(h)p Fo(\()p Fn(q)q Fo(\))11 b(=)h Fn(h)p Fo(\()p Fn(p)p Fo(\)])f Fm(\024)h Fn(p)792 778 y Fj(2)810 772 y Fk(.)-30 860 y Fo(In)21 b(the)g(ab)q(o)o(v)o(e)f (de\014nition,)h(probabilities)f(are)h(considered)-30 907 y(with)11 b(resp)q(ect)i(to)f(the)f(random)f(c)o(hoice)i(of)f(a)g (function)g Fn(h)g Fo(from)-30 954 y(the)17 b(family)d Fm(H)p Fo(.)26 b(In)17 b(order)g(for)f(a)g(lo)q(calit)o(y-sensitiv)o(e) g(family)-30 1002 y(to)f(b)q(e)h(useful,)f(it)g(has)h(to)f(satisfy)g (the)h(inequalities)e Fn(p)827 1008 y Fj(1)859 1002 y Fn(>)g(p)926 1008 y Fj(2)-30 1049 y Fo(and)g Fn(r)70 1055 y Fj(1)100 1049 y Fn(<)d(r)162 1055 y Fj(2)181 1049 y Fo(.)20 1096 y(Observ)o(e)24 b(that)f(if)f Fn(D)q Fo(\()p Fm(\001)p Fn(;)7 b Fm(\001)p Fo(\))23 b(is)f(the)i(Hammi)o(ng)c (distance)-30 1144 y Fn(d)-8 1150 y Fl(H)23 1144 y Fo(\()p Fm(\001)p Fn(;)7 b Fm(\001)p Fo(\),)19 b(then)h(the)g(family)c(of)i (pro)r(jections)i(on)f(one)h(co)q(or-)-30 1191 y(dinate)14 b(is)g(lo)q(calit)o(y-sensitiv)o(e.)j(More)d(sp)q(eci\014cally:)-30 1278 y Fi(F)l(act)i(2)21 b Fk(L)n(et)33 b Fn(S)j Fk(b)n(e)d Fn(H)378 1263 y Fl(d)395 1251 y Fh(0)442 1278 y Fk(\(the)g Fn(d)568 1263 y Fg(0)580 1278 y Fk(-dimensional)g(Ham-)-30 1326 y(ming)f(cub)n(e\))g(and)g Fn(D)q Fo(\()p Fn(p;)7 b(q)q Fo(\))42 b(=)g Fn(d)579 1332 y Fl(H)610 1326 y Fo(\()p Fn(p;)7 b(q)q Fo(\))31 b Fk(for)g Fn(p;)7 b(q)42 b Fm(2)-30 1379 y Fn(H)8 1364 y Fl(d)25 1352 y Fh(0)38 1379 y Fk(.)47 b(Then)25 b(for)e(any)i Fn(r)q Fk(,)h Fn(\017)i(>)h Fo(0)p Fk(,)d(the)e(family)g Fm(H)854 1385 y Fl(d)871 1377 y Fh(0)913 1379 y Fo(=)-30 1427 y Fm(f)p Fn(h)15 1433 y Fl(i)59 1427 y Fo(:)30 b Fn(h)125 1433 y Fl(i)139 1427 y Fo(\(\()p Fn(b)189 1433 y Fj(1)207 1427 y Fn(;)7 b(:)g(:)g(:)e(;)i(b)318 1433 y Fl(d)335 1425 y Fh(0)348 1427 y Fo(\)\))31 b(=)g Fn(b)492 1433 y Fl(i)505 1427 y Fn(;)h Fk(for)14 b Fn(i)31 b Fo(=)g(1)p Fn(;)7 b(:)g(:)g(:)e(;)i(d)859 1412 y Fg(0)869 1427 y Fm(g)25 b Fk(is)-30 1443 y Fe(\020)-5 1489 y Fn(r)o(;)7 b(r)q Fo(\(1)h(+)h Fn(\017)p Fo(\))p Fn(;)e Fo(1)i Fm(\000)272 1473 y Fl(r)p 266 1480 29 2 v 266 1503 a(d)283 1495 y Fh(0)300 1489 y Fn(;)e Fo(1)h Fm(\000)395 1469 y Fl(r)q Fj(\(1+)p Fl(\017)p Fj(\))p 395 1479 99 2 v 429 1503 a Fl(d)446 1495 y Fh(0)498 1443 y Fe(\021)523 1489 y Fk(-sensitive.)20 1589 y Fo(W)m(e)19 b(no)o(w)g(generalize)i(the)f (algorithm)d(from)h(the)i(previ-)-30 1636 y(ous)j(section)h(to)f(an)g Fk(arbitr)n(ary)f Fo(lo)q(calit)o(y-sensitiv)o(e)g(family)-30 1684 y Fm(H)p Fo(.)e(Th)o(us,)15 b(the)g(algorithm)d(is)i(equally)f (applicable)h(to)g(other)-30 1731 y(lo)q(calit)o(y-sensitiv)o(e)19 b(hash)i(functions)g(\(e.g.,)f(see)i([5)o(]\).)38 b(The)-30 1778 y(generalization)16 b(is)h(simple:)22 b(the)17 b(functions)g Fn(g)h Fo(are)f(no)o(w)g(de-)-30 1826 y(\014ned)d(to)g(b)q(e)h(of)e (the)h(form)148 1913 y Fn(g)168 1919 y Fl(i)181 1913 y Fo(\()p Fn(p)p Fo(\))e(=)g(\()p Fn(h)330 1919 y Fl(i)342 1923 y Fd(1)360 1913 y Fo(\()p Fn(p)p Fo(\))p Fn(;)7 b(h)456 1919 y Fl(i)468 1923 y Fd(2)485 1913 y Fo(\()p Fn(p)p Fo(\))p Fn(;)g(:)g(:)g(:)e(;)i(h)655 1919 y Fl(i)667 1923 y Ff(k)687 1913 y Fo(\()p Fn(p)p Fo(\)\))p Fn(;)-30 2000 y Fo(where)17 b(the)g(functions)g Fn(h)372 2006 y Fl(i)384 2010 y Fd(1)402 2000 y Fn(;)7 b(:)g(:)g(:)t(;)g(h)518 2006 y Fl(i)530 2010 y Ff(k)566 2000 y Fo(are)17 b(randomly)d(c)o (hosen)-30 2048 y(from)d Fm(H)h Fo(with)g(replacemen)o(t.)18 b(As)13 b(b)q(efore,)g(w)o(e)f(c)o(ho)q(ose)i Fn(l)f Fo(suc)o(h)-30 2095 y(functions)e Fn(g)166 2101 y Fj(1)185 2095 y Fn(;)c(:)g(:)g(:)t(;)g(g)297 2101 y Fl(l)309 2095 y Fo(.)18 b(In)11 b(the)h(case)g(when)f(the)h(family)c Fm(H)875 2101 y Fl(d)892 2093 y Fh(0)917 2095 y Fo(is)-30 2142 y(used,)k(i.e.,)e(eac)o(h)i(function)f(selects)h(one)g(bit)e(of)h (an)g(argumen)o(t,)-30 2190 y(the)k(resulting)g(v)n(alues)g(of)f Fn(g)407 2196 y Fl(j)424 2190 y Fo(\()p Fn(p)p Fo(\))h(are)g(essen)o (tially)g(equiv)n(alen)o(t)-30 2237 y(to)f Fn(p)42 2244 y Fg(j)p Fl(I)67 2248 y Ff(j)84 2237 y Fo(.)20 2284 y(W)m(e)9 b(no)o(w)g(sho)o(w)g(that)h(the)g(LSH)f(algorithm)e(can)j(b)q(e)g(used) g(to)-30 2332 y(solv)o(e)k(what)f(w)o(e)h(call)f(the)h(\()p Fn(r)o(;)7 b(\017)p Fo(\)-Neigh)o(b)q(or)13 b(problem:)k(deter-)-30 2379 y(mine)11 b(whether)j(there)g(exists)f(a)g(p)q(oin)o(t)f Fn(p)g Fo(within)g(a)g(\014xed)h(dis-)-30 2426 y(tance)f Fn(r)97 2432 y Fj(1)127 2426 y Fo(=)g Fn(r)g Fo(of)e Fn(q)q Fo(,)h(or)h(whether)g(all)e(p)q(oin)o(ts)h(in)g(the)h(database) -30 2473 y(are)f(at)g(least)h(a)e(distance)i Fn(r)388 2479 y Fj(2)418 2473 y Fo(=)g Fn(r)q Fo(\(1)t(+)t Fn(\017)p Fo(\))e(a)o(w)o(a)o(y)g(from)f Fn(q)q Fo(;)j(in)e(the)-30 2521 y(\014rst)17 b(case,)f(the)h(algorithm)c(is)j(required)g(to)g (return)h(a)e(p)q(oin)o(t)-30 2568 y Fn(p)-9 2553 y Fg(0)21 2568 y Fo(within)j(distance)h(at)f(most)g(\(1)12 b(+)h Fn(\017)p Fo(\))p Fn(r)19 b Fo(from)e Fn(q)q Fo(.)32 b(In)18 b(par-)-30 2615 y(ticular,)f(w)o(e)h(argue)f(that)h(the)f(LSH)h (algorithm)c(solv)o(es)k(this)-30 2663 y(problem)c(for)h(a)f(prop)q(er) i(c)o(hoice)g(of)e Fn(k)i Fo(and)f Fn(l)q Fo(,)g(dep)q(ending)h(on)1020 -33 y Fn(r)h Fo(and)e Fn(\017)p Fo(.)24 b(Then)16 b(w)o(e)g(sho)o(w)g (ho)o(w)f(to)h(apply)f(the)h(solution)f(to)1020 14 y(this)f(problem)e (to)i(solv)o(e)g Fn(\017)p Fo(-NNS.)1070 63 y(Denote)g(b)o(y)g Fn(P)1303 48 y Fg(0)1328 63 y Fo(the)h(set)g(of)e(all)g(p)q(oin)o(ts)h Fn(p)1716 48 y Fg(0)1739 63 y Fm(2)e Fn(P)19 b Fo(suc)o(h)c(that)1020 110 y Fn(d)p Fo(\()p Fn(q)q(;)7 b(p)1118 95 y Fg(0)1129 110 y Fo(\))k Fn(>)h(r)1219 116 y Fj(2)1238 110 y Fo(.)18 b(W)m(e)13 b(observ)o(e)i(that)e(the)i(algorithm)c(correctly)1020 157 y(solv)o(es)j(the)g(\()p Fn(r)o(;)7 b(\017)p Fo(\)-Neigh)o(b)q(or) 12 b(problem)g(if)h(the)h(follo)o(wing)c(t)o(w)o(o)1020 205 y(prop)q(erties)15 b(hold:)1020 289 y Fi(P1)20 b Fo(If)g(there)h(exists)g Fn(p)1399 274 y Fg(\003)1438 289 y Fo(suc)o(h)g(that)f Fn(p)1655 274 y Fg(\003)1695 289 y Fm(2)i(B)q Fo(\()p Fn(q)q(;)7 b(r)1847 295 y Fj(1)1865 289 y Fo(\),)21 b(then)1103 336 y Fn(g)1123 342 y Fl(j)1140 336 y Fo(\()p Fn(p)1177 321 y Fg(\003)1196 336 y Fo(\))12 b(=)g Fn(g)1288 342 y Fl(j)1305 336 y Fo(\()p Fn(q)q Fo(\))i(for)g(some)f Fn(j)h Fo(=)e(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(l)q Fo(.)1020 422 y Fi(P2)20 b Fo(The)e(total)e(n)o(um)o(b)q(er)h(of)f(blo) q(c)o(ks)i(p)q(oin)o(ted)f(to)g(b)o(y)g Fn(q)h Fo(and)1103 469 y(con)o(taining)13 b(only)g(p)q(oin)o(ts)h(from)e Fn(P)1652 454 y Fg(0)1677 469 y Fo(is)h(less)i(than)f Fn(cl)q Fo(.)1070 554 y(Assume)h(that)h Fm(H)g Fo(is)f(a)g(\()p Fn(r)1482 560 y Fj(1)1501 554 y Fn(;)7 b(r)1539 560 y Fj(2)1556 554 y Fn(;)g(p)1596 560 y Fj(1)1614 554 y Fn(;)g(p)1654 560 y Fj(2)1672 554 y Fo(\)-sensitiv)o(e)16 b(family;)1020 610 y(de\014ne)i Fn(\032)f Fo(=)1236 590 y Fj(ln)6 b(1)p Fl(=p)1320 594 y Fd(1)p 1236 601 101 2 v 1236 625 a Fj(ln)g(1)p Fl(=p)1320 629 y Fd(2)1341 610 y Fo(.)27 b(The)18 b(correctness)i(of)c (the)i(LSH)f(algo-)1020 665 y(rithm)12 b(follo)o(ws)h(from)f(the)i (follo)o(wing)e(theorem.)1020 730 y Fi(Theorem)j(1)20 b Fk(Setting)d Fn(k)f Fo(=)e(log)1547 740 y Fj(1)p Fl(=p)1598 744 y Fd(2)1616 730 y Fo(\()p Fn(n=B)r Fo(\))j Fk(and)g Fn(l)f Fo(=)1901 696 y Fe(\000)1928 714 y Fl(n)p 1925 721 27 2 v 1925 744 a(B)1957 696 y Fe(\001)1976 705 y Fl(\032)1020 777 y Fk(guar)n(ante)n(es)f(that)g(pr)n(op)n(erties)f Fi(P1)h Fk(and)g Fi(P2)g Fk(hold)g(with)f(pr)n(ob-)1020 825 y(ability)g(at)h(le)n(ast)1294 808 y Fj(1)p 1294 815 17 2 v 1294 839 a(2)1325 825 y Fm(\000)1372 808 y Fj(1)p 1372 815 V 1372 839 a Fl(e)1405 825 y Fm(\025)d Fo(0)p Fn(:)p Fo(132)p Fk(.)1020 919 y Fi(Remark)j(1)21 b Fk(Note)15 b(that)f(by)h(r)n(ep)n(e)n(ating)g(the)g(LSH)g(algorithm) 1020 966 y Fn(O)q Fo(\(1)p Fn(=\016)r Fo(\))d Fk(times,)g(we)g(c)n(an)h (amplify)f(the)g(pr)n(ob)n(ability)g(of)g(suc)n(c)n(ess)1020 1013 y(in)j(at)g(le)n(ast)f(one)i(trial)d(to)i Fo(1)9 b Fm(\000)h Fn(\016)r Fk(,)k(for)h(any)g Fn(\016)f(>)e Fo(0)p Fk(.)1070 1107 y Fi(Pro)q(of:)19 b Fo(Let)d(prop)q(ert)o(y)f Fi(P1)f Fo(hold)h(with)f(probabilit)o(y)f Fn(P)1965 1113 y Fj(1)1983 1107 y Fo(,)1020 1155 y(and)18 b(prop)q(ert)o(y)h Fi(P2)f Fo(hold)g(with)f(probabilit)o(y)g Fn(P)1793 1161 y Fj(2)1811 1155 y Fo(.)32 b(W)m(e)17 b(will)1020 1202 y(sho)o(w)10 b(that)g(b)q(oth)g Fn(P)1326 1208 y Fj(1)1354 1202 y Fo(and)f Fn(P)1457 1208 y Fj(2)1485 1202 y Fo(are)i(large.)16 b(Assume)10 b(that)g(there)1020 1249 y(exists)16 b(a)g(p)q(oin)o(t)f Fn(p)1304 1234 y Fg(\003)1339 1249 y Fo(within)g(distance)h Fn(r)1652 1255 y Fj(1)1686 1249 y Fo(of)f Fn(q)q Fo(;)h(the)h(pro)q(of) e(is)1020 1297 y(quite)j(similar)d(otherwise.)31 b(Set)18 b Fn(k)h Fo(=)f(log)1702 1307 y Fj(1)p Fl(=p)1753 1311 y Fd(2)1770 1297 y Fo(\()p Fn(n=B)r Fo(\).)31 b(The)1020 1349 y(probabilit)o(y)12 b(that)i Fn(g)q Fo(\()p Fn(p)1380 1334 y Fg(0)1391 1349 y Fo(\))e(=)g Fn(g)q Fo(\()p Fn(q)q Fo(\))i(for)f Fn(p)f Fm(2)f Fn(P)j Fm(\000)9 b(B)q Fo(\()p Fn(q)q(;)e(r)1869 1355 y Fj(2)1887 1349 y Fo(\))14 b(is)f(at)1020 1397 y(most)d Fn(p)1140 1382 y Fl(k)1140 1407 y Fj(2)1172 1397 y Fo(=)1221 1380 y Fl(B)p 1221 1387 27 2 v 1224 1411 a(n)1252 1397 y Fo(.)18 b(Denote)11 b(the)h(set)h(of)e(all)f(p)q (oin)o(ts)h Fn(p)1796 1382 y Fg(0)1824 1397 y Fn(=)-26 b Fm(2)11 b(B)q Fo(\()p Fn(q)q(;)c(r)1960 1403 y Fj(2)1979 1397 y Fo(\))1020 1444 y(b)o(y)12 b Fn(P)1109 1429 y Fg(0)1120 1444 y Fo(.)17 b(The)12 b(exp)q(ected)i(n)o(um)o(b)q(er)e(of) f(blo)q(c)o(ks)h(allo)q(cated)f(for)h Fn(g)1978 1450 y Fl(j)1020 1491 y Fo(whic)o(h)i(con)o(tain)g Fk(exclusively)h Fo(p)q(oin)o(ts)f(from)f Fn(P)1747 1476 y Fg(0)1772 1491 y Fo(do)q(es)i(not)g(ex-)1020 1539 y(ceed)f(2.)k(The)13 b(exp)q(ected)i(n)o(um)o(b)q(er)d(of)h(suc)o(h)g(blo)q(c)o(ks)g(allo)q (cated)1020 1586 y(for)h(all)f Fn(g)1162 1592 y Fl(j)1193 1586 y Fo(is)h(at)g(most)f(2)p Fn(l)q Fo(.)19 b(Th)o(us,)14 b(b)o(y)g(the)g(Mark)o(o)o(v)g(inequal-)1020 1633 y(it)o(y)e([35)o(],)h (the)g(probabilit)o(y)f(that)h(this)g(n)o(um)o(b)q(er)f(exceeds)j(4)p Fn(l)f Fo(is)1020 1681 y(less)g(than)f(1)p Fn(=)p Fo(2.)k(If)c(w)o(e)h (c)o(ho)q(ose)g Fn(c)e Fo(=)f(4,)i(the)h(probabilit)o(y)e(that)1020 1728 y(the)i(prop)q(ert)o(y)h(P2)f(holds)g(is)f Fn(P)1501 1734 y Fj(2)1531 1728 y Fn(>)f Fo(1)p Fn(=)p Fo(2.)1070 1777 y(Consider)22 b(no)o(w)g(the)h(probabilit)o(y)d(of)i Fn(g)1722 1783 y Fl(j)1739 1777 y Fo(\()p Fn(p)1776 1761 y Fg(\003)1795 1777 y Fo(\))j(=)h Fn(g)1914 1783 y Fl(j)1931 1777 y Fo(\()p Fn(q)q Fo(\).)1020 1824 y(Clearly)m(,)12 b(it)i(is)g(b)q(ounded)g(from)e(b)q(elo)o(w)i(b)o(y)1094 1933 y Fn(p)1115 1916 y Fl(k)1115 1943 y Fj(1)1147 1933 y Fo(=)e Fn(p)1212 1905 y Fj(log)1255 1913 y Fd(1)p Ff(=p)1299 1919 y Fd(2)1323 1905 y Fl(n=B)1212 1944 y Fj(1)1401 1933 y Fo(=)g(\()p Fn(n=B)r Fo(\))1556 1912 y Fg(\000)1587 1896 y Fd(log)6 b(1)p Ff(=p)1673 1902 y Fd(1)p 1587 1905 104 2 v 1587 1922 a(log)g(1)p Ff(=p)1673 1928 y Fd(2)1709 1933 y Fo(=)12 b(\()p Fn(n=B)r Fo(\))1864 1916 y Fg(\000)p Fl(\032)1909 1933 y Fn(:)1020 2030 y Fo(By)j(setting)h Fn(l)e Fo(=)1296 1996 y Fe(\000)1323 2014 y Fl(n)p 1320 2021 27 2 v 1320 2044 a(B)1351 1996 y Fe(\001)1370 2005 y Fl(\032)1390 2030 y Fo(,)g(w)o(e)h(b)q(ound)h(from)d(ab)q(o)o(v)o(e)h (the)i(prob-)1020 2077 y(abilit)o(y)e(that)h Fn(g)1262 2083 y Fl(j)1280 2077 y Fo(\()p Fn(p)1317 2062 y Fg(\003)1336 2077 y Fo(\))f Fm(6)p Fo(=)g Fn(g)1432 2083 y Fl(j)1450 2077 y Fo(\()p Fn(q)q Fo(\))h(for)g(all)g Fn(j)h Fo(=)f(1)p Fn(;)7 b(:)g(:)g(:)t(;)g(l)16 b Fo(b)o(y)g(1)p Fn(=e)p Fo(.)1020 2125 y(Th)o(us)d(the)g(probabilit)o(y)d(that)j(one)f(suc)o(h) i Fn(g)1680 2131 y Fl(j)1709 2125 y Fo(exists)f(is)g(at)f(least)1020 2172 y Fn(P)1047 2178 y Fj(1)1077 2172 y Fm(\025)g Fo(1)d Fm(\000)g Fo(1)p Fn(=e)p Fo(.)1070 2221 y(Therefore,)16 b(the)g(probabilit)o(y)e(that)i(b)q(oth)f(prop)q(erties)i(P1)1020 2268 y(and)i(P2)f(hold)g(is)h(at)f(least)h(1)12 b Fm(\000)h Fo([\(1)f Fm(\000)h Fn(P)1685 2274 y Fj(1)1703 2268 y Fo(\))f(+)h(\(1)g Fm(\000)f Fn(P)1897 2274 y Fj(2)1916 2268 y Fo(\)])19 b(=)1020 2315 y Fn(P)1047 2321 y Fj(1)1074 2315 y Fo(+)10 b Fn(P)1143 2321 y Fj(2)1171 2315 y Fm(\000)f Fo(1)i Fm(\025)1293 2299 y Fj(1)p 1293 2306 17 2 v 1293 2330 a(2)1324 2315 y Fm(\000)1370 2299 y Fj(1)p 1370 2306 V 1370 2330 a Fl(e)1392 2315 y Fo(.)18 b(The)c(theorem)g(follo)o (ws.)159 b Fc(2)1070 2364 y Fo(In)13 b(the)g(follo)o(wing)d(w)o(e)j (consider)h(the)f(LSH)g(family)d(for)i(the)1020 2411 y(Hamming)7 b(metric)k(of)f(dimension)f Fn(d)1593 2396 y Fg(0)1615 2411 y Fo(as)i(sp)q(eci\014ed)h(in)f(F)m(act)g(2.)1020 2458 y(F)m(or)18 b(this)h(case,)h(w)o(e)f(sho)o(w)f(that)h Fn(\032)g Fm(\024)1674 2442 y Fj(1)p 1655 2449 56 2 v 1655 2473 a(1+)p Fl(\017)1734 2458 y Fo(assuming)e(that)1020 2521 y Fn(r)e(<)1117 2504 y Fl(d)1134 2492 y Fh(0)p 1104 2511 54 2 v 1104 2535 a Fj(ln)7 b Fl(n)1163 2521 y Fo(;)16 b(the)f(latter)h(assumption)d(can)j(b)q(e)g(easily)e(satis\014ed)1020 2568 y(b)o(y)19 b(increasing)g(the)g(dimensionalit)o(y)e(b)o(y)h (padding)g(a)h(su\016-)1020 2615 y(cien)o(tly)12 b(long)g(string)h(of)f (0s)g(at)h(the)g(end)g(of)f(eac)o(h)h(p)q(oin)o(t's)f(rep-)1020 2663 y(resen)o(tation.)p eop %%Page: 6 6 6 5 bop -30 -33 a Fi(F)l(act)16 b(3)21 b Fk(L)n(et)11 b Fn(r)h(<)280 -49 y Fl(d)297 -62 y Fh(0)p 268 -42 54 2 v 268 -18 a Fj(ln)6 b Fl(n)327 -33 y Fk(.)17 b(If)11 b Fn(p)418 -27 y Fj(1)448 -33 y Fo(=)h(1)q Fm(\000)558 -49 y Fl(r)p 552 -42 29 2 v 552 -18 a(d)569 -27 y Fh(0)597 -33 y Fk(and)g Fn(p)695 -27 y Fj(2)725 -33 y Fo(=)g(1)q Fm(\000)829 -53 y Fl(r)q Fj(\(1+)p Fl(\017)p Fj(\))p 829 -42 99 2 v 864 -18 a Fl(d)881 -27 y Fh(0)932 -33 y Fk(,)-30 28 y(then)j Fn(\032)d Fo(=)144 7 y Fj(ln)6 b(1)p Fl(=p)228 11 y Fd(1)p 144 18 101 2 v 144 42 a Fj(ln)g(1)p Fl(=p)228 46 y Fd(2)261 28 y Fm(\024)329 11 y Fj(1)p 310 18 56 2 v 310 42 a(1+)p Fl(\017)371 28 y Fk(.)20 121 y Fi(Pro)q(of:)17 b Fo(Observ)o(e)f(that)-3 236 y Fn(\032)c Fo(=)78 208 y(ln)7 b(1)p Fn(=p)183 214 y Fj(1)p 78 227 123 2 v 78 265 a Fo(ln)g(1)p Fn(=p)183 271 y Fj(2)217 236 y Fo(=)307 198 y(ln)397 182 y Fj(1)p 354 189 105 2 v 354 213 a(1)p Fg(\000)p Fl(r)q(=d)447 204 y Fh(0)p 266 227 238 2 v 266 267 a Fo(ln)397 250 y Fj(1)p 313 257 187 2 v 313 281 a(1)p Fg(\000)p Fj(\(1+)p Fl(\017)p Fj(\))p Fl(r)q(=d)488 273 y Fh(0)520 236 y Fo(=)629 208 y(ln\(1)i Fm(\000)h Fn(r)q(=d)815 193 y Fg(0)825 208 y Fo(\))p 569 227 333 2 v 569 265 a(ln\(1)f Fm(\000)g Fo(\(1)g(+)h Fn(\017)p Fo(\))p Fn(r)q(=d)875 253 y Fg(0)886 265 y Fo(\))907 236 y Fn(:)-30 360 y Fo(Multiplying)h(b)q(oth)j(the)g (n)o(umerator)e(and)i(the)g(denominator)-30 412 y(b)o(y)33 396 y Fl(d)50 384 y Fh(0)p 33 403 29 2 v 39 427 a Fl(r)66 412 y Fo(,)f(w)o(e)i(obtain:)137 540 y Fn(\032)42 b Fo(=)344 493 y Fl(d)361 481 y Fh(0)p 344 500 V 350 524 a Fl(r)384 510 y Fo(ln\(1)9 b Fm(\000)g Fn(r)q(=d)569 495 y Fg(0)580 510 y Fo(\))p 278 531 379 2 v 283 556 a Fl(d)300 548 y Fh(0)p 283 563 29 2 v 289 587 a Fl(r)324 572 y Fo(ln)o(\(1)h Fm(\000)f Fo(\(1)g(+)h Fn(\017)p Fo(\))p Fn(r)q(=d)630 560 y Fg(0)641 572 y Fo(\))200 670 y(=)339 642 y(ln)o(\(1)f Fm(\000)h Fn(r)q(=d)524 627 y Fg(0)535 642 y Fo(\))551 627 y Fl(d)568 615 y Fh(0)579 627 y Fl(=r)p 278 661 397 2 v 278 700 a Fo(ln\(1)f Fm(\000)h Fo(\(1)f(+)g Fn(\017)p Fo(\))p Fn(r)q(=d)584 688 y Fg(0)595 700 y Fo(\))611 688 y Fl(d)628 679 y Fh(0)640 688 y Fl(=r)692 670 y Fo(=)740 642 y Fn(U)p 740 661 33 2 v 742 699 a(L)-30 781 y Fo(In)14 b(order)h(to)f(upp)q(er)h(b)q(ound)f Fn(\032)p Fo(,)g(w)o(e)h(need)g (to)f(b)q(ound)g Fn(U)19 b Fo(from)-30 829 y(b)q(elo)o(w)14 b(and)g Fn(L)g Fo(from)e(ab)q(o)o(v)o(e;)i(note)g(that)g(b)q(oth)g Fn(U)19 b Fo(and)14 b Fn(L)h Fo(are)-30 876 y(negativ)o(e.)30 b(T)m(o)17 b(this)h(end)g(w)o(e)g(use)h(the)f(follo)o(wing)d(inequali-) -30 923 y(ties)f([35)o(]:)184 1018 y(\(1)9 b Fm(\000)h Fo(\(1)f(+)g Fn(\017)p Fo(\))p Fn(r)q(=d)455 1001 y Fg(0)466 1018 y Fo(\))482 1001 y Fl(d)499 988 y Fh(0)511 1001 y Fl(=r)558 1018 y Fn(<)j(e)621 1001 y Fg(\000)p Fj(\(1+)p Fl(\017)p Fj(\))-30 1103 y Fo(and)163 1123 y Fe(\020)188 1169 y Fo(1)d Fm(\000)271 1141 y Fn(r)p 264 1160 34 2 v 264 1198 a(d)286 1186 y Fg(0)302 1123 y Fe(\021)327 1132 y Fl(d)344 1119 y Fh(0)356 1132 y Fl(=r)403 1169 y Fn(>)i(e)465 1152 y Fg(\000)p Fj(1)517 1111 y Fe(\022)548 1169 y Fo(1)e Fm(\000)651 1141 y Fo(1)p 624 1160 74 2 v 624 1198 a Fn(d)646 1186 y Fg(0)658 1198 y Fn(=r)703 1111 y Fe(\023)741 1169 y Fn(:)-30 1265 y Fo(Therefore,)150 1360 y Fn(U)p 150 1379 33 2 v 152 1417 a(L)230 1388 y(<)308 1351 y Fo(ln\()p Fn(e)378 1335 y Fg(\000)p Fj(1)423 1351 y Fo(\(1)g Fm(\000)538 1334 y Fj(1)p 516 1341 62 2 v 516 1365 a Fl(d)533 1357 y Fh(0)544 1365 y Fl(=r)583 1351 y Fo(\)\))p 308 1379 307 2 v 376 1418 a(ln)e Fn(e)437 1406 y Fg(\000)p Fj(\(1+)p Fl(\017)p Fj(\))230 1507 y Fo(=)308 1469 y Fm(\000)p Fo(1)j(+)f(ln\(1)g Fm(\000)562 1453 y Fj(1)p 539 1460 62 2 v 539 1483 a Fl(d)556 1475 y Fh(0)568 1483 y Fl(=r)606 1469 y Fo(\))p 308 1497 314 2 v 389 1535 a Fm(\000)p Fo(\(1)g(+)h Fn(\017)p Fo(\))230 1635 y(=)42 b(1)p Fn(=)p Fo(\(1)8 b(+)i Fn(\017)p Fo(\))f Fm(\000)521 1597 y Fo(ln\(1)g Fm(\000)671 1581 y Fj(1)p 649 1588 62 2 v 649 1612 a Fl(d)666 1603 y Fh(0)677 1612 y Fl(=r)715 1597 y Fo(\))p 521 1626 211 2 v 582 1664 a(1)g(+)h Fn(\017)230 1715 y(<)42 b Fo(1)p Fn(=)p Fo(\(1)8 b(+)i Fn(\017)p Fo(\))f Fm(\000)g Fo(ln\(1)g Fm(\000)h Fo(1)p Fn(=)d Fo(ln)f Fn(n)p Fo(\))-30 1801 y(where)18 b(the)g(last)f(step)i(uses)f(the)g(assumptions)e(that)i Fn(\017)f(>)g Fo(0)-30 1853 y(and)d Fn(r)e(<)144 1837 y Fl(d)161 1825 y Fh(0)p 131 1844 54 2 v 131 1868 a Fj(ln)6 b Fl(n)190 1853 y Fo(.)18 b(W)m(e)13 b(conclude)i(that)22 1949 y Fn(n)47 1932 y Fl(\032)108 1949 y Fn(<)42 b(n)207 1932 y Fj(1)p Fl(=)p Fj(\(1+)p Fl(\017)p Fj(\))324 1949 y Fn(n)349 1932 y Fg(\000)5 b Fj(ln)q(\(1)p Fg(\000)p Fj(1)p Fl(=)g Fj(ln)h Fl(n)p Fj(\))108 2009 y Fo(=)42 b Fn(n)207 1992 y Fj(1)p Fl(=)p Fj(\(1+)p Fl(\017)p Fj(\))324 2009 y Fo(\(1)9 b Fm(\000)h Fo(1)p Fn(=)d Fo(ln)e Fn(n)p Fo(\))542 1992 y Fg(\000)h Fj(ln)h Fl(n)642 2009 y Fo(=)12 b Fn(O)q Fo(\()p Fn(n)760 1992 y Fj(1)p Fl(=)p Fj(\(1+)p Fl(\017)p Fj(\))877 2009 y Fo(\))914 2095 y Fc(2)20 2142 y Fo(W)m(e)18 b(no)o(w)h(return)h(to)e(the)i Fn(\017)p Fo(-NNS)e(problem.)32 b(First,)20 b(w)o(e)-30 2190 y(observ)o(e)c(that) e(w)o(e)h(could)g(reduce)h(it)e(to)h(the)g(\()p Fn(r)o(;)7 b(\017)p Fo(\)-Neigh)o(b)q(or)-30 2237 y(problem)19 b(b)o(y)i(building) e(sev)o(eral)i(data)g(structures)i(for)d(the)-30 2284 y(latter)14 b(problem)e(with)i(di\013eren)o(t)g(v)n(alues)g(of)f Fn(r)q Fo(.)k(More)d(sp)q(ecif-)-30 2332 y(ically)m(,)26 b(w)o(e)g(could)f(explore)h Fn(r)h Fo(equal)e(to)h Fn(r)702 2338 y Fj(0)720 2332 y Fo(,)i Fn(r)779 2338 y Fj(0)797 2332 y Fo(\(1)17 b(+)g Fn(\017)p Fo(\),)-30 2379 y Fn(r)-11 2385 y Fj(0)7 2379 y Fo(\(1)8 b(+)g Fn(\017)p Fo(\))125 2364 y Fj(2)143 2379 y Fn(;)f(:)g(:)g(:)e(;)i(r)255 2385 y Fl(max)323 2379 y Fo(,)12 b(where)j Fn(r)486 2385 y Fj(0)517 2379 y Fo(and)e Fn(r)616 2385 y Fl(max)697 2379 y Fo(are)h(the)f(small-)-30 2426 y(est)e(and)g(the)g(largest)f(p)q (ossible)h(distance)g(b)q(et)o(w)o(een)h(the)f(query)-30 2473 y(and)k(the)g(data)f(p)q(oin)o(t,)g(resp)q(ectiv)o(ely)m(.)21 b(W)m(e)15 b(remark)f(that)g(the)-30 2521 y(n)o(um)o(b)q(er)e(of)g (di\013eren)o(t)h(radii)f(could)g(b)q(e)i(further)f(reduced)h([24)o(]) -30 2568 y(at)k(the)g(cost)h(of)e(increasing)h(running)g(time)f(and)g (space)i(re-)-30 2615 y(quiremen)o(t.)29 b(On)19 b(the)f(other)h(hand,) f(w)o(e)g(observ)o(ed)h(that)f(in)-30 2663 y(practice)h(c)o(ho)q(osing) f(only)g Fk(one)h Fo(v)n(alue)e(of)h Fn(r)h Fo(is)f(su\016cien)o(t)h (to)1020 -33 y(pro)q(duce)k(answ)o(ers)g(of)e(go)q(o)q(d)h(qualit)o(y)m (.)40 b(This)22 b(can)g(b)q(e)h(ex-)1020 14 y(plained)10 b(as)h(in)g([10)o(])f(where)i(it)e(w)o(as)h(observ)o(ed)h(that)f(the)g (distri-)1020 61 y(bution)f(of)f(distances)j(b)q(et)o(w)o(een)f(a)f (query)h(p)q(oin)o(t)e(and)h(the)h(data)1020 109 y(set)h(in)e(most)g (cases)i(do)q(es)f(not)g(dep)q(end)h(on)f(the)g(sp)q(eci\014c)h(query) 1020 156 y(p)q(oin)o(t,)j(but)h(on)g(the)g(in)o(trinsic)g(prop)q (erties)h(of)f(the)g(data)g(set.)1020 203 y(Under)i(the)f(assumption)f (of)g(distribution)h(in)o(v)n(ariance,)f(the)1020 251 y(same)c(parameter)h Fn(r)i Fo(is)e(lik)o(ely)f(to)h(w)o(ork)g(for)g(a) g(v)n(ast)g(ma)r(jorit)o(y)1020 298 y(of)k(queries.)28 b(Therefore)19 b(in)e(the)g(exp)q(erimen)o(tal)g(section)h(w)o(e)1020 345 y(adopt)13 b(a)g(\014xed)g(c)o(hoice)h(of)e Fn(r)i Fo(and)f(therefore)h(also)f(of)f Fn(k)i Fo(and)f Fn(l)q Fo(.)1020 455 y Fv(4)56 b(Exp)r(erime)o(n)n(ts)1020 529 y Fo(In)11 b(this)g(section)h(w)o(e)f(rep)q(ort)h(the)f(results)h(of)f (our)g(exp)q(erimen)o(ts)1020 576 y(with)h(lo)q(calit)o(y-sensitiv)o(e) f(hashing)g(metho)q(d.)17 b(W)m(e)11 b(p)q(erformed)1020 623 y(exp)q(erimen)o(ts)16 b(on)g(t)o(w)o(o)f(data)g(sets.)25 b(The)16 b(\014rst)h(one)f(con)o(tains)1020 671 y(up)d(to)g(20,000)f (histograms)f(of)i(color)g(images)f(from)f(COREL)1020 718 y(Dra)o(w)18 b(library)m(,)f(where)i(eac)o(h)g(histogram)d(w)o(as)i (represen)o(ted)1020 765 y(as)13 b(a)g(p)q(oin)o(t)g(in)g Fn(d)p Fo(-dimensional)d(space,)k(for)f Fn(d)g Fo(up)g(to)g(64.)18 b(The)1020 813 y(second)f(one)f(con)o(tains)g(around)g(270,000)e(p)q (oin)o(ts)i(of)f(dimen-)1020 860 y(sion)i(60)g(represen)o(ting)i (texture)f(information)d(of)h(blo)q(c)o(ks)i(of)1020 907 y(large)e(large)g(aerial)f(photographs.)25 b(W)m(e)16 b(describ)q(e)i(the)e(data)1020 955 y(sets)f(in)f(more)e(detail)i (later)g(in)f(the)i(section.)1070 1004 y(W)m(e)i(decided)j(not)d(to)h (use)h(randomly-c)o(hosen)e(syn)o(thetic)1020 1051 y(data)g(in)g(our)g (exp)q(erimen)o(ts.)29 b(Though)17 b(suc)o(h)h(data)f(is)g(often)1020 1098 y(used)j(to)e(measure)h(the)g(p)q(erformance)g(of)f Fk(exact)h Fo(similarit)o(y)1020 1146 y(searc)o(h)f(algorithms,)c(w)o (e)j(found)g(it)f(unsuitable)h(for)f(ev)n(alua-)1020 1193 y(tion)h(of)h Fk(appr)n(oximate)g Fo(algorithms)e(for)h(the)i (high)e(data)h(di-)1020 1240 y(mensionalit)o(y)m(.)h(The)d(main)e (reason)h(is)h(as)f(follo)o(ws.)21 b(Assume)1020 1288 y(a)d(data)g(set)h(consisting)f(of)f(p)q(oin)o(ts)h(c)o(hosen)h(indep)q (enden)o(tly)1020 1335 y(at)f(random)f(from)g(the)i(same)f (distribution.)31 b(Most)19 b(distri-)1020 1382 y(butions)h(\(notably)e (uniform\))g(used)j(in)e(the)h(literature)g(as-)1020 1430 y(sume)13 b(that)h(all)f(co)q(ordinates)i(of)e(eac)o(h)i(p)q(oin)o (t)e(are)i(c)o(hosen)g(in-)1020 1477 y(dep)q(enden)o(tly)m(.)j(In)13 b(suc)o(h)h(a)e(case,)i(for)e(an)o(y)g(pair)h(of)f(p)q(oin)o(ts)h Fn(p;)7 b(q)1020 1524 y Fo(the)12 b(distance)g Fn(d)p Fo(\()p Fn(p;)7 b(q)q Fo(\))k(is)g(sharply)g(concen)o(trated)i(around)e (the)1020 1572 y(mean;)17 b(for)f(example,)g(for)g(the)i(uniform)d (distribution)h(o)o(v)o(er)1020 1619 y(the)21 b(unit)e(cub)q(e,)j(the)f (exp)q(ected)h(distance)f(is)f Fn(O)q Fo(\()p Fn(d)p Fo(\),)g(while)1020 1672 y(the)f(standard)g(deviation)e(is)h(only)g Fn(O)q Fo(\()1650 1637 y Fm(p)p 1684 1637 22 2 v 1684 1672 a Fn(d)p Fo(\).)31 b(Th)o(us)19 b(almost)1020 1719 y(all)14 b(pairs)h(are)h(appro)o(ximately)c(within)j(the)g(same)g (distance,)1020 1767 y(so)i(the)g(notion)e(of)h(appro)o(ximate)f (nearest)i(neigh)o(b)q(or)g(is)f(not)1020 1814 y(meaningful)g(|)i (almost)f Fk(every)i(p)n(oint)g Fo(is)f(an)h(appro)o(ximate)1020 1861 y(nearest)c(neigh)o(b)q(or.)1070 1910 y Fi(Implemen)o(tati)o(on)o (.)31 b Fo(W)m(e)18 b(implemen)o(t)e(the)j(LSH)g(algo-)1020 1958 y(rithm)f(as)h(sp)q(eci\014ed)i(in)d(Section)i(3.)33 b(The)20 b(LSH)f(functions)1020 2005 y(can)c(b)q(e)h(computed)e(as)h (describ)q(ed)i(in)e(Section)g(3.1.)20 b(Denote)1020 2052 y(the)13 b(resulting)f(v)o(ector)g(of)g(co)q(ordinates)g(b)o(y)g (\()p Fn(v)1737 2058 y Fj(1)1756 2052 y Fn(;)7 b(:)g(:)g(:)e(;)i(v)1869 2058 y Fl(k)1889 2052 y Fo(\).)17 b(F)m(or)1020 2100 y(the)c(second)h(lev)o(el)f(mapping)d(w)o(e)j(use)g(functions)g(of)f (the)i(form)1093 2192 y Fn(h)p Fo(\()p Fn(v)1153 2198 y Fj(1)1172 2192 y Fn(;)7 b(:)g(:)g(:)e(;)i(v)1285 2198 y Fl(k)1304 2192 y Fo(\))12 b(=)g Fn(a)1398 2198 y Fj(1)1426 2192 y Fm(\001)d Fn(v)1467 2198 y Fj(1)1495 2192 y Fo(+)g Fm(\001)e(\001)g(\001)h Fo(+)h Fn(a)1657 2198 y Fl(d)1686 2192 y Fm(\001)f Fn(v)1726 2198 y Fl(k)1761 2192 y Fo(mo)q(d)26 b Fn(M)r(;)1020 2284 y Fo(where)18 b Fn(M)j Fo(is)c(the)g(size)h(of)e (the)i(hash)f(table)f(and)h Fn(a)1842 2290 y Fj(1)1860 2284 y Fn(;)7 b(:)g(:)g(:)e(;)i(a)1975 2290 y Fl(k)1020 2332 y Fo(are)12 b(random)e(n)o(um)o(b)q(ers)h(from)f(in)o(terv)n(al)h ([0)c Fn(:)g(:)g(:)e(M)k Fm(\000)c Fo(1].)17 b(These)1020 2379 y(functions)j(can)g(b)q(e)g(computed)g(using)f(only)h(2)p Fn(k)13 b Fm(\000)h Fo(1)19 b(op)q(er-)1020 2426 y(ations,)e(and)g(are) g(su\016cien)o(tly)g(random)f(for)h(our)g(purp)q(oses,)1020 2473 y(i.e.,)11 b(giv)o(e)g(lo)o(w)f(probabilit)o(y)g(of)h(collision.) 16 b(Eac)o(h)c(second)g(lev)o(el)1020 2521 y(buc)o(k)o(et)g(is)f(then)h (directly)g(mapp)q(ed)f(to)g(a)g(disk)g(blo)q(c)o(k.)17 b(W)m(e)11 b(as-)1020 2568 y(sumed)g(that)g(eac)o(h)h(blo)q(c)o(k)f(is) g(8KB)h(of)f(data.)17 b(As)11 b(eac)o(h)h(co)q(ordi-)1020 2615 y(nate)i(in)f(our)h(data)f(sets)i(can)f(b)q(e)h(represen)o(ted)h (using)e(1)f(b)o(yte,)1020 2663 y(w)o(e)19 b(can)g(store)g(up)g(to)g (8192)p Fn(=d)e(d)p Fo(-dimensional)f(p)q(oin)o(ts)i(p)q(er)p eop %%Page: 7 7 7 6 bop -30 -33 a Fo(blo)q(c)o(k.)20 b(Therefore,)c(w)o(e)f(assume)f (the)i(buc)o(k)o(et)f(size)h Fn(B)f Fo(=)f(100)-30 14 y(for)c Fn(d)h Fo(=)h(64)d(or)i Fn(d)g Fo(=)g(60,)f Fn(B)k Fo(=)e(300)e(for)f Fn(d)j Fo(=)f(27)f(and)g Fn(B)k Fo(=)e(1000)-30 61 y(for)i Fn(d)d Fo(=)g(8.)20 109 y(F)m(or)20 b(the)h(SR-tree,)i(w)o (e)e(used)h(the)f(implemen)o(tatio)o(n)d(b)o(y)-30 156 y(Kata)o(y)o(ama,)9 b(a)o(v)n(ailable)f(from)h(his)i(w)o(eb)g(page)g ([28)o(].)16 b(As)c(ab)q(o)o(v)o(e,)-30 203 y(w)o(e)18 b(allo)o(w)d(it)i(to)g(store)h(ab)q(out)g(8192)e(co)q(ordinates)i(p)q (er)g(disk)-30 251 y(blo)q(c)o(k.)20 298 y Fi(P)o(erformance)e (measures.)23 b Fo(The)16 b(goal)f(of)g(our)h(exp)q(er-)-30 345 y(imen)o(ts)22 b(w)o(as)g(to)h(estimate)f(t)o(w)o(o)g(p)q (erformance)h(measures:)-30 393 y(sp)q(eed)18 b(\(for)f(b)q(oth)g (SR-tree)g(and)g(LSH\))g(and)f(accuracy)i(\(for)-30 440 y(LSH\).)f(The)g(sp)q(eed)h(is)f(measured)g(b)o(y)f(the)i(n)o(um)o(b)q (er)e(of)g(disk)-30 487 y(blo)q(c)o(ks)e(accessed)j(in)d(order)h(to)f (answ)o(er)h(a)f(query)m(.)20 b(W)m(e)14 b(coun)o(t)-30 535 y(all)k(disk)h(accesses,)k(th)o(us)c(ignoring)f(the)i(issue)g(of)f (cac)o(hing.)-30 582 y(Observ)o(e)h(that)f(in)f(case)h(of)f(LSH)h(this) f(n)o(um)o(b)q(er)g(is)g(easy)h(to)-30 629 y(predict)d(as)g(it)f(is)g (clearly)g(equal)g(to)g(the)h(n)o(um)o(b)q(er)f(of)g(indices)-30 677 y(used.)33 b(As)18 b(the)h(n)o(um)o(b)q(er)f(of)g(indices)h(also)f (determines)g(the)-30 724 y(storage)i(o)o(v)o(erhead,)h(it)f(is)f(a)h (natural)f(parameter)g(to)h(opti-)-30 771 y(mize.)20 819 y(The)e(error)g(of)f(LSH)h(is)f(measured)g(as)h(follo)o(ws.)27 b(F)m(ollo)o(w-)-30 866 y(ing)9 b([2)o(])h(w)o(e)g(de\014ne)h(\(for)f (the)g(Appro)o(ximate)e(1-NNS)i(problem\))-30 913 y(the)k Fk(e\013e)n(ctive)i(err)n(or)c Fo(as)217 1018 y Fn(E)h Fo(=)328 990 y(1)p 310 1008 56 2 v 310 1046 a Fm(j)p Fn(Q)p Fm(j)445 979 y Fe(X)392 1068 y Fo(query)h Fl(q)q Fg(2)p Fl(Q)584 990 y Fn(d)606 996 y Fl(LS)r(H)p 584 1008 98 2 v 612 1046 a Fn(d)634 1034 y Fg(\003)687 1018 y Fn(;)-30 1148 y Fo(where)f Fn(d)110 1154 y Fl(LS)r(H)198 1148 y Fo(denotes)g(the)g(distance)g(from)d(a)i(query)g(p)q(oin)o(t)g Fn(q)-30 1196 y Fo(to)i(a)g(p)q(oin)o(t)g(found)f(b)o(y)h(LSH,)g Fn(d)467 1181 y Fg(\003)500 1196 y Fo(is)g(the)h(distance)g(from)d Fn(q)j Fo(to)-30 1243 y(the)i(closest)g(p)q(oin)o(t,)g(and)f(the)h(sum) e(is)h(tak)o(en)h(of)f(all)f(queries)-30 1290 y(for)e(whic)o(h)h(a)f (nonempt)o(y)g(index)g(w)o(as)h(found.)j(W)m(e)c(also)g(mea-)-30 1338 y(sure)j(the)g(\(small\))e(fraction)h(of)g(queries)h(for)f(whic)o (h)g(no)g(non-)-30 1385 y(empt)o(y)j(buc)o(k)o(et)i(w)o(as)e(found;)j (w)o(e)e(call)f(this)h(quan)o(tit)o(y)f Fk(miss)-30 1432 y(r)n(atio)p Fo(.)23 b(F)m(or)16 b(the)h(Appro)o(ximate)d Fn(K)s Fo(-NNS)i(w)o(e)g(measure)g(sep-)-30 1480 y(arately)f(the)i (distance)f(ratios)f(b)q(et)o(w)o(een)i(the)f(closest)h(p)q(oin)o(ts) -30 1527 y(found)e(to)g(the)h(nearest)h(neigh)o(b)q(or,)e(the)h(2nd)f (closest)i(one)e(to)-30 1574 y(the)h(2nd)g(nearest)h(neigh)o(b)q(or)f (and)f(so)h(on;)g(then)g(w)o(e)g(a)o(v)o(erage)-30 1622 y(the)e(ratios.)j(The)c(miss)f(ratio)h(is)f(de\014ned)i(to)f(b)q(e)h (the)f(fraction)-30 1669 y(of)g(cases)j(when)e Fk(less)g(than)i Fn(K)h Fo(p)q(oin)o(ts)d(w)o(ere)g(found.)20 1716 y Fi(Data)20 b(Sets.)31 b Fo(Our)19 b(\014rst)g(data)f(set)h(consists)g(of)f(20,000) -30 1764 y(histograms)j(of)g(color)h(th)o(um)o(bnail-sized)f(images)f (of)i(v)n(ari-)-30 1811 y(ous)g(con)o(ten)o(ts)h(tak)o(en)g(from)d(the) j(COREL)f(library)m(.)42 b(The)-30 1858 y(histograms)12 b(w)o(ere)j(extracted)g(after)f(transforming)e(the)j(pix-)-30 1906 y(els)h(of)f(the)i(images)d(to)i(the)g(3-dimensional)e(CIE-Lab)h (color)-30 1953 y(space)i([44)o(];)g(the)g(prop)q(ert)o(y)g(of)f(this)g (space)i(is)e(that)g(the)h(dis-)-30 2000 y(tance)g(b)q(et)o(w)o(een)g (eac)o(h)g(pair)e(of)h(p)q(oin)o(ts)g(corresp)q(onds)h(to)f(the)-30 2048 y(p)q(erceptual)21 b(dissimilarit)o(y)d(b)q(et)o(w)o(een)j(the)g (colors)f(that)g(the)-30 2095 y(t)o(w)o(o)d(p)q(oin)o(ts)h(represen)o (t.)31 b(Then)18 b(w)o(e)g(partitioned)f(the)h(color)-30 2142 y(space)13 b(in)o(to)e(a)g(grid)h(of)f(smaller)f(cub)q(es,)j(and)f (giv)o(en)f(an)g(image,)-30 2190 y(w)o(e)g(create)h(the)g(color)e (histogram)f(of)h(the)i(image)d(b)o(y)h(coun)o(ting)-30 2237 y(ho)o(w)15 b(man)o(y)e(pixels)h(fall)g(in)o(to)g(eac)o(h)i(of)e (these)i(cub)q(es.)23 b(By)16 b(di-)-30 2284 y(viding)10 b(eac)o(h)i(axis)e(in)o(to)h Fn(u)f Fo(in)o(terv)n(als)h(w)o(e)g (obtain)g(a)g(total)f(of)h Fn(u)927 2269 y Fj(3)-30 2332 y Fo(cub)q(es.)26 b(F)m(or)15 b(most)g(exp)q(erimen)o(ts,)h(w)o(e)g (assumed)g Fn(u)e Fo(=)i(4)f(ob-)-30 2379 y(taining)f(a)g (64-dimensional)e(space.)22 b(Eac)o(h)15 b(histogram)e(cub)q(e)-30 2426 y(\(i.e.,)i(color\))g(then)h(corresp)q(onds)i(to)e(a)f(dimension)f (of)h(space)-30 2473 y(represen)o(ting)i(the)f(images.)21 b(Finally)m(,)14 b(quan)o(tization)g(is)h(p)q(er-)-30 2521 y(formed)g(in)g(order)i(to)f(\014t)g(eac)o(h)g(co)q(ordinate)h(in) e(1)h(b)o(yte.)24 b(F)m(or)-30 2568 y(eac)o(h)17 b(p)q(oin)o(t)f (represen)o(ting)i(an)f(image)d(eac)o(h)j(co)q(ordinate)g(ef-)-30 2615 y(fectiv)o(ely)d(coun)o(ts)i(the)f(n)o(um)o(b)q(er)f(of)g(the)h (image's)e(pixels)h(of)g(a)-30 2663 y(sp)q(eci\014c)g(color.)j(All)11 b(co)q(ordinates)i(are)g(clearly)f(non-negativ)o(e)1057 -33 y 14208860 11651264 3289088 12564316 36772003 40061091 startTexFig 1057 -33 a %%BeginDocument: distribution.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 392 30 6111 5013 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 898 4613 mt 898 388 L 898 388 mt 898 388 L 1568 4613 mt 1568 388 L 1568 388 mt 1568 388 L 2237 4613 mt 2237 388 L 2237 388 mt 2237 388 L 2907 4613 mt 2907 388 L 2907 388 mt 2907 388 L 3576 4613 mt 3576 388 L 3576 388 mt 3576 388 L 4246 4613 mt 4246 388 L 4246 388 mt 4246 388 L 4915 4613 mt 4915 388 L 4915 388 mt 4915 388 L 5585 4613 mt 5585 388 L 5585 388 mt 5585 388 L 6254 4613 mt 6254 388 L 6254 388 mt 6254 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3845 mt 6254 3845 L 6254 3845 mt 6254 3845 L 898 3077 mt 6254 3077 L 6254 3077 mt 6254 3077 L 898 2308 mt 6254 2308 L 6254 2308 mt 6254 2308 L 898 1540 mt 6254 1540 L 6254 1540 mt 6254 1540 L 898 772 mt 6254 772 L 6254 772 mt 6254 772 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 898 4613 mt 898 4559 L 898 388 mt 898 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 848 4815 mt (0) s 1568 4613 mt 1568 4559 L 1568 388 mt 1568 442 L 1368 4815 mt (1000) s 2237 4613 mt 2237 4559 L 2237 388 mt 2237 442 L 2037 4815 mt (2000) s 2907 4613 mt 2907 4559 L 2907 388 mt 2907 442 L 2707 4815 mt (3000) s 3576 4613 mt 3576 4559 L 3576 388 mt 3576 442 L 3376 4815 mt (4000) s 4246 4613 mt 4246 4559 L 4246 388 mt 4246 442 L 4046 4815 mt (5000) s 4915 4613 mt 4915 4559 L 4915 388 mt 4915 442 L 4715 4815 mt (6000) s 5585 4613 mt 5585 4559 L 5585 388 mt 5585 442 L 5385 4815 mt (7000) s 6254 4613 mt 6254 4559 L 6254 388 mt 6254 442 L 6054 4815 mt (8000) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 3845 mt 952 3845 L 6254 3845 mt 6200 3845 L 613 3911 mt (0.2) s 898 3077 mt 952 3077 L 6254 3077 mt 6200 3077 L 613 3143 mt (0.4) s 898 2308 mt 952 2308 L 6254 2308 mt 6200 2308 L 613 2374 mt (0.6) s 898 1540 mt 952 1540 L 6254 1540 mt 6200 1540 L 613 1606 mt (0.8) s 898 772 mt 952 772 L 6254 772 mt 6200 772 L 763 838 mt (1) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np 178 326 179 322 178 77 178 216 179 529 178 -4 178 246 179 190 178 399 178 17 178 229 179 70 178 -53 178 -200 179 -149 178 33 178 -389 179 -216 178 -380 178 -439 179 -262 178 -413 178 -192 178 -350 179 -189 178 -240 178 -229 179 -57 178 -99 1018 4596 30 MP stroke gr 2842 5001 mt (Interpoint distance) s 559 3373 mt -90 rotate (Normalized frequency) s 90 rotate 2409 245 mt (Point set distance distribution) s end eplot epage end showpage %%EndDocument endTexFig 1319 752 a Fo(\(a\))i(Color)f(histograms)1057 767 y 14208860 11935440 2960179 12235407 36508876 40455782 startTexFig 1057 767 a %%BeginDocument: texture-profile.eps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 330 -38 6124 5142 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5357 0 0 -4226 898 4615 4 MP PP -5357 0 0 4226 5357 0 0 -4226 898 4615 5 MP stroke 4 w DO 0 sg 898 4615 mt 898 389 L 898 389 mt 898 389 L 1663 4615 mt 1663 389 L 1663 389 mt 1663 389 L 2429 4615 mt 2429 389 L 2429 389 mt 2429 389 L 3194 4615 mt 3194 389 L 3194 389 mt 3194 389 L 3959 4615 mt 3959 389 L 3959 389 mt 3959 389 L 4724 4615 mt 4724 389 L 4724 389 mt 4724 389 L 5490 4615 mt 5490 389 L 5490 389 mt 5490 389 L 6255 4615 mt 6255 389 L 6255 389 mt 6255 389 L 898 4615 mt 6255 4615 L 6255 4615 mt 6255 4615 L 898 3847 mt 6255 3847 L 6255 3847 mt 6255 3847 L 898 3078 mt 6255 3078 L 6255 3078 mt 6255 3078 L 898 2310 mt 6255 2310 L 6255 2310 mt 6255 2310 L 898 1542 mt 6255 1542 L 6255 1542 mt 6255 1542 L 898 773 mt 6255 773 L 6255 773 mt 6255 773 L SO 6 w 898 4615 mt 6255 4615 L 898 389 mt 6255 389 L 6255 4615 mt 6255 389 L 898 4615 mt 898 389 L 898 4615 mt 6255 4615 L 898 4615 mt 898 389 L 898 4615 mt 898 4561 L 898 389 mt 898 443 L /Helvetica /ISOLatin1Encoding 180 FMSR 848 4817 mt (0) s 1663 4615 mt 1663 4561 L 1663 389 mt 1663 443 L 1513 4817 mt (100) s 2429 4615 mt 2429 4561 L 2429 389 mt 2429 443 L 2279 4817 mt (200) s 3194 4615 mt 3194 4561 L 3194 389 mt 3194 443 L 3044 4817 mt (300) s 3959 4615 mt 3959 4561 L 3959 389 mt 3959 443 L 3809 4817 mt (400) s 4724 4615 mt 4724 4561 L 4724 389 mt 4724 443 L 4574 4817 mt (500) s 5490 4615 mt 5490 4561 L 5490 389 mt 5490 443 L 5340 4817 mt (600) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L 6105 4817 mt (700) s 898 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 763 4681 mt (0) s 898 3847 mt 952 3847 L 6255 3847 mt 6201 3847 L 613 3913 mt (0.2) s 898 3078 mt 952 3078 L 6255 3078 mt 6201 3078 L 613 3144 mt (0.4) s 898 2310 mt 952 2310 L 6255 2310 mt 6201 2310 L 613 2376 mt (0.6) s 898 1542 mt 952 1542 L 6255 1542 mt 6201 1542 L 613 1608 mt (0.8) s 898 773 mt 952 773 L 6255 773 mt 6201 773 L 763 839 mt (1) s 898 389 mt 6255 389 L 898 4615 mt 6255 4615 L 898 4615 mt 898 389 L 6255 4615 mt 6255 389 L gs 898 389 5358 4227 MR c np 129 41 130 -21 129 16 130 8 129 -5 130 10 129 33 130 7 129 17 130 11 130 16 129 88 130 4 129 51 130 162 129 37 130 166 129 151 130 143 129 60 130 180 129 309 130 233 129 109 130 317 129 -15 130 376 129 416 130 130 130 109 129 252 130 250 129 169 130 -72 129 -179 130 -569 129 -1034 130 -795 129 -900 963 4322 40 MP stroke gr /Helvetica /ISOLatin1Encoding 240 FMSR 558 3666 mt -90 rotate (Normalized frequency) s 90 rotate 2600 5048 mt (Interpoint distance) s 1810 233 mt (Texture data set point distribution) s end eplot epage end showpage %%EndDocument endTexFig 1325 1570 a Fo(\(b\))h(T)m(exture)g(features)1156 1700 y(Figure)g(3:)k(The)c(pro\014les)h(of)e(the)h(data)g(sets.)1020 1801 y(in)o(tegers,)h(as)f(assumed)g(in)f(Section)i(3.)j(The)d (distribution)f(of)1020 1848 y(in)o(terp)q(oin)o(t)g(distances)h(in)f (our)g(p)q(oin)o(t)g(sets)i(is)e(sho)o(wn)g(in)g(Fig-)1020 1895 y(ure)19 b(3.)30 b(Both)19 b(graphs)f(w)o(ere)h(obtained)f(b)o(y)g (computing)e(all)1020 1943 y(in)o(terp)q(oin)o(t)g(distances)h(of)e (random)g(subsets)j(of)d(200)h(p)q(oin)o(ts,)1020 1990 y(normalizing)11 b(the)k(maxim)o(al)10 b(v)n(alue)j(to)h(1.)1070 2048 y(The)j(second)i(data)d(set)i(con)o(tains)f(275,465)f(feature)i(v) o(ec-)1020 2095 y(tors)d(of)f(dimension)e(60)i(represen)o(ting)i (texture)g(information)1020 2142 y(of)i(blo)q(c)o(ks)g(of)g(large)g (aerial)g(photographs.)32 b(This)18 b(data)g(set)1020 2190 y(w)o(as)c(pro)o(vided)g(b)o(y)g(B.S.)g(Manjunath)f([31,)g(32];)g (its)h(size)h(and)1020 2237 y(dimensionalit)o(y)10 b(\\pro)o(vides)i(c) o(hallenging)f(problems)h(in)g(high)1020 2284 y(dimensional)19 b(indexing")i([31)o(].)40 b(These)23 b(features)f(are)g(ob-)1020 2332 y(tained)d(from)f(Gab)q(or)g(\014ltering)h(of)g(the)g(image)f (tiles.)34 b(The)1020 2379 y(Gab)q(or)22 b(\014lter)g(bank)g(consists)i (of)d(5)h(scales)h(and)f(6)g(orien-)1020 2426 y(tations)e(of)f (\014lters,)j(th)o(us)f(the)g(total)e(n)o(um)o(b)q(er)h(of)f(\014lters) i(is)1020 2473 y(5)9 b Fm(\002)g Fo(6)i(=)h(30.)17 b(The)d(mean)f(and)g (standard)i(deviation)d(of)h(eac)o(h)1020 2521 y(\014ltered)f(output)f (are)g(used)h(to)f(constructed)i(the)f(feature)g(v)o(ec-)1020 2568 y(tor)j(\()p Fn(d)e Fo(=)g(30)d Fm(\002)g Fo(2)j(=)g(60\).)20 b(These)c(texture)h(features)f(are)f(ex-)1020 2615 y(tracted)k(from)d (40)h(large)g(air)g(photos.)29 b(Before)19 b(the)f(feature)1020 2663 y(extraction,)c(eac)o(h)g(airphoto)f(is)h(\014rst)h(partitioned)e (in)o(to)g(non-)p eop %%Page: 8 8 8 7 bop -30 -33 a Fo(o)o(v)o(erlapping)16 b(tiles)h(of)g(size)h(64)e (times)h(64,)g(from)e(whic)o(h)i(the)-30 14 y(feature)e(v)o(ectors)g (are)f(computed.)20 62 y Fi(Query)e(Sets.)k Fo(The)c(di\016cult)o(y)e (in)h(ev)n(aluating)f(similarit)o(y)-30 109 y(searc)o(hing)j (algorithms)e(is)i(the)h(lac)o(k)e(of)g(a)h(publicly)f(a)o(v)n(ailable) -30 156 y(database)17 b(con)o(taining)f(t)o(ypical)g(query)i(p)q(oin)o (ts.)27 b(Therefore,)-30 204 y(w)o(e)17 b(had)g(to)g(construct)i(the)f (query)g(set)g(from)d(the)j(data)f(set)-30 251 y(itself.)g(Our)12 b(construction)h(is)f(as)g(follo)o(ws:)j(w)o(e)d(split)g(the)g(data)-30 298 y(set)h(randomly)d(in)o(to)i(t)o(w)o(o)f(disjoin)o(t)g(parts)i (\(call)f(them)f Fn(S)847 304 y Fj(1)878 298 y Fo(and)-30 346 y Fn(S)-5 352 y Fj(2)14 346 y Fo(\).)17 b(F)m(or)12 b(the)g(\014rst)g(data)f(set)i(the)f(size)g(of)f Fn(S)654 352 y Fj(1)684 346 y Fo(is)h(19,000)d(while)-30 393 y(the)15 b(size)f(of)g Fn(S)193 399 y Fj(2)226 393 y Fo(is)f(1000.)18 b(The)c(set)h Fn(S)556 399 y Fj(1)589 393 y Fo(forms)e(a)g(database)i (of)-30 440 y(images,)k(while)g(the)h(\014rst)g(500)e(p)q(oin)o(ts)i (from)d Fn(S)747 446 y Fj(2)786 440 y Fo(\(denoted)-30 488 y(b)o(y)d Fn(Q)p Fo(\))h(are)g(used)g(as)g(query)g(p)q(oin)o(ts)g (\(w)o(e)g(use)g(the)g(other)h(500)-30 535 y(p)q(oin)o(ts)f(for)g(v)n (arious)g(v)o(eri\014cation)f(purp)q(oses\).)24 b(F)m(or)15 b(the)h(sec-)-30 582 y(ond)i(data)f(set)i(w)o(e)f(c)o(hose)h Fn(S)426 588 y Fj(1)463 582 y Fo(to)e(b)q(e)i(of)e(size)i(270,000,)d (and)-30 630 y(w)o(e)f(use)h(1000)e(of)h(the)h(remaining)d(5,465)g(p)q (oin)o(ts)i(as)h(a)e(query)-30 677 y(set.)27 b(The)17 b(n)o(um)o(b)q(ers)f(are)h(sligh)o(tly)e(di\013eren)o(t)i(for)f(the)h (scala-)-30 724 y(bilit)o(y)e(exp)q(erimen)o(ts)h(as)h(they)f(require)i (v)n(arying)d(the)i(size)g(of)-30 771 y(the)c(data)g(set.)18 b(In)13 b(this)g(case)h(w)o(e)f(c)o(hose)h(a)e(random)f(subset)k(of)-30 819 y Fn(S)-5 825 y Fj(1)28 819 y Fo(of)e(required)i(size.)-30 913 y Fi(4.1)48 b(Exp)q(erimen)o(tal)13 b(Results)-30 983 y Fo(In)k(this)h(section)g(w)o(e)g(describ)q(e)h(the)f(results)h (of)e(our)g(exp)q(eri-)-30 1031 y(men)o(ts.)26 b(F)m(or)16 b(b)q(oth)h(data)f(sets)i(they)f(consist)g(essen)o(tially)g(of)-30 1078 y(the)c(follo)o(wing)c(three)14 b(steps.)k(In)12 b(the)h(\014rst)g(phase)g(w)o(e)f(ha)o(v)o(e)g(to)-30 1125 y(mak)o(e)g(the)h(follo)o(wing)e(c)o(hoice:)18 b(the)13 b(v)n(alue)g(of)f Fn(k)i Fo(\(the)g(n)o(um)o(b)q(er)-30 1173 y(of)e(sampled)f(bits\))h(to)g(c)o(ho)q(ose)h(for)f(a)g(giv)o(en)g (data)g(set)h(and)f(the)-30 1220 y(giv)o(en)i(n)o(um)o(b)q(er)f(of)h (indices)g Fn(l)i Fo(in)d(order)i(to)f(minimi)o(ze)e(the)j(ef-)-30 1267 y(fectiv)o(e)d(error.)19 b(It)12 b(turned)h(out)f(that)g(the)h (optimal)c(v)n(alue)j(of)f Fn(k)-30 1315 y Fo(is)j(essen)o(tially)g (indep)q(enden)o(t)h(of)f Fn(n)g Fo(and)f Fn(d)h Fo(and)g(th)o(us)g(w)o (e)h(can)-30 1362 y(use)e(the)g(same)f(v)n(alue)g(for)g(di\013eren)o(t) h(v)n(alues)f(of)g(these)i(param-)-30 1409 y(eters.)25 b(In)15 b(the)h(second)h(phase,)f(w)o(e)g(estimate)f(the)h(in\015uence) -30 1457 y(of)h(the)i(n)o(um)o(b)q(er)e(of)h(indices)g Fn(l)h Fo(on)f(the)h(error.)31 b(Finally)m(,)17 b(w)o(e)-30 1504 y(measure)i(the)g(p)q(erformance)g(of)f(LSH)h(b)o(y)f(computing)f (\(for)-30 1551 y(a)e(v)n(ariet)o(y)f(of)h(data)f(sets\))j(the)e (minima)o(l)d(n)o(um)o(b)q(er)i(of)h(indices)-30 1599 y(needed)f(to)e(ac)o(hiev)o(e)h(a)f(sp)q(eci\014ed)i(v)n(alue)e(of)g (error.)19 b(When)12 b(ap-)-30 1646 y(plicable,)j(w)o(e)g(also)g (compare)g(this)g(p)q(erformance)h(with)f(that)-30 1693 y(of)e(SR-trees.)-30 1787 y Fi(4.2)48 b(Color)15 b(histograms)-30 1858 y Fo(F)m(or)j(this)h(data)f(set,)i(w)o(e)f(p)q(erformed)g(sev)o (eral)g(exp)q(erimen)o(ts)-30 1905 y(aimed)8 b(at)i(understanding)g (the)h(b)q(eha)o(vior)e(of)h(LSH)g(algorithm)-30 1953 y(and)h(its)h(p)q(erformance)f(relativ)o(e)g(to)g(SR-tree.)18 b(As)12 b(men)o(tioned)-30 2000 y(ab)q(o)o(v)o(e,)e(w)o(e)h(started)g (with)f(an)g(observ)n(ation)g(that)g(the)h(optimal)-30 2047 y(v)n(alue)h(of)h(sampled)f(bits)h Fn(k)h Fo(is)f(essen)o(tially)g (indep)q(enden)o(t)i(of)d Fn(n)-30 2094 y Fo(and)g Fn(d)g Fo(and)f(appro)o(ximately)f(equal)i(to)g(700)f(for)h Fn(d)f Fo(=)h(64.)17 b(The)-30 2142 y(lac)o(k)d(of)g(dep)q(endence)k (on)c Fn(n)h Fo(can)f(b)q(e)i(explained)e(b)o(y)h(the)g(fact)-30 2189 y(that)g(the)h(smaller)d(data)i(sets)h(w)o(ere)g(obtained)f(b)o(y) f(sampling)-30 2236 y(the)g(large)f(one)h(and)f(therefore)i(all)d(of)h (the)h(sets)g(ha)o(v)o(e)f(similar)-30 2284 y(structure;)g(w)o(e)c(b)q (eliev)o(e)h(the)g(lac)o(k)f(of)f(dep)q(endence)k(on)d Fn(d)g Fo(is)h(also)-30 2331 y(in\015uenced)15 b(b)o(y)e(the)h (structure)h(of)e(the)h(data.)k(Therefore)c(the)-30 2378 y(follo)o(wing)d(exp)q(erimen)o(ts)j(w)o(ere)h(done)f(assuming)f Fn(k)f Fo(=)g(700.)20 2426 y(Our)20 b(next)g(observ)n(ation)g(w)o(as)f (that)h(the)g(v)n(alue)g(of)f(stor-)-30 2473 y(age)12 b(o)o(v)o(erhead)g Fn(\013)g Fo(do)q(es)h(not)f(exert)h(m)o(uc)o(h)e (in\015uence)i(o)o(v)o(er)f(the)-30 2521 y(p)q(erformance)17 b(of)g(the)h(algorithm)d(\(w)o(e)j(tried)g Fn(\013)p Fo('s)f(from)e(the)-30 2568 y(in)o(terv)n(al)e([2)p Fn(;)7 b Fo(5]\);)12 b(th)o(us,)h(in)h(the)g(follo)o(wing)e(w)o(e)i(set)h Fn(\013)c Fo(=)h(2.)20 2615 y(In)i(the)i(next)f(step)g(w)o(e)g (estimated)f(the)i(in\015uence)f(of)f Fn(l)i Fo(on)-30 2663 y Fn(E)r Fo(.)25 b(The)16 b(results)i(\(for)d Fn(n)h Fo(=)f(19)p Fn(;)7 b Fo(000,)15 b Fn(d)f Fo(=)i(64,)g Fn(K)i Fo(=)e(1\))g(are)1020 -33 y 14208860 11935440 3289088 12564316 35719495 40061091 startTexFig 1020 -33 a %%BeginDocument: error_vs_numofindices.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 392 30 5912 5013 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 1385 4613 mt 1385 388 L 1385 388 mt 1385 388 L 1872 4613 mt 1872 388 L 1872 388 mt 1872 388 L 2359 4613 mt 2359 388 L 2359 388 mt 2359 388 L 2846 4613 mt 2846 388 L 2846 388 mt 2846 388 L 3333 4613 mt 3333 388 L 3333 388 mt 3333 388 L 3819 4613 mt 3819 388 L 3819 388 mt 3819 388 L 4306 4613 mt 4306 388 L 4306 388 mt 4306 388 L 4793 4613 mt 4793 388 L 4793 388 mt 4793 388 L 5280 4613 mt 5280 388 L 5280 388 mt 5280 388 L 5767 4613 mt 5767 388 L 5767 388 mt 5767 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3963 mt 6254 3963 L 6254 3963 mt 6254 3963 L 898 3313 mt 6254 3313 L 6254 3313 mt 6254 3313 L 898 2663 mt 6254 2663 L 6254 2663 mt 6254 2663 L 898 2013 mt 6254 2013 L 6254 2013 mt 6254 2013 L 898 1363 mt 6254 1363 L 6254 1363 mt 6254 1363 L 898 713 mt 6254 713 L 6254 713 mt 6254 713 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 1385 4613 mt 1385 4559 L 1385 388 mt 1385 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 1335 4815 mt (1) s 1872 4613 mt 1872 4559 L 1872 388 mt 1872 442 L 1822 4815 mt (2) s 2359 4613 mt 2359 4559 L 2359 388 mt 2359 442 L 2309 4815 mt (3) s 2846 4613 mt 2846 4559 L 2846 388 mt 2846 442 L 2796 4815 mt (4) s 3333 4613 mt 3333 4559 L 3333 388 mt 3333 442 L 3283 4815 mt (5) s 3819 4613 mt 3819 4559 L 3819 388 mt 3819 442 L 3769 4815 mt (6) s 4306 4613 mt 4306 4559 L 4306 388 mt 4306 442 L 4256 4815 mt (7) s 4793 4613 mt 4793 4559 L 4793 388 mt 4793 442 L 4743 4815 mt (8) s 5280 4613 mt 5280 4559 L 5280 388 mt 5280 442 L 5230 4815 mt (9) s 5767 4613 mt 5767 4559 L 5767 388 mt 5767 442 L 5667 4815 mt (10) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 3963 mt 952 3963 L 6254 3963 mt 6200 3963 L 613 4029 mt (0.1) s 898 3313 mt 952 3313 L 6254 3313 mt 6200 3313 L 613 3379 mt (0.2) s 898 2663 mt 952 2663 L 6254 2663 mt 6200 2663 L 613 2729 mt (0.3) s 898 2013 mt 952 2013 L 6254 2013 mt 6200 2013 L 613 2079 mt (0.4) s 898 1363 mt 952 1363 L 6254 1363 mt 6200 1363 L 613 1429 mt (0.5) s 898 713 mt 952 713 L 6254 713 mt 6200 713 L 613 779 mt (0.6) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np 487 45 487 20 487 -7 487 102 486 67 487 255 487 199 487 894 487 2000 1385 948 10 MP stroke gr 0 j 0 -58 -58 0 0 58 58 0 1356 919 5 MP DP 0 -58 -58 0 0 58 58 0 1843 2919 5 MP DP 0 -58 -58 0 0 58 58 0 2330 3813 5 MP DP 0 -58 -58 0 0 58 58 0 2817 4012 5 MP DP 0 -58 -58 0 0 58 58 0 3304 4267 5 MP DP 0 -58 -58 0 0 58 58 0 3790 4334 5 MP DP 0 -58 -58 0 0 58 58 0 4277 4436 5 MP DP 0 -58 -58 0 0 58 58 0 4764 4429 5 MP DP 0 -58 -58 0 0 58 58 0 5251 4449 5 MP DP 0 -58 -58 0 0 58 58 0 5738 4494 5 MP DP gs 898 388 5357 4226 MR c np gr 2853 5001 mt (Number of indices) s 559 2700 mt -90 rotate (Error) s 90 rotate 2301 245 mt (alpha=2, n=19000, d=64, k=700) s end eplot epage end showpage %%EndDocument endTexFig 1125 900 a Fo(Figure)e(4:)j(Error)e(vs.)j(the)d(n)o(um)o(b)q(er)e(of)g (indices.)1020 999 y(sho)o(wn)e(on)g(Figure)g(4.)16 b(As)c(exp)q (ected,)h(one)e(index)g(is)f(not)h(su\016-)1020 1046 y(cien)o(t)i(to)f(ac)o(hiev)o(e)h(reasonably)g(small)d(error)j(|)f(the) i(e\013ectiv)o(e)1020 1094 y(error)19 b(can)f(easily)g(exceed)i (50\045.)30 b(The)18 b(error)h(ho)o(w)o(ev)o(er)g(de-)1020 1141 y(creases)c(v)o(ery)f(fast)f(as)g Fn(l)h Fo(increases.)20 b(This)13 b(is)g(due)g(to)g(the)h(fact)1020 1188 y(that)i(the)g (probabilities)e(of)h(\014nding)g(empt)o(y)f(buc)o(k)o(et)j(are)f(in-) 1020 1235 y(dep)q(enden)o(t)e(for)d(di\013eren)o(t)j(indices)e(and)g (therefore)h(the)g(prob-)1020 1283 y(abilit)o(y)c(that)i(all)f(buc)o(k) o(ets)i(are)f(empt)o(y)f(deca)o(ys)i(exp)q(onen)o(tially)1020 1330 y(in)h Fn(l)q Fo(.)1070 1379 y(In)j(order)g(to)g(compare)f(the)i (p)q(erformance)f(of)f(LSH)h(with)1020 1427 y(SR-tree,)g(w)o(e)f (computed)g(\(for)g(a)g(v)n(ariet)o(y)f(of)h(data)g(sets\))h(the)1020 1474 y(minim)o(al)11 b(n)o(um)o(b)q(er)k(of)f(indices)h(needed)i(to)d (ac)o(hiev)o(e)i(a)e(sp)q(eci-)1020 1521 y(\014ed)j(v)n(alue)e(of)h (error)h Fn(E)h Fo(equal)e(to)g(2\045,)g(5\045)g(,)g(10\045)f(or)h (20\045.)1020 1569 y(Then)h(w)o(e)g(in)o(v)o(estigated)g(the)g(p)q (erformance)g(of)f(the)i(t)o(w)o(o)e(al-)1020 1616 y(gorithms)c(while)i (v)n(arying)e(the)j(dimension)d(and)i(data)f(size.)1070 1665 y Fi(Dep)q(endence)18 b(on)h(Data)h(Size.)28 b Fo(W)m(e)17 b(p)q(erformed)g(the)1020 1713 y(sim)o(ulations)11 b(for)i Fn(d)e Fo(=)h(64)h(and)g(the)h(data)f(sets)i(of)e(sizes)h(1000,)1020 1760 y(2000,)c(5000,)g(10000)g(and)h(19000.)k(T)m(o)c(ac)o(hiev)o(e)g (b)q(etter)h(under-)1020 1807 y(standing)17 b(of)g(scalabilit)o(y)f(of) h(our)g(algorithms,)f(w)o(e)i(did)f(run)1020 1854 y(the)i(exp)q(erimen) o(ts)g(t)o(wice:)28 b(for)18 b(Appro)o(ximate)g(1-NNS)g(and)1020 1902 y(for)i(Appro)o(ximate)e(10-NNS.)h(The)h(results)h(are)g(presen)o (ted)1020 1949 y(on)14 b(Figure)g(5.)1070 1998 y(Notice)f(the)h (strongly)e(sublinear)h(dep)q(endence)j(exhibited)1020 2046 y(b)o(y)10 b(LSH:)g(although)f(for)h(small)e Fn(E)13 b Fo(=)f(2\045)e(it)f(matc)o(hes)h(SR-tree)1020 2093 y(for)17 b Fn(n)g Fo(=)h(1000)e(with)h(5)g(blo)q(c)o(ks)h(accessed)h (\(for)f Fn(K)i Fo(=)e(1\),)f(it)1020 2140 y(requires)c(3)f(accesses)j (more)d(for)f(a)h(data)g(set)h(19)f(times)f(larger.)1020 2188 y(A)o(t)h(the)h(same)e(time)f(the)j(I/O)f(activit)o(y)f(of)h (SR-tree)g(increases)1020 2235 y(b)o(y)i(more)f(than)h(200\045.)i(F)m (or)e(larger)g(errors)h(the)g(LSH)f(curv)o(es)1020 2282 y(are)c(nearly)f(\015at,)h(i.e.,)f(exhibit)f(little)h(dep)q(endence)j (on)d(the)h(size)1020 2330 y(of)i(the)g(data.)18 b(Similar)9 b(or)j(ev)o(en)h(b)q(etter)h(b)q(eha)o(vior)e(o)q(ccurs)h(for)1020 2377 y(Appro)o(ximate)f(10-NNS.)1070 2426 y(W)m(e)h(also)g(computed)g (the)h(miss)f(ratios,)g(i.e.,)f(the)i(fraction)1020 2473 y(of)f(queries)h(for)f(whic)o(h)g(no)g(answ)o(er)h(w)o(as)f(found.)k (The)d(results)1020 2521 y(are)k(presen)o(ted)j(on)c(Figure)i(6.)30 b(W)m(e)17 b(used)i(the)g(parameters)1020 2568 y(from)d(the)j(previous) g(exp)q(erimen)o(t.)31 b(On)18 b(can)h(observ)o(e)g(that)1020 2615 y(for)g(sa)o(y)g Fn(E)k Fo(=)d(5\045)f(and)g(Appro)o(ximate)f (1-NNS,)g(the)i(miss)1020 2663 y(ratios)c(are)h(quite)f(high)g (\(10\045\))f(for)h(small)e Fn(n)p Fo(,)i(but)h(decrease)p eop %%Page: 9 9 9 8 bop 7 -33 a 14208860 12219616 3552215 12103843 35719495 40061091 startTexFig 7 -33 a %%BeginDocument: size_2_1.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 442 30 5862 5102 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 981 4613 mt 981 388 L 981 388 mt 981 388 L 1259 4613 mt 1259 388 L 1259 388 mt 1259 388 L 2091 4613 mt 2091 388 L 2091 388 mt 2091 388 L 3479 4613 mt 3479 388 L 3479 388 mt 3479 388 L 5976 4613 mt 5976 388 L 5976 388 mt 5976 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3733 mt 6254 3733 L 6254 3733 mt 6254 3733 L 898 2853 mt 6254 2853 L 6254 2853 mt 6254 2853 L 898 1972 mt 6254 1972 L 6254 1972 mt 6254 1972 L 898 1092 mt 6254 1092 L 6254 1092 mt 6254 1092 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 981 4613 mt 981 4559 L 981 388 mt 981 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 856 4815 mt (0.1) s 1259 4613 mt 1259 4559 L 1259 388 mt 1259 442 L 1134 4815 mt (0.2) s 2091 4613 mt 2091 4559 L 2091 388 mt 2091 442 L 1966 4815 mt (0.5) s 3479 4613 mt 3479 4559 L 3479 388 mt 3479 442 L 3429 4815 mt (1) s 5976 4613 mt 5976 4559 L 5976 388 mt 5976 442 L 5851 4815 mt (1.9) s 5848 5091 mt (x 10) s /Helvetica /ISOLatin1Encoding 120 FMSR 6188 4980 mt (4) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L /Helvetica /ISOLatin1Encoding 180 FMSR 763 4679 mt (0) s 898 3733 mt 952 3733 L 6254 3733 mt 6200 3733 L 763 3799 mt (5) s 898 2853 mt 952 2853 L 6254 2853 mt 6200 2853 L 663 2919 mt (10) s 898 1972 mt 952 1972 L 6254 1972 mt 6200 1972 L 663 2038 mt (15) s 898 1092 mt 952 1092 L 6254 1092 mt 6200 1092 L 663 1158 mt (20) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np DA 2497 -600 1388 -613 832 -732 278 -410 981 3787 5 MP stroke gr DA SO 0 j 0 -58 -58 0 0 58 58 0 952 3758 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1230 3348 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 2062 2616 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 3450 2003 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 5947 1403 5 MP DP DA gs 898 388 5357 4226 MR c np SO 2497 -176 1388 -352 832 0 278 -176 981 3733 5 MP stroke gr SO 0 -58 -58 0 0 58 58 0 952 3704 5 MP DP 0 -58 -58 0 0 58 58 0 1230 3528 5 MP DP 0 -58 -58 0 0 58 58 0 2062 3528 5 MP DP 0 -58 -58 0 0 58 58 0 3450 3176 5 MP DP 0 -58 -58 0 0 58 58 0 5947 3000 5 MP DP gs 898 388 5357 4226 MR c np 2497 0 1388 0 832 -176 278 0 981 3909 5 MP stroke gr 945 3909 mt 1017 3909 L 981 3873 mt 981 3945 L 956 3884 mt 1006 3934 L 1006 3884 mt 956 3934 L 1223 3909 mt 1295 3909 L 1259 3873 mt 1259 3945 L 1234 3884 mt 1284 3934 L 1284 3884 mt 1234 3934 L 2055 3733 mt 2127 3733 L 2091 3697 mt 2091 3769 L 2066 3708 mt 2116 3758 L 2116 3708 mt 2066 3758 L 3443 3733 mt 3515 3733 L 3479 3697 mt 3479 3769 L 3454 3708 mt 3504 3758 L 3504 3708 mt 3454 3758 L 5940 3733 mt 6012 3733 L 5976 3697 mt 5976 3769 L 5951 3708 mt 6001 3758 L 6001 3708 mt 5951 3758 L gs 898 388 5357 4226 MR c np 2497 -176 1388 0 832 0 278 -176 981 4261 5 MP stroke gr 36 47 -36 47 -36 -47 36 -47 981 4308 5 MP DP 36 47 -36 47 -36 -47 36 -47 1259 4132 5 MP DP 36 47 -36 47 -36 -47 36 -47 2091 4132 5 MP DP 36 47 -36 47 -36 -47 36 -47 3479 4132 5 MP DP 36 47 -36 47 -36 -47 36 -47 5976 3956 5 MP DP gs 898 388 5357 4226 MR c np 2497 -176 1388 0 832 0 278 0 981 4261 5 MP stroke gr 36 36 981 4261 FO 36 36 1259 4261 FO 36 36 2091 4261 FO 36 36 3479 4261 FO 36 36 5976 4085 FO gs 898 388 5357 4226 MR c np gr 2497 5001 mt (Number of database points) s 609 3075 mt -90 rotate (Disk accesses) s 90 rotate 2861 245 mt (alpha = 2, 1-NNS) s 1 sg 0 1449 2374 0 0 -1449 1034 1982 4 MP PP -2374 0 0 1449 2374 0 0 -1449 1034 1982 5 MP stroke DO 4 w SO 6 w 0 sg 1034 1982 mt 3408 1982 L 1034 533 mt 3408 533 L 3408 1982 mt 3408 533 L 1034 1982 mt 1034 533 L 1034 1982 mt 3408 1982 L 1034 1982 mt 1034 533 L 1034 533 mt 3408 533 L 1034 1982 mt 3408 1982 L 1034 1982 mt 1034 533 L 3408 1982 mt 3408 533 L 1837 841 mt (SR-Tree ) s gs 1034 533 2375 1450 MR c np DA 428 0 1141 775 2 MP stroke gr DA SO 0 -58 -58 0 0 58 58 0 1112 746 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1540 746 5 MP DP DA gs 1034 533 2375 1450 MR c np gr 1837 1082 mt (LSH, error=.02) s gs 1034 533 2375 1450 MR c np SO 428 0 1141 1016 2 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1112 987 5 MP DP 0 -58 -58 0 0 58 58 0 1540 987 5 MP DP gs 1034 533 2375 1450 MR c np gr 1837 1324 mt (LSH, error=.05) s gs 1034 533 2375 1450 MR c np 428 0 1141 1258 2 MP stroke gr 1105 1258 mt 1177 1258 L 1141 1222 mt 1141 1294 L 1116 1233 mt 1166 1283 L 1166 1233 mt 1116 1283 L 1533 1258 mt 1605 1258 L 1569 1222 mt 1569 1294 L 1544 1233 mt 1594 1283 L 1594 1233 mt 1544 1283 L gs 1034 533 2375 1450 MR c np gr 1837 1565 mt (LSH, error=.1 ) s gs 1034 533 2375 1450 MR c np 428 0 1141 1499 2 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1141 1546 5 MP DP 36 47 -36 47 -36 -47 36 -47 1569 1546 5 MP DP gs 1034 533 2375 1450 MR c np gr 1837 1807 mt (LSH, error=.2 ) s gs 1034 533 2375 1450 MR c np 428 0 1141 1741 2 MP stroke gr 36 36 1141 1741 FO 36 36 1569 1741 FO gs 1034 533 2375 1450 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 239 788 a Fo(\(a\))14 b(Appro)o(ximate)e(1-NNS)7 803 y 14208860 12219616 3552215 12103843 35719495 40061091 startTexFig 7 803 a %%BeginDocument: size_2_10.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 442 29 5862 5103 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 981 4613 mt 981 388 L 981 388 mt 981 388 L 1259 4613 mt 1259 388 L 1259 388 mt 1259 388 L 2091 4613 mt 2091 388 L 2091 388 mt 2091 388 L 3479 4613 mt 3479 388 L 3479 388 mt 3479 388 L 5976 4613 mt 5976 388 L 5976 388 mt 5976 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 4085 mt 6254 4085 L 6254 4085 mt 6254 4085 L 898 3557 mt 6254 3557 L 6254 3557 mt 6254 3557 L 898 3029 mt 6254 3029 L 6254 3029 mt 6254 3029 L 898 2501 mt 6254 2501 L 6254 2501 mt 6254 2501 L 898 1972 mt 6254 1972 L 6254 1972 mt 6254 1972 L 898 1444 mt 6254 1444 L 6254 1444 mt 6254 1444 L 898 916 mt 6254 916 L 6254 916 mt 6254 916 L 898 388 mt 6254 388 L 6254 388 mt 6254 388 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 981 4613 mt 981 4559 L 981 388 mt 981 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 856 4815 mt (0.1) s 1259 4613 mt 1259 4559 L 1259 388 mt 1259 442 L 1134 4815 mt (0.2) s 2091 4613 mt 2091 4559 L 2091 388 mt 2091 442 L 1966 4815 mt (0.5) s 3479 4613 mt 3479 4559 L 3479 388 mt 3479 442 L 3429 4815 mt (1) s 5976 4613 mt 5976 4559 L 5976 388 mt 5976 442 L 5851 4815 mt (1.9) s 5848 5091 mt (x 10) s /Helvetica /ISOLatin1Encoding 120 FMSR 6188 4980 mt (4) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L /Helvetica /ISOLatin1Encoding 180 FMSR 763 4679 mt (0) s 898 4085 mt 952 4085 L 6254 4085 mt 6200 4085 L 763 4151 mt (5) s 898 3557 mt 952 3557 L 6254 3557 mt 6200 3557 L 663 3623 mt (10) s 898 3029 mt 952 3029 L 6254 3029 mt 6200 3029 L 663 3095 mt (15) s 898 2501 mt 952 2501 L 6254 2501 mt 6200 2501 L 663 2567 mt (20) s 898 1972 mt 952 1972 L 6254 1972 mt 6200 1972 L 663 2038 mt (25) s 898 1444 mt 952 1444 L 6254 1444 mt 6200 1444 L 663 1510 mt (30) s 898 916 mt 952 916 L 6254 916 mt 6200 916 L 663 982 mt (35) s 898 388 mt 952 388 L 6254 388 mt 6200 388 L 663 454 mt (40) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np DA 2497 -664 1388 -646 832 -834 278 -268 981 3879 5 MP stroke gr DA SO 0 j 0 -58 -58 0 0 58 58 0 952 3850 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1230 3582 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 2062 2748 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 3450 2102 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 5947 1438 5 MP DP DA gs 898 388 5357 4226 MR c np SO 2497 0 1388 0 832 -317 278 -211 981 4085 5 MP stroke gr SO 0 -58 -58 0 0 58 58 0 952 4056 5 MP DP 0 -58 -58 0 0 58 58 0 1230 3845 5 MP DP 0 -58 -58 0 0 58 58 0 2062 3528 5 MP DP 0 -58 -58 0 0 58 58 0 3450 3528 5 MP DP 0 -58 -58 0 0 58 58 0 5947 3528 5 MP DP gs 898 388 5357 4226 MR c np 2497 0 1388 -211 832 0 278 -106 981 4191 5 MP stroke gr 945 4191 mt 1017 4191 L 981 4155 mt 981 4227 L 956 4166 mt 1006 4216 L 1006 4166 mt 956 4216 L 1223 4085 mt 1295 4085 L 1259 4049 mt 1259 4121 L 1234 4060 mt 1284 4110 L 1284 4060 mt 1234 4110 L 2055 4085 mt 2127 4085 L 2091 4049 mt 2091 4121 L 2066 4060 mt 2116 4110 L 2116 4060 mt 2066 4110 L 3443 3874 mt 3515 3874 L 3479 3838 mt 3479 3910 L 3454 3849 mt 3504 3899 L 3504 3849 mt 3454 3899 L 5940 3874 mt 6012 3874 L 5976 3838 mt 5976 3910 L 5951 3849 mt 6001 3899 L 6001 3849 mt 5951 3899 L gs 898 388 5357 4226 MR c np 2497 0 1388 -106 832 0 278 -105 981 4296 5 MP stroke gr 36 47 -36 47 -36 -47 36 -47 981 4343 5 MP DP 36 47 -36 47 -36 -47 36 -47 1259 4238 5 MP DP 36 47 -36 47 -36 -47 36 -47 2091 4238 5 MP DP 36 47 -36 47 -36 -47 36 -47 3479 4132 5 MP DP 36 47 -36 47 -36 -47 36 -47 5976 4132 5 MP DP gs 898 388 5357 4226 MR c np 2497 0 1388 0 832 0 278 -106 981 4402 5 MP stroke gr 36 36 981 4402 FO 36 36 1259 4296 FO 36 36 2091 4296 FO 36 36 3479 4296 FO 36 36 5976 4296 FO gs 898 388 5357 4226 MR c np gr 2497 5001 mt (Number of database points) s 609 3074 mt -90 rotate (Disk accesses) s 90 rotate 2811 245 mt (alpha = 2, 10-NNS) s 1 sg 0 1448 2374 0 0 -1448 1083 1932 4 MP PP -2374 0 0 1448 2374 0 0 -1448 1083 1932 5 MP stroke DO 4 w SO 6 w 0 sg 1083 1932 mt 3457 1932 L 1083 484 mt 3457 484 L 3457 1932 mt 3457 484 L 1083 1932 mt 1083 484 L 1083 1932 mt 3457 1932 L 1083 1932 mt 1083 484 L 1083 484 mt 3457 484 L 1083 1932 mt 3457 1932 L 1083 1932 mt 1083 484 L 3457 1932 mt 3457 484 L 1886 791 mt (SR-Tree ) s gs 1083 484 2375 1449 MR c np DA 428 0 1190 725 2 MP stroke gr DA SO 0 -58 -58 0 0 58 58 0 1161 696 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1589 696 5 MP DP DA gs 1083 484 2375 1449 MR c np gr 1886 1033 mt (LSH, error=.02) s gs 1083 484 2375 1449 MR c np SO 428 0 1190 967 2 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1161 938 5 MP DP 0 -58 -58 0 0 58 58 0 1589 938 5 MP DP gs 1083 484 2375 1449 MR c np gr 1886 1274 mt (LSH, error=.05) s gs 1083 484 2375 1449 MR c np 428 0 1190 1208 2 MP stroke gr 1154 1208 mt 1226 1208 L 1190 1172 mt 1190 1244 L 1165 1183 mt 1215 1233 L 1215 1183 mt 1165 1233 L 1582 1208 mt 1654 1208 L 1618 1172 mt 1618 1244 L 1593 1183 mt 1643 1233 L 1643 1183 mt 1593 1233 L gs 1083 484 2375 1449 MR c np gr 1886 1515 mt (LSH, error=.1 ) s gs 1083 484 2375 1449 MR c np 428 0 1190 1449 2 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1190 1496 5 MP DP 36 47 -36 47 -36 -47 36 -47 1618 1496 5 MP DP gs 1083 484 2375 1449 MR c np gr 1886 1757 mt (LSH, error=.2 ) s gs 1083 484 2375 1449 MR c np 428 0 1190 1691 2 MP stroke gr 36 36 1190 1691 FO 36 36 1618 1691 FO gs 1083 484 2375 1449 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 227 1624 a Fo(\(b\))j(Appro)o(ximate)d(10-NNS)75 1754 y(Figure)i(5:)k(Num)o(b)q(er)13 b(of)g(indices)i(vs.)j(data)c(size.)-30 1870 y(to)g(around)g(1\045)f(for)g Fn(n)f Fo(=)g(19)p Fn(;)7 b Fo(000)12 b(.)20 1935 y Fi(Dep)q(endence)j(on)i(Dimension.)i Fo(W)m(e)c(p)q(erformed)g(the)-30 1982 y(sim)o(ulations)10 b(for)i Fn(d)f Fo(=)h(2)347 1967 y Fj(3)366 1982 y Fn(;)7 b Fo(3)406 1967 y Fj(3)436 1982 y Fo(and)12 b(4)536 1967 y Fj(3)555 1982 y Fo(;)g(the)h(c)o(hoice)g(of)f Fn(d)p Fo('s)g(w)o(as)-30 2030 y(limited)c(to)i(cub)q(es)i(of)d(natural)h(n)o (um)o(b)q(ers)g(b)q(ecause)h(of)f(the)h(w)o(a)o(y)-30 2077 y(the)17 b(data)f(has)h(b)q(een)g(created.)27 b(Again,)16 b(w)o(e)g(p)q(erformed)g(the)-30 2124 y(comparison)11 b(for)g(Appro)o(ximate)g(1-NNS)h(and)g(Appro)o(ximate)-30 2172 y(10-NNS;)g(the)i(results)h(are)e(sho)o(wn)g(on)g(Figure)h(7.)j (Note)d(that)-30 2219 y(LSH)j(scales)h(v)o(ery)f(w)o(ell)f(with)g(the)i (increase)g(of)e(dimension-)-30 2266 y(alit)o(y:)23 b(for)16 b Fn(E)i Fo(=)f(5\045)f(the)i(c)o(hange)f(from)e Fn(d)h Fo(=)h(8)f(to)h Fn(d)f Fo(=)g(64)-30 2314 y(increases)i(the)g(n)o(um)o (b)q(er)e(of)g(indices)h(only)f(b)o(y)h(2.)26 b(The)17 b(miss)-30 2361 y(ratio)c(w)o(as)h(alw)o(a)o(ys)f(b)q(elo)o(w)h(6\045)f (for)g(all)g(dimensions.)20 2426 y(This)20 b(completes)g(the)h (comparison)e(of)g(LSH)i(with)f(SR-)-30 2473 y(tree.)34 b(F)m(or)18 b(a)h(b)q(etter)h(understanding)g(of)e(the)h(b)q(eha)o (vior)g(of)-30 2521 y(LSH,)14 b(w)o(e)g(p)q(erformed)g(an)g(additional) f(exp)q(erimen)o(t)h(on)g(LSH)-30 2568 y(only)m(.)j(Figure)d(8)f (presen)o(ts)j(the)f(p)q(erformance)e(of)h(LSH)f(when)-30 2615 y(the)h(n)o(um)o(b)q(er)f(of)g(nearest)i(neigh)o(b)q(ors)f(to)f (retriev)o(e)i(v)n(ary)e(from)-30 2663 y(1)h(to)f(100.)1057 -33 y 14208860 11935440 2762833 12103843 35719495 40061091 startTexFig 1057 -33 a %%BeginDocument: miss_1.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 292 30 6012 5102 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 981 4613 mt 981 388 L 981 388 mt 981 388 L 1259 4613 mt 1259 388 L 1259 388 mt 1259 388 L 2091 4613 mt 2091 388 L 2091 388 mt 2091 388 L 3479 4613 mt 3479 388 L 3479 388 mt 3479 388 L 5976 4613 mt 5976 388 L 5976 388 mt 5976 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3909 mt 6254 3909 L 6254 3909 mt 6254 3909 L 898 3205 mt 6254 3205 L 6254 3205 mt 6254 3205 L 898 2500 mt 6254 2500 L 6254 2500 mt 6254 2500 L 898 1796 mt 6254 1796 L 6254 1796 mt 6254 1796 L 898 1092 mt 6254 1092 L 6254 1092 mt 6254 1092 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 981 4613 mt 981 4559 L 981 388 mt 981 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 856 4815 mt (0.1) s 1259 4613 mt 1259 4559 L 1259 388 mt 1259 442 L 1134 4815 mt (0.2) s 2091 4613 mt 2091 4559 L 2091 388 mt 2091 442 L 1966 4815 mt (0.5) s 3479 4613 mt 3479 4559 L 3479 388 mt 3479 442 L 3429 4815 mt (1) s 5976 4613 mt 5976 4559 L 5976 388 mt 5976 442 L 5851 4815 mt (1.9) s 5848 5091 mt (x 10) s /Helvetica /ISOLatin1Encoding 120 FMSR 6188 4980 mt (4) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L /Helvetica /ISOLatin1Encoding 180 FMSR 763 4679 mt (0) s 898 3909 mt 952 3909 L 6254 3909 mt 6200 3909 L 513 3975 mt (0.05) s 898 3205 mt 952 3205 L 6254 3205 mt 6200 3205 L 613 3271 mt (0.1) s 898 2500 mt 952 2500 L 6254 2500 mt 6200 2500 L 513 2566 mt (0.15) s 898 1796 mt 952 1796 L 6254 1796 mt 6200 1796 L 613 1862 mt (0.2) s 898 1092 mt 952 1092 L 6254 1092 mt 6200 1092 L 513 1158 mt (0.25) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np 2497 85 1388 394 832 226 278 1352 981 2303 5 MP stroke gr 945 2303 mt 1017 2303 L 981 2267 mt 981 2339 L 956 2278 mt 1006 2328 L 1006 2278 mt 956 2328 L 1223 3655 mt 1295 3655 L 1259 3619 mt 1259 3691 L 1234 3630 mt 1284 3680 L 1284 3630 mt 1234 3680 L 2055 3881 mt 2127 3881 L 2091 3845 mt 2091 3917 L 2066 3856 mt 2116 3906 L 2116 3856 mt 2066 3906 L 3443 4275 mt 3515 4275 L 3479 4239 mt 3479 4311 L 3454 4250 mt 3504 4300 L 3504 4250 mt 3454 4300 L 5940 4360 mt 6012 4360 L 5976 4324 mt 5976 4396 L 5951 4335 mt 6001 4385 L 6001 4335 mt 5951 4385 L gs 898 388 5357 4226 MR c np 2497 113 1388 394 832 676 278 1380 981 1571 5 MP stroke gr 0 j 0 -58 -58 0 0 58 58 0 952 1542 5 MP DP 0 -58 -58 0 0 58 58 0 1230 2922 5 MP DP 0 -58 -58 0 0 58 58 0 2062 3598 5 MP DP 0 -58 -58 0 0 58 58 0 3450 3992 5 MP DP 0 -58 -58 0 0 58 58 0 5947 4105 5 MP DP gs 898 388 5357 4226 MR c np gr 2497 5001 mt (Number of database points) s 459 2884 mt -90 rotate (Miss ratio) s 90 rotate 2256 245 mt (alpha=2, n=19000, d=64, 1-NNS) s 1 sg 0 596 1819 0 0 -596 3618 1915 4 MP PP -1819 0 0 596 1819 0 0 -596 3618 1915 5 MP stroke DO 4 w SO 6 w 0 sg 3618 1915 mt 5437 1915 L 3618 1319 mt 5437 1319 L 5437 1915 mt 5437 1319 L 3618 1915 mt 3618 1319 L 3618 1915 mt 5437 1915 L 3618 1915 mt 3618 1319 L 3618 1319 mt 5437 1319 L 3618 1915 mt 5437 1915 L 3618 1915 mt 3618 1319 L 5437 1915 mt 5437 1319 L 4421 1584 mt (Error=.05) s gs 3618 1319 1820 597 MR c np 428 0 3725 1518 2 MP stroke gr 3689 1518 mt 3761 1518 L 3725 1482 mt 3725 1554 L 3700 1493 mt 3750 1543 L 3750 1493 mt 3700 1543 L 4117 1518 mt 4189 1518 L 4153 1482 mt 4153 1554 L 4128 1493 mt 4178 1543 L 4178 1493 mt 4128 1543 L gs 3618 1319 1820 597 MR c np gr 4421 1782 mt (Error=.1 ) s gs 3618 1319 1820 597 MR c np 428 0 3725 1716 2 MP stroke gr 0 -58 -58 0 0 58 58 0 3696 1687 5 MP DP 0 -58 -58 0 0 58 58 0 4124 1687 5 MP DP gs 3618 1319 1820 597 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 1289 770 a Fo(\(a\))h(Appro)o(ximate)e(1-NNS)1057 785 y 14208860 12219616 3289088 12103843 35719495 40061091 startTexFig 1057 785 a %%BeginDocument: miss_10.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 392 30 5912 5102 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 981 4613 mt 981 388 L 981 388 mt 981 388 L 1259 4613 mt 1259 388 L 1259 388 mt 1259 388 L 2091 4613 mt 2091 388 L 2091 388 mt 2091 388 L 3479 4613 mt 3479 388 L 3479 388 mt 3479 388 L 5976 4613 mt 5976 388 L 5976 388 mt 5976 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3909 mt 6254 3909 L 6254 3909 mt 6254 3909 L 898 3205 mt 6254 3205 L 6254 3205 mt 6254 3205 L 898 2500 mt 6254 2500 L 6254 2500 mt 6254 2500 L 898 1796 mt 6254 1796 L 6254 1796 mt 6254 1796 L 898 1092 mt 6254 1092 L 6254 1092 mt 6254 1092 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 981 4613 mt 981 4559 L 981 388 mt 981 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 856 4815 mt (0.1) s 1259 4613 mt 1259 4559 L 1259 388 mt 1259 442 L 1134 4815 mt (0.2) s 2091 4613 mt 2091 4559 L 2091 388 mt 2091 442 L 1966 4815 mt (0.5) s 3479 4613 mt 3479 4559 L 3479 388 mt 3479 442 L 3429 4815 mt (1) s 5976 4613 mt 5976 4559 L 5976 388 mt 5976 442 L 5851 4815 mt (1.9) s 5848 5091 mt (x 10) s /Helvetica /ISOLatin1Encoding 120 FMSR 6188 4980 mt (4) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L /Helvetica /ISOLatin1Encoding 180 FMSR 763 4679 mt (0) s 898 3909 mt 952 3909 L 6254 3909 mt 6200 3909 L 613 3975 mt (0.1) s 898 3205 mt 952 3205 L 6254 3205 mt 6200 3205 L 613 3271 mt (0.2) s 898 2500 mt 952 2500 L 6254 2500 mt 6200 2500 L 613 2566 mt (0.3) s 898 1796 mt 952 1796 L 6254 1796 mt 6200 1796 L 613 1862 mt (0.4) s 898 1092 mt 952 1092 L 6254 1092 mt 6200 1092 L 613 1158 mt (0.5) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np 2497 169 1388 212 832 225 278 155 981 3444 5 MP stroke gr 945 3444 mt 1017 3444 L 981 3408 mt 981 3480 L 956 3419 mt 1006 3469 L 1006 3419 mt 956 3469 L 1223 3599 mt 1295 3599 L 1259 3563 mt 1259 3635 L 1234 3574 mt 1284 3624 L 1284 3574 mt 1234 3624 L 2055 3824 mt 2127 3824 L 2091 3788 mt 2091 3860 L 2066 3799 mt 2116 3849 L 2116 3799 mt 2066 3849 L 3443 4036 mt 3515 4036 L 3479 4000 mt 3479 4072 L 3454 4011 mt 3504 4061 L 3504 4011 mt 3454 4061 L 5940 4205 mt 6012 4205 L 5976 4169 mt 5976 4241 L 5951 4180 mt 6001 4230 L 6001 4180 mt 5951 4230 L gs 898 388 5357 4226 MR c np 2497 140 1388 240 832 1113 278 478 981 2022 5 MP stroke gr 0 j 0 -58 -58 0 0 58 58 0 952 1993 5 MP DP 0 -58 -58 0 0 58 58 0 1230 2471 5 MP DP 0 -58 -58 0 0 58 58 0 2062 3584 5 MP DP 0 -58 -58 0 0 58 58 0 3450 3824 5 MP DP 0 -58 -58 0 0 58 58 0 5947 3964 5 MP DP gs 898 388 5357 4226 MR c np gr 2497 5001 mt (Number of database points) s 559 2884 mt -90 rotate (Miss ratio) s 90 rotate 2206 245 mt (alpha=2, n=19000, d=64, 10-NNS) s 1 sg 0 595 1819 0 0 -595 3285 1890 4 MP PP -1819 0 0 595 1819 0 0 -595 3285 1890 5 MP stroke DO 4 w SO 6 w 0 sg 3285 1890 mt 5104 1890 L 3285 1295 mt 5104 1295 L 5104 1890 mt 5104 1295 L 3285 1890 mt 3285 1295 L 3285 1890 mt 5104 1890 L 3285 1890 mt 3285 1295 L 3285 1295 mt 5104 1295 L 3285 1890 mt 5104 1890 L 3285 1890 mt 3285 1295 L 5104 1890 mt 5104 1295 L 4088 1559 mt (Error=.05) s gs 3285 1295 1820 596 MR c np 428 0 3392 1493 2 MP stroke gr 3356 1493 mt 3428 1493 L 3392 1457 mt 3392 1529 L 3367 1468 mt 3417 1518 L 3417 1468 mt 3367 1518 L 3784 1493 mt 3856 1493 L 3820 1457 mt 3820 1529 L 3795 1468 mt 3845 1518 L 3845 1468 mt 3795 1518 L gs 3285 1295 1820 596 MR c np gr 4088 1758 mt (Error=.1 ) s gs 3285 1295 1820 596 MR c np 428 0 3392 1692 2 MP stroke gr 0 -58 -58 0 0 58 58 0 3363 1663 5 MP DP 0 -58 -58 0 0 58 58 0 3791 1663 5 MP DP gs 3285 1295 1820 596 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 1277 1606 a Fo(\(b\))j(Appro)o(ximate)d(10-NNS)1199 1736 y(Figure)i(6:)k(Miss)c(ratio)g(vs.)k(data)13 b(size.)1020 1824 y Fi(4.3)48 b(T)l(exture)15 b(features)1020 1902 y Fo(The)k(exp)q(erimen)o(ts)g(with)f(texture)i(feature)g(data)e(w)o (ere)i(de-)1020 1949 y(signed)d(to)g(measure)g(the)h(p)q(erformance)f (of)f(the)i(LSH)f(algo-)1020 1996 y(rithm)j(on)h(large)f(data)h(sets;) 26 b(note)21 b(that)g(the)h(size)g(of)e(the)1020 2044 y(texture)d(\014le)e(\(270,000)e(p)q(oin)o(ts\))i(is)g(an)g(order)h(of) f(magnitude)1020 2091 y(larger)g(than)f(the)h(size)h(of)e(the)h (histogram)e(data)h(set)i(\(20,000)1020 2138 y(p)q(oin)o(ts\).)22 b(The)15 b(\014rst)h(step)g(\(i.e.,)e(the)i(c)o(hoice)f(of)g(the)h(n)o (um)o(b)q(er)1020 2186 y(of)j(sampled)f(bits)i Fn(k)q Fo(\))g(w)o(as)f(v)o(ery)h(similar)d(to)i(the)i(previous)1020 2233 y(exp)q(erimen)o(t,)13 b(therefore)i(w)o(e)f(skip)g(the)g (detailed)f(description)1020 2280 y(here.)19 b(W)m(e)13 b(just)h(state)g(that)g(w)o(e)g(assumed)f(that)h(the)g(n)o(um)o(b)q(er) 1020 2327 y(of)g(sampled)g(bits)h Fn(k)g Fo(=)f(65,)g(with)h(other)g (parameters)g(b)q(eing:)1020 2375 y(the)h(storage)h(o)o(v)o(erhead)f Fn(\013)e Fo(=)h(1,)g(blo)q(c)o(k)h(size)g Fn(B)h Fo(=)e(100,)g(and) 1020 2422 y(the)c(n)o(um)o(b)q(er)f(of)f(nearest)j(neigh)o(b)q(ors)f (equal)f(to)g(10.)16 b(As)11 b(stated)1020 2469 y(ab)q(o)o(v)o(e,)i (the)i(v)n(alue)e(of)g Fn(n)h Fo(w)o(as)g(equal)f(to)h(270,000.)1070 2521 y(W)m(e)21 b(v)n(aried)g(the)h(n)o(um)o(b)q(er)e(of)h(indices)h (from)d(3)i(to)g(100,)1020 2568 y(whic)o(h)g(resulted)h(in)f(error)g (from)f(50\045)g(to)g(15\045)g(\(see)j(Fig-)1020 2615 y(ure)d(9)g(\(a\)\).)35 b(The)20 b(shap)q(e)g(of)f(the)i(curv)o(e)f(is) g(similar)d(as)i(in)1020 2663 y(the)d(previous)f(exp)q(erimen)o(t.)22 b(The)16 b(miss)e(ratio)h(w)o(as)g(roughly)p eop %%Page: 10 10 10 9 bop 7 -33 a 14208860 12077528 3552215 12564316 35719495 40061091 startTexFig 7 -33 a %%BeginDocument: dim_2_1.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 442 30 5862 5013 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 1510 4613 mt 1510 388 L 1510 388 mt 1510 388 L 2964 4613 mt 2964 388 L 2964 388 mt 2964 388 L 5795 4613 mt 5795 388 L 5795 388 mt 5795 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 4191 mt 6254 4191 L 6254 4191 mt 6254 4191 L 898 3768 mt 6254 3768 L 6254 3768 mt 6254 3768 L 898 3346 mt 6254 3346 L 6254 3346 mt 6254 3346 L 898 2923 mt 6254 2923 L 6254 2923 mt 6254 2923 L 898 2501 mt 6254 2501 L 6254 2501 mt 6254 2501 L 898 2078 mt 6254 2078 L 6254 2078 mt 6254 2078 L 898 1656 mt 6254 1656 L 6254 1656 mt 6254 1656 L 898 1233 mt 6254 1233 L 6254 1233 mt 6254 1233 L 898 811 mt 6254 811 L 6254 811 mt 6254 811 L 898 388 mt 6254 388 L 6254 388 mt 6254 388 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 1510 4613 mt 1510 4559 L 1510 388 mt 1510 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 1460 4815 mt (8) s 2964 4613 mt 2964 4559 L 2964 388 mt 2964 442 L 2864 4815 mt (27) s 5795 4613 mt 5795 4559 L 5795 388 mt 5795 442 L 5695 4815 mt (64) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 4191 mt 952 4191 L 6254 4191 mt 6200 4191 L 763 4257 mt (2) s 898 3768 mt 952 3768 L 6254 3768 mt 6200 3768 L 763 3834 mt (4) s 898 3346 mt 952 3346 L 6254 3346 mt 6200 3346 L 763 3412 mt (6) s 898 2923 mt 952 2923 L 6254 2923 mt 6200 2923 L 763 2989 mt (8) s 898 2501 mt 952 2501 L 6254 2501 mt 6200 2501 L 663 2567 mt (10) s 898 2078 mt 952 2078 L 6254 2078 mt 6200 2078 L 663 2144 mt (12) s 898 1656 mt 952 1656 L 6254 1656 mt 6200 1656 L 663 1722 mt (14) s 898 1233 mt 952 1233 L 6254 1233 mt 6200 1233 L 663 1299 mt (16) s 898 811 mt 952 811 L 6254 811 mt 6200 811 L 663 877 mt (18) s 898 388 mt 952 388 L 6254 388 mt 6200 388 L 663 454 mt (20) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np DA 2831 -1829 1454 -997 1510 3622 3 MP stroke gr DA SO 0 j 0 -58 -58 0 0 58 58 0 1481 3593 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 2935 2596 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 5766 767 5 MP DP DA gs 898 388 5357 4226 MR c np SO 2831 -634 1454 -422 1510 3768 3 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1481 3739 5 MP DP 0 -58 -58 0 0 58 58 0 2935 3317 5 MP DP 0 -58 -58 0 0 58 58 0 5766 2683 5 MP DP gs 898 388 5357 4226 MR c np 2831 -422 1454 0 1510 3979 3 MP stroke gr 1474 3979 mt 1546 3979 L 1510 3943 mt 1510 4015 L 1485 3954 mt 1535 4004 L 1535 3954 mt 1485 4004 L 2928 3979 mt 3000 3979 L 2964 3943 mt 2964 4015 L 2939 3954 mt 2989 4004 L 2989 3954 mt 2939 4004 L 5759 3557 mt 5831 3557 L 5795 3521 mt 5795 3593 L 5770 3532 mt 5820 3582 L 5820 3532 mt 5770 3582 L gs 898 388 5357 4226 MR c np 2831 -211 1454 -212 1510 4191 3 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1510 4238 5 MP DP 36 47 -36 47 -36 -47 36 -47 2964 4026 5 MP DP 36 47 -36 47 -36 -47 36 -47 5795 3815 5 MP DP gs 898 388 5357 4226 MR c np 2831 -212 1454 0 1510 4191 3 MP stroke gr 36 36 1510 4191 FO 36 36 2964 4191 FO 36 36 5795 3979 FO gs 898 388 5357 4226 MR c np gr 3108 5001 mt (Dimensions) s 609 3074 mt -90 rotate (Disk accesses) s 90 rotate 2862 245 mt (alpha = 2, 1-NNS) s 1 sg 0 1448 2375 0 0 -1448 1149 2064 4 MP PP -2375 0 0 1448 2375 0 0 -1448 1149 2064 5 MP stroke DO 4 w SO 6 w 0 sg 1149 2064 mt 3524 2064 L 1149 616 mt 3524 616 L 3524 2064 mt 3524 616 L 1149 2064 mt 1149 616 L 1149 2064 mt 3524 2064 L 1149 2064 mt 1149 616 L 1149 616 mt 3524 616 L 1149 2064 mt 3524 2064 L 1149 2064 mt 1149 616 L 3524 2064 mt 3524 616 L 1952 923 mt (SR-Tree ) s gs 1149 616 2376 1449 MR c np DA 429 0 1256 857 2 MP stroke gr DA SO 0 -58 -58 0 0 58 58 0 1227 828 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1656 828 5 MP DP DA gs 1149 616 2376 1449 MR c np gr 1952 1165 mt (LSH, error=.02) s gs 1149 616 2376 1449 MR c np SO 429 0 1256 1099 2 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1227 1070 5 MP DP 0 -58 -58 0 0 58 58 0 1656 1070 5 MP DP gs 1149 616 2376 1449 MR c np gr 1952 1406 mt (LSH, error=.05) s gs 1149 616 2376 1449 MR c np 429 0 1256 1340 2 MP stroke gr 1220 1340 mt 1292 1340 L 1256 1304 mt 1256 1376 L 1231 1315 mt 1281 1365 L 1281 1315 mt 1231 1365 L 1649 1340 mt 1721 1340 L 1685 1304 mt 1685 1376 L 1660 1315 mt 1710 1365 L 1710 1315 mt 1660 1365 L gs 1149 616 2376 1449 MR c np gr 1952 1647 mt (LSH, error=.1 ) s gs 1149 616 2376 1449 MR c np 429 0 1256 1581 2 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1256 1628 5 MP DP 36 47 -36 47 -36 -47 36 -47 1685 1628 5 MP DP gs 1149 616 2376 1449 MR c np gr 1952 1889 mt (LSH, error=.2 ) s gs 1149 616 2376 1449 MR c np 429 0 1256 1823 2 MP stroke gr 36 36 1256 1823 FO 36 36 1685 1823 FO gs 1149 616 2376 1449 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 239 779 a Fo(\(a\))14 b(Appro)o(ximate)e(1-NNS)7 794 y 14208860 12077528 3552215 12564316 35719495 40061091 startTexFig 7 794 a %%BeginDocument: dim_2_10.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 442 30 5862 5013 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 1510 4613 mt 1510 388 L 1510 388 mt 1510 388 L 2964 4613 mt 2964 388 L 2964 388 mt 2964 388 L 5795 4613 mt 5795 388 L 5795 388 mt 5795 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 4009 mt 6254 4009 L 6254 4009 mt 6254 4009 L 898 3406 mt 6254 3406 L 6254 3406 mt 6254 3406 L 898 2802 mt 6254 2802 L 6254 2802 mt 6254 2802 L 898 2199 mt 6254 2199 L 6254 2199 mt 6254 2199 L 898 1595 mt 6254 1595 L 6254 1595 mt 6254 1595 L 898 992 mt 6254 992 L 6254 992 mt 6254 992 L 898 388 mt 6254 388 L 6254 388 mt 6254 388 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 1510 4613 mt 1510 4559 L 1510 388 mt 1510 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 1460 4815 mt (8) s 2964 4613 mt 2964 4559 L 2964 388 mt 2964 442 L 2864 4815 mt (27) s 5795 4613 mt 5795 4559 L 5795 388 mt 5795 442 L 5695 4815 mt (64) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 4009 mt 952 4009 L 6254 4009 mt 6200 4009 L 763 4075 mt (5) s 898 3406 mt 952 3406 L 6254 3406 mt 6200 3406 L 663 3472 mt (10) s 898 2802 mt 952 2802 L 6254 2802 mt 6200 2802 L 663 2868 mt (15) s 898 2199 mt 952 2199 L 6254 2199 mt 6200 2199 L 663 2265 mt (20) s 898 1595 mt 952 1595 L 6254 1595 mt 6200 1595 L 663 1661 mt (25) s 898 992 mt 952 992 L 6254 992 mt 6200 992 L 663 1058 mt (30) s 898 388 mt 952 388 L 6254 388 mt 6200 388 L 663 454 mt (35) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np DA 2831 -1790 1454 -901 1510 3709 3 MP stroke gr DA SO 0 j 0 -58 -58 0 0 58 58 0 1481 3680 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 2935 2779 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 5766 989 5 MP DP DA gs 898 388 5357 4226 MR c np SO 2831 -121 1454 -724 1510 4251 3 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1481 4222 5 MP DP 0 -58 -58 0 0 58 58 0 2935 3498 5 MP DP 0 -58 -58 0 0 58 58 0 5766 3377 5 MP DP gs 898 388 5357 4226 MR c np 2831 -242 1454 -362 1510 4251 3 MP stroke gr 1474 4251 mt 1546 4251 L 1510 4215 mt 1510 4287 L 1485 4226 mt 1535 4276 L 1535 4226 mt 1485 4276 L 2928 3889 mt 3000 3889 L 2964 3853 mt 2964 3925 L 2939 3864 mt 2989 3914 L 2989 3864 mt 2939 3914 L 5759 3647 mt 5831 3647 L 5795 3611 mt 5795 3683 L 5770 3622 mt 5820 3672 L 5820 3622 mt 5770 3672 L gs 898 388 5357 4226 MR c np 2831 -121 1454 -242 1510 4372 3 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1510 4419 5 MP DP 36 47 -36 47 -36 -47 36 -47 2964 4177 5 MP DP 36 47 -36 47 -36 -47 36 -47 5795 4056 5 MP DP gs 898 388 5357 4226 MR c np 2831 -121 1454 0 1510 4372 3 MP stroke gr 36 36 1510 4372 FO 36 36 2964 4372 FO 36 36 5795 4251 FO gs 898 388 5357 4226 MR c np gr 3108 5001 mt (Dimensions) s 609 3075 mt -90 rotate (Disk accesses) s 90 rotate 2812 245 mt (alpha = 2, 10-NNS) s 1 sg 0 1448 2374 0 0 -1448 1174 2224 4 MP PP -2374 0 0 1448 2374 0 0 -1448 1174 2224 5 MP stroke DO 4 w SO 6 w 0 sg 1174 2224 mt 3548 2224 L 1174 776 mt 3548 776 L 3548 2224 mt 3548 776 L 1174 2224 mt 1174 776 L 1174 2224 mt 3548 2224 L 1174 2224 mt 1174 776 L 1174 776 mt 3548 776 L 1174 2224 mt 3548 2224 L 1174 2224 mt 1174 776 L 3548 2224 mt 3548 776 L 1977 1083 mt (SR-Tree ) s gs 1174 776 2375 1449 MR c np DA 428 0 1281 1017 2 MP stroke gr DA SO 0 -58 -58 0 0 58 58 0 1252 988 5 MP DP DA SO 0 -58 -58 0 0 58 58 0 1680 988 5 MP DP DA gs 1174 776 2375 1449 MR c np gr 1977 1325 mt (LSH, error=.02) s gs 1174 776 2375 1449 MR c np SO 428 0 1281 1259 2 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1252 1230 5 MP DP 0 -58 -58 0 0 58 58 0 1680 1230 5 MP DP gs 1174 776 2375 1449 MR c np gr 1977 1566 mt (LSH, error=.05) s gs 1174 776 2375 1449 MR c np 428 0 1281 1500 2 MP stroke gr 1245 1500 mt 1317 1500 L 1281 1464 mt 1281 1536 L 1256 1475 mt 1306 1525 L 1306 1475 mt 1256 1525 L 1673 1500 mt 1745 1500 L 1709 1464 mt 1709 1536 L 1684 1475 mt 1734 1525 L 1734 1475 mt 1684 1525 L gs 1174 776 2375 1449 MR c np gr 1977 1807 mt (LSH, error=.1 ) s gs 1174 776 2375 1449 MR c np 428 0 1281 1741 2 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1281 1788 5 MP DP 36 47 -36 47 -36 -47 36 -47 1709 1788 5 MP DP gs 1174 776 2375 1449 MR c np gr 1977 2049 mt (LSH, error=.2 ) s gs 1174 776 2375 1449 MR c np 428 0 1281 1983 2 MP stroke gr 36 36 1281 1983 FO 36 36 1709 1983 FO gs 1174 776 2375 1449 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 227 1606 a Fo(\(b\))j(Appro)o(ximate)d(10-NNS)64 1736 y(Figure)i(7:)j(Num)o(b)q(er)d(of)f(indices)h(vs.)19 b(dimension.)-30 1844 y(4\045)13 b(for)h(3)f(indices,)h(1\045)f(for)h (5)f(indices,)h(and)g(0\045)f(otherwise.)20 1906 y(T)m(o)d(compare)h (with)g(SR-tree,)h(w)o(e)f(implemen)o(ted)e(that)j(lat-)-30 1953 y(ter)18 b(on)f(random)e(subsets)k(of)e(the)h(whole)e(data)h(set)h (of)f(sizes)-30 2000 y(from)12 b(10)p Fn(;)7 b Fo(000)12 b(to)h(200)p Fn(;)7 b Fo(000.)16 b(F)m(or)e Fn(n)d Fo(=)h(200)p Fn(;)7 b Fo(000)12 b(the)i(a)o(v)o(erage)-30 2048 y(n)o(um)o(b)q(er)i (of)g(blo)q(c)o(ks)h(accessed)j(p)q(er)d(query)h(b)o(y)e(SR-tree)i(w)o (as)-30 2095 y(1310,)d(whic)o(h)h(is)g(one)g(to)g(t)o(w)o(o)f(orders)i (of)f(magnitude)e(larger)-30 2142 y(than)h(the)h(n)o(um)o(b)q(er)f(of)f (blo)q(c)o(ks)i(accessed)h(b)o(y)e(our)h(algorithm)-30 2190 y(\(see)f(Figure)g(9)f(\(b\))g(where)h(w)o(e)g(sho)o(w)f(the)h (running)e(times)h(of)-30 2237 y(LSH)g(for)f(e\013ectiv)o(e)i(error)g (15\045\).)i(Observ)o(e)f(though)d(that)h(an)-30 2284 y(SR-tree)19 b(computes)e(exact)i(answ)o(ers)g(while)e(LSH)h(pro)o (vides)-30 2332 y(only)d(an)h(appro)o(ximation.)k(Th)o(us)d(in)e(order) i(to)e(p)q(erform)g(an)-30 2379 y(accurate)21 b(ev)n(aluation)e(of)g (LSH,)h(w)o(e)g(decided)h(to)f(compare)-30 2426 y(it)c(with)g(a)f(mo)q (di\014ed)g(SR-tree)i(algorithm)c(whic)o(h)j(pro)q(duces)-30 2473 y Fk(appr)n(oximate)f Fo(answ)o(ers.)23 b(The)16 b(mo)q(di\014cation)d(is)i(simple:)k(in-)-30 2521 y(stead)12 b(of)e(running)h(SR-tree)h(on)f(the)h(whole)e(data)h(set,)h(w)o(e)g (run)-30 2568 y(it)k(on)f(a)h(randomly)e(c)o(hosen)i(subset)i(of)d(it.) 24 b(In)16 b(this)g(w)o(a)o(y)f(w)o(e)-30 2615 y(ac)o(hiev)o(e)k(a)f (sp)q(eed-up)i(of)e(the)h(algorithm)d(\(as)i(the)i(random)-30 2663 y(sample)11 b(of)h(the)i(data)e(set)i(is)e(smaller)f(than)i(the)g (original)e(set\))1057 -33 y 14208860 12077528 3552215 12564316 35719495 40061091 startTexFig 1057 -33 a %%BeginDocument: knn.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 442 30 5862 5013 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6912 5185 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke DO 4 w 0 sg 1389 4613 mt 1389 388 L 1389 388 mt 1389 388 L 1791 4613 mt 1791 388 L 1791 388 mt 1791 388 L 2237 4613 mt 2237 388 L 2237 388 mt 2237 388 L 3576 4613 mt 3576 388 L 3576 388 mt 3576 388 L 5808 4613 mt 5808 388 L 5808 388 mt 5808 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 3205 mt 6254 3205 L 6254 3205 mt 6254 3205 L 898 1796 mt 6254 1796 L 6254 1796 mt 6254 1796 L 898 388 mt 6254 388 L 6254 388 mt 6254 388 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 1389 4613 mt 1389 4559 L 1389 388 mt 1389 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 1339 4815 mt (1) s 1791 4613 mt 1791 4559 L 1791 388 mt 1791 442 L 1691 4815 mt (10) s 2237 4613 mt 2237 4559 L 2237 388 mt 2237 442 L 2137 4815 mt (20) s 3576 4613 mt 3576 4559 L 3576 388 mt 3576 442 L 3476 4815 mt (50) s 5808 4613 mt 5808 4559 L 5808 388 mt 5808 442 L 5658 4815 mt (100) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 3205 mt 952 3205 L 6254 3205 mt 6200 3205 L 763 3271 mt (5) s 898 1796 mt 952 1796 L 6254 1796 mt 6200 1796 L 663 1862 mt (10) s 898 388 mt 952 388 L 6254 388 mt 6200 388 L 663 454 mt (15) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np 2232 -1126 1339 -282 446 -282 402 -845 1389 3205 5 MP stroke gr 1353 3205 mt 1425 3205 L 1389 3169 mt 1389 3241 L 1364 3180 mt 1414 3230 L 1414 3180 mt 1364 3230 L 1755 2360 mt 1827 2360 L 1791 2324 mt 1791 2396 L 1766 2335 mt 1816 2385 L 1816 2335 mt 1766 2385 L 2201 2078 mt 2273 2078 L 2237 2042 mt 2237 2114 L 2212 2053 mt 2262 2103 L 2262 2053 mt 2212 2103 L 3540 1796 mt 3612 1796 L 3576 1760 mt 3576 1832 L 3551 1771 mt 3601 1821 L 3601 1771 mt 3551 1821 L 5772 670 mt 5844 670 L 5808 634 mt 5808 706 L 5783 645 mt 5833 695 L 5833 645 mt 5783 695 L gs 898 388 5357 4226 MR c np 2232 -563 1339 0 446 -282 402 -563 1389 3486 5 MP stroke gr 0 j 0 -58 -58 0 0 58 58 0 1360 3457 5 MP DP 0 -58 -58 0 0 58 58 0 1762 2894 5 MP DP 0 -58 -58 0 0 58 58 0 2208 2612 5 MP DP 0 -58 -58 0 0 58 58 0 3547 2612 5 MP DP 0 -58 -58 0 0 58 58 0 5779 2049 5 MP DP gs 898 388 5357 4226 MR c np 2232 -282 1339 -281 446 -282 402 0 1389 3768 5 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1389 3815 5 MP DP 36 47 -36 47 -36 -47 36 -47 1791 3815 5 MP DP 36 47 -36 47 -36 -47 36 -47 2237 3533 5 MP DP 36 47 -36 47 -36 -47 36 -47 3576 3252 5 MP DP 36 47 -36 47 -36 -47 36 -47 5808 2970 5 MP DP gs 898 388 5357 4226 MR c np gr 2414 5001 mt (Number of nearest neighbors) s 609 3075 mt -90 rotate (Disk accesses) s 90 rotate 2599 245 mt (alpha=2, n=19000, d=64) s 1 sg 0 880 1819 0 0 -880 1458 1520 4 MP PP -1819 0 0 880 1819 0 0 -880 1458 1520 5 MP stroke DO 4 w SO 6 w 0 sg 1458 1520 mt 3277 1520 L 1458 640 mt 3277 640 L 3277 1520 mt 3277 640 L 1458 1520 mt 1458 640 L 1458 1520 mt 3277 1520 L 1458 1520 mt 1458 640 L 1458 640 mt 3277 640 L 1458 1520 mt 3277 1520 L 1458 1520 mt 1458 640 L 3277 1520 mt 3277 640 L 2261 926 mt (Error=.05) s gs 1458 640 1820 881 MR c np 428 0 1565 860 2 MP stroke gr 1529 860 mt 1601 860 L 1565 824 mt 1565 896 L 1540 835 mt 1590 885 L 1590 835 mt 1540 885 L 1957 860 mt 2029 860 L 1993 824 mt 1993 896 L 1968 835 mt 2018 885 L 2018 835 mt 1968 885 L gs 1458 640 1820 881 MR c np gr 2261 1146 mt (Error=.1 ) s gs 1458 640 1820 881 MR c np 428 0 1565 1080 2 MP stroke gr 0 -58 -58 0 0 58 58 0 1536 1051 5 MP DP 0 -58 -58 0 0 58 58 0 1964 1051 5 MP DP gs 1458 640 1820 881 MR c np gr 2261 1366 mt (Error=.2 ) s gs 1458 640 1820 881 MR c np 428 0 1565 1300 2 MP stroke gr 36 47 -36 47 -36 -47 36 -47 1565 1347 5 MP DP 36 47 -36 47 -36 -47 36 -47 1993 1347 5 MP DP gs 1458 640 1820 881 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 1020 862 a Fo(Figure)19 b(8:)27 b(Num)o(b)q(er)18 b(of)h(indices)g (vs.)32 b(n)o(um)o(b)q(er)18 b(of)h(nearest)1020 909 y(neigh)o(b)q(ors.)1020 1008 y(while)13 b(incurring)h(some)f(error.) 1070 1058 y(The)e(query)g(cost)g(v)o(ersus)g(error)h(tradeo\013)f (obtained)f(in)g(this)1020 1105 y(w)o(a)o(y)15 b(\(for)h(the)g(en)o (tire)g(data)g(set\))h(is)e(depicted)i(on)f(Figure)g(9;)1020 1152 y(w)o(e)e(also)f(include)h(a)g(similar)d(graph)j(for)g(LSH.)1070 1202 y(Observ)o(e)f(that)e(using)g(random)f(sampling)f(results)k(in)e (con-)1020 1249 y(siderable)22 b(sp)q(eed-up)h(for)d(the)i(SR-tree)g (algorithm,)e(while)1020 1296 y(k)o(eeping)i(the)g(error)g(relativ)o (ely)f(lo)o(w.)41 b(Ho)o(w)o(ev)o(er,)23 b(ev)o(en)f(in)1020 1344 y(this)c(case)h(the)g(LSH)f(algorithm)d(o\013ers)k(considerably)f (out-)1020 1391 y(p)q(erforms)13 b(SR-trees,)h(b)q(eing)f(up)h(to)f(an) g(order)h(of)f(magnitude)1020 1438 y(faster.)1020 1551 y Fv(5)56 b(Previous)18 b(W)-5 b(ork)1020 1625 y Fo(There)16 b(is)e(considerable)h(literature)g(on)g(v)n(arious)f(v)o(ersions)h(of) 1020 1672 y(the)d(nearest)i(neigh)o(b)q(or)e(problem.)k(Due)c(to)f(lac) o(k)h(of)f(space)i(w)o(e)1020 1719 y(omit)i(detailed)i(description)h (of)f(related)h(w)o(ork;)g(the)g(reader)1020 1767 y(is)13 b(advised)g(to)f(read)h([39)o(])g(for)f(a)h(surv)o(ey)g(of)f(a)h(v)n (ariet)o(y)f(of)g(data)1020 1814 y(structures)20 b(for)e(nearest)h (neigh)o(b)q(ors)f(in)g(geometric)f(spaces,)1020 1861 y(including)11 b(v)n(arian)o(ts)g(of)h Fn(k)q Fo(-d)g(trees,)h Fn(R)p Fo(-trees,)h(and)d(structures)1020 1909 y(based)k(on)e (space-\014lling)h(curv)o(es.)20 b(The)14 b(more)f(recen)o(t)j(results) 1020 1956 y(are)c(surv)o(ey)o(ed)g(in)f([41)o(];)g(see)h(also)f(an)g (excellen)o(t)h(surv)o(ey)g(b)o(y)f([4)o(].)1020 2003 y(Recen)o(t)17 b(theoretical)f(w)o(ork)g(in)f(nearest)i(neigh)o(b)q(or) f(searc)o(h)h(is)1020 2051 y(brie\015y)d(surv)o(ey)o(ed)h(in)f([24)o (].)1020 2163 y Fv(6)56 b(Conclusions)1020 2237 y Fo(W)m(e)20 b(presen)o(ted)i(a)d(no)o(v)o(el)g(sc)o(heme)i(for)e(appro)o(ximate)f (simi-)1020 2284 y(larit)o(y)e(searc)o(h)i(based)g(on)f(lo)q(calit)o (y-sensitiv)o(e)f(hashing.)27 b(W)m(e)1020 2332 y(compared)15 b(the)i(p)q(erformance)e(of)g(this)h(tec)o(hnique)h(and)f(SR-)1020 2379 y(tree,)i(a)d(go)q(o)q(d)h(represen)o(tativ)o(e)i(of)e(tree-based) i(spatial)d(data)1020 2426 y(structures.)35 b(W)m(e)18 b(sho)o(w)o(ed)h(that)g(b)o(y)f(allo)o(wing)f(small)f(error)1020 2473 y(and)f(additional)e(storage)i(o)o(v)o(erhead,)g(w)o(e)h(can)f (considerably)1020 2521 y(impro)o(v)o(e)d(the)i(query)g(time.)j(Exp)q (erimen)o(tal)12 b(results)j(also)e(in-)1020 2568 y(dicate)e(that)h (our)f(sc)o(heme)g(scales)h(w)o(ell)e(to)h(ev)o(en)h(a)f(large)g(n)o (um-)1020 2615 y(b)q(er)i(of)f(dimensions)e(and)i(data)g(size.)18 b(An)13 b(additional)d(adv)n(an-)1020 2663 y(tage)16 b(of)g(our)h(data)f(structure)i(is)e(that)h(its)f(running)g(time)f(is)p eop %%Page: 11 11 11 10 bop 7 -33 a 14208860 11935440 2697052 12235407 36245749 40455782 startTexFig 7 -33 a %%BeginDocument: LSH-SR.eps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 280 -38 6124 5142 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5357 0 0 -4226 898 4615 4 MP PP -5357 0 0 4226 5357 0 0 -4226 898 4615 5 MP stroke 4 w DO 0 sg 898 4615 mt 898 389 L 898 389 mt 898 389 L 1568 4615 mt 1568 389 L 1568 389 mt 1568 389 L 2237 4615 mt 2237 389 L 2237 389 mt 2237 389 L 2907 4615 mt 2907 389 L 2907 389 mt 2907 389 L 3577 4615 mt 3577 389 L 3577 389 mt 3577 389 L 4246 4615 mt 4246 389 L 4246 389 mt 4246 389 L 4916 4615 mt 4916 389 L 4916 389 mt 4916 389 L 5585 4615 mt 5585 389 L 5585 389 mt 5585 389 L 6255 4615 mt 6255 389 L 6255 389 mt 6255 389 L 898 4615 mt 6255 4615 L 6255 4615 mt 6255 4615 L 898 4145 mt 6255 4145 L 6255 4145 mt 6255 4145 L 898 3676 mt 6255 3676 L 6255 3676 mt 6255 3676 L 898 3206 mt 6255 3206 L 6255 3206 mt 6255 3206 L 898 2737 mt 6255 2737 L 6255 2737 mt 6255 2737 L 898 2267 mt 6255 2267 L 6255 2267 mt 6255 2267 L 898 1798 mt 6255 1798 L 6255 1798 mt 6255 1798 L 898 1328 mt 6255 1328 L 6255 1328 mt 6255 1328 L 898 859 mt 6255 859 L 6255 859 mt 6255 859 L 898 389 mt 6255 389 L 6255 389 mt 6255 389 L SO 6 w 898 4615 mt 6255 4615 L 898 389 mt 6255 389 L 6255 4615 mt 6255 389 L 898 4615 mt 898 389 L 898 4615 mt 6255 4615 L 898 4615 mt 898 389 L 898 4615 mt 898 4561 L 898 389 mt 898 443 L /Helvetica /ISOLatin1Encoding 180 FMSR 798 4817 mt (10) s 1568 4615 mt 1568 4561 L 1568 389 mt 1568 443 L 1468 4817 mt (15) s 2237 4615 mt 2237 4561 L 2237 389 mt 2237 443 L 2137 4817 mt (20) s 2907 4615 mt 2907 4561 L 2907 389 mt 2907 443 L 2807 4817 mt (25) s 3577 4615 mt 3577 4561 L 3577 389 mt 3577 443 L 3477 4817 mt (30) s 4246 4615 mt 4246 4561 L 4246 389 mt 4246 443 L 4146 4817 mt (35) s 4916 4615 mt 4916 4561 L 4916 389 mt 4916 443 L 4816 4817 mt (40) s 5585 4615 mt 5585 4561 L 5585 389 mt 5585 443 L 5485 4817 mt (45) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L 6155 4817 mt (50) s 898 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 763 4681 mt (0) s 898 4145 mt 952 4145 L 6255 4145 mt 6201 4145 L 663 4211 mt (50) s 898 3676 mt 952 3676 L 6255 3676 mt 6201 3676 L 563 3742 mt (100) s 898 3206 mt 952 3206 L 6255 3206 mt 6201 3206 L 563 3272 mt (150) s 898 2737 mt 952 2737 L 6255 2737 mt 6201 2737 L 563 2803 mt (200) s 898 2267 mt 952 2267 L 6255 2267 mt 6201 2267 L 563 2333 mt (250) s 898 1798 mt 952 1798 L 6255 1798 mt 6201 1798 L 563 1864 mt (300) s 898 1328 mt 952 1328 L 6255 1328 mt 6201 1328 L 563 1394 mt (350) s 898 859 mt 952 859 L 6255 859 mt 6201 859 L 563 925 mt (400) s 898 389 mt 952 389 L 6255 389 mt 6201 389 L 563 455 mt (450) s 898 389 mt 6255 389 L 898 4615 mt 6255 4615 L 898 4615 mt 898 389 L 6255 4615 mt 6255 389 L gs 898 389 5358 4227 MR c np 1339 460 603 366 469 385 1071 2132 1300 624 5 MP stroke gr 0 j 0 -58 -58 0 0 58 58 0 1271 595 5 MP DP 0 -58 -58 0 0 58 58 0 2342 2727 5 MP DP 0 -58 -58 0 0 58 58 0 2811 3112 5 MP DP 0 -58 -58 0 0 58 58 0 3414 3478 5 MP DP 0 -58 -58 0 0 58 58 0 4753 3938 5 MP DP gs 898 389 5358 4227 MR c np -134 -282 -267 -187 -536 -188 -402 -94 -1071 -94 -1474 -47 5318 4568 7 MP stroke gr 5282 4568 mt 5354 4568 L 5318 4532 mt 5318 4604 L 5293 4543 mt 5343 4593 L 5343 4543 mt 5293 4593 L 3808 4521 mt 3880 4521 L 3844 4485 mt 3844 4557 L 3819 4496 mt 3869 4546 L 3869 4496 mt 3819 4546 L 2737 4427 mt 2809 4427 L 2773 4391 mt 2773 4463 L 2748 4402 mt 2798 4452 L 2798 4402 mt 2748 4452 L 2335 4333 mt 2407 4333 L 2371 4297 mt 2371 4369 L 2346 4308 mt 2396 4358 L 2396 4308 mt 2346 4358 L 1799 4145 mt 1871 4145 L 1835 4109 mt 1835 4181 L 1810 4120 mt 1860 4170 L 1860 4120 mt 1810 4170 L 1532 3958 mt 1604 3958 L 1568 3922 mt 1568 3994 L 1543 3933 mt 1593 3983 L 1593 3933 mt 1543 3983 L 1398 3676 mt 1470 3676 L 1434 3640 mt 1434 3712 L 1409 3651 mt 1459 3701 L 1459 3651 mt 1409 3701 L gs 898 389 5358 4227 MR c np gr /Helvetica /ISOLatin1Encoding 240 FMSR 3092 5048 mt (Error \(%\)) s 508 3269 mt -90 rotate (Disk accesses) s 90 rotate 2454 233 mt (Performance vs error) s 1 sg 0 568 1696 0 0 -568 4421 1060 4 MP PP -1696 0 0 568 1696 0 0 -568 4421 1060 5 MP stroke 4 w DO SO 6 w 0 sg 4421 1060 mt 6117 1060 L 4421 492 mt 6117 492 L 6117 1060 mt 6117 492 L 4421 1060 mt 4421 492 L 4421 1060 mt 6117 1060 L 4421 1060 mt 4421 492 L 4421 492 mt 6117 492 L 4421 1060 mt 6117 1060 L 4421 1060 mt 4421 492 L 6117 1060 mt 6117 492 L /Helvetica /ISOLatin1Encoding 192 FMSR 5224 749 mt (SR-Tree) s gs 4421 492 1697 569 MR c np 429 0 4528 681 2 MP stroke gr 0 -58 -58 0 0 58 58 0 4499 652 5 MP DP 0 -58 -58 0 0 58 58 0 4928 652 5 MP DP gs 4421 492 1697 569 MR c np gr 5224 939 mt (LSH ) s gs 4421 492 1697 569 MR c np 429 0 4528 871 2 MP stroke gr 4492 871 mt 4564 871 L 4528 835 mt 4528 907 L 4503 846 mt 4553 896 L 4553 846 mt 4503 896 L 4921 871 mt 4993 871 L 4957 835 mt 4957 907 L 4932 846 mt 4982 896 L 4982 846 mt 4932 896 L gs 4421 492 1697 569 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 431 770 a Fo(\(a\))7 785 y 14208860 10940818 2170798 12235407 36508876 38942801 startTexFig 7 785 a %%BeginDocument: scaling.eps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 180 239 6273 4863 MR c np 88 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4225 5356 0 0 -4225 898 4613 4 MP PP -5356 0 0 4225 5356 0 0 -4225 898 4613 5 MP stroke 4 w DO 0 sg 898 4613 mt 898 388 L 898 388 mt 898 388 L 2237 4613 mt 2237 388 L 2237 388 mt 2237 388 L 3576 4613 mt 3576 388 L 3576 388 mt 3576 388 L 4915 4613 mt 4915 388 L 4915 388 mt 4915 388 L 6254 4613 mt 6254 388 L 6254 388 mt 6254 388 L 898 4613 mt 6254 4613 L 6254 4613 mt 6254 4613 L 898 4009 mt 6254 4009 L 6254 4009 mt 6254 4009 L 898 3406 mt 6254 3406 L 6254 3406 mt 6254 3406 L 898 2802 mt 6254 2802 L 6254 2802 mt 6254 2802 L 898 2199 mt 6254 2199 L 6254 2199 mt 6254 2199 L 898 1595 mt 6254 1595 L 6254 1595 mt 6254 1595 L 898 992 mt 6254 992 L 6254 992 mt 6254 992 L 898 388 mt 6254 388 L 6254 388 mt 6254 388 L SO 6 w 898 4613 mt 6254 4613 L 898 388 mt 6254 388 L 6254 4613 mt 6254 388 L 898 4613 mt 898 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 898 4613 mt 898 4559 L 898 388 mt 898 442 L /Helvetica /ISOLatin1Encoding 180 FMSR 848 4815 mt (0) s 2237 4613 mt 2237 4559 L 2237 388 mt 2237 442 L 2137 4815 mt (50) s 3576 4613 mt 3576 4559 L 3576 388 mt 3576 442 L 3426 4815 mt (100) s 4915 4613 mt 4915 4559 L 4915 388 mt 4915 442 L 4765 4815 mt (150) s 6254 4613 mt 6254 4559 L 6254 388 mt 6254 442 L 6104 4815 mt (200) s 898 4613 mt 952 4613 L 6254 4613 mt 6200 4613 L 763 4679 mt (0) s 898 4009 mt 952 4009 L 6254 4009 mt 6200 4009 L 563 4075 mt (200) s 898 3406 mt 952 3406 L 6254 3406 mt 6200 3406 L 563 3472 mt (400) s 898 2802 mt 952 2802 L 6254 2802 mt 6200 2802 L 563 2868 mt (600) s 898 2199 mt 952 2199 L 6254 2199 mt 6200 2199 L 563 2265 mt (800) s 898 1595 mt 952 1595 L 6254 1595 mt 6200 1595 L 463 1661 mt (1000) s 898 992 mt 952 992 L 6254 992 mt 6200 992 L 463 1058 mt (1200) s 898 388 mt 952 388 L 6254 388 mt 6200 388 L 463 454 mt (1400) s 898 388 mt 6254 388 L 898 4613 mt 6254 4613 L 898 4613 mt 898 388 L 6254 4613 mt 6254 388 L gs 898 388 5357 4226 MR c np DA 2678 -1656 1339 -1014 803 -685 268 -242 1166 4257 5 MP stroke gr DA SO 0 j 36 47 -36 47 -36 -47 36 -47 1166 4304 5 MP DP DA SO 36 47 -36 47 -36 -47 36 -47 1434 4062 5 MP DP DA SO 36 47 -36 47 -36 -47 36 -47 2237 3377 5 MP DP DA SO 36 47 -36 47 -36 -47 36 -47 3576 2363 5 MP DP DA SO 36 47 -36 47 -36 -47 36 -47 6254 707 5 MP DP DA gs 898 388 5357 4226 MR c np SO 2678 -91 1339 0 803 -15 268 0 1166 4598 5 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1137 4569 5 MP DP 0 -58 -58 0 0 58 58 0 1405 4569 5 MP DP 0 -58 -58 0 0 58 58 0 2208 4554 5 MP DP 0 -58 -58 0 0 58 58 0 3547 4554 5 MP DP 0 -58 -58 0 0 58 58 0 6225 4463 5 MP DP gs 898 388 5357 4226 MR c np gr /Helvetica /ISOLatin1Encoding 240 FMSR 408 3872 mt -90 rotate (Number of Disk Accesses) s 90 rotate 2844 5046 mt (Data Set Size) s 1 sg 0 652 1794 0 0 -652 1285 1342 4 MP PP -1794 0 0 652 1794 0 0 -652 1285 1342 5 MP stroke 4 w DO SO 6 w 0 sg 1285 1342 mt 3079 1342 L 1285 690 mt 3079 690 L 3079 1342 mt 3079 690 L 1285 1342 mt 1285 690 L 1285 1342 mt 3079 1342 L 1285 1342 mt 1285 690 L 1285 690 mt 3079 690 L 1285 1342 mt 3079 1342 L 1285 1342 mt 1285 690 L 3079 1342 mt 3079 690 L /Helvetica /ISOLatin1Encoding 228 FMSR 2088 988 mt (SR-Tree) s gs 1285 690 1795 653 MR c np DA 428 0 1392 907 2 MP stroke gr DA SO 36 47 -36 47 -36 -47 36 -47 1392 954 5 MP DP DA SO 36 47 -36 47 -36 -47 36 -47 1820 954 5 MP DP DA gs 1285 690 1795 653 MR c np gr 2088 1206 mt (LSH ) s gs 1285 690 1795 653 MR c np SO 428 0 1392 1125 2 MP stroke gr SO 0 -58 -58 0 0 58 58 0 1363 1096 5 MP DP 0 -58 -58 0 0 58 58 0 1791 1096 5 MP DP gs 1285 690 1795 653 MR c np gr end eplot epage end showpage %%EndDocument endTexFig 430 1525 a Fo(\(b\))-30 1655 y(Figure)12 b(9:)k(\(a\))11 b(n)o(um)o(b)q(er)g(of)g(indices)h(vs.)17 b(error)c(and)e(\(b\))h(n)o (um-)-30 1703 y(b)q(er)j(of)e(indices)h(vs.)19 b(size.)-30 1774 y(essen)o(tially)10 b(determined)h(in)e(adv)n(ance.)17 b(All)10 b(these)i(prop)q(erties)-30 1821 y(mak)o(e)j(LSH)h(a)g (suitable)g(candidate)h(for)f(high-p)q(erformance)-30 1869 y(and)e(real-time)e(systems.)20 1916 y(In)19 b(recen)o(t)h(w)o (ork)f([5,)f(23],)h(w)o(e)g(explore)h(applications)e(of)-30 1963 y(LSH-t)o(yp)q(e)j(tec)o(hniques)g(to)f(data)g(mining)e(and)i (searc)o(h)h(for)-30 2011 y(cop)o(yrigh)o(ted)14 b(video)g(data.)19 b(Our)c(exp)q(erience)h(suggests)g(that)-30 2058 y(there)i(is)e(a)g (lot)f(of)h(p)q(oten)o(tial)g(for)g(further)h(impro)o(v)o(emen)o(t)d (of)-30 2105 y(the)i(p)q(erformance)f(of)g(the)h(LSH)f(algorithm.)20 b(F)m(or)15 b(example,)-30 2153 y(our)20 b(data)g(structures)j(are)d (created)i(using)e(a)g(randomized)-30 2200 y(pro)q(cedure.)g(It)12 b(w)o(ould)g(b)q(e)i(in)o(teresting)f(if)f(there)i(w)o(as)f(a)f(more) -30 2247 y(systematic)20 b(metho)q(d)f(for)h(p)q(erforming)f(this)h (task;)j(suc)o(h)e(a)-30 2295 y(metho)q(d)14 b(could)h(tak)o(e)g (additional)e(adv)n(an)o(tage)i(of)f(the)i(struc-)-30 2342 y(ture)i(of)f(the)h(data)f(set.)30 b(W)m(e)17 b(also)g(b)q(eliev)o (e)g(that)h(in)o(v)o(estiga-)-30 2389 y(tion)11 b(of)g Fk(hybrid)h Fo(data)f(structures)j(obtained)e(b)o(y)f(merging)f(the)-30 2437 y(tree-based)17 b(and)f(hashing-based)g(approac)o(hes)g(is)g(a)f (fruitful)-30 2484 y(direction)f(for)g(further)g(researc)o(h.)1020 -33 y Fv(References)1039 35 y Fu([1])20 b(S.)g(Ary)o(a,)h(D.M.)e(Moun)o (t,)i(and)f(O.)f(Nara)o(y)o(an,)39 b(Accoun)o(t-)1100 81 y(ing)11 b(for)f(b)q(oundary)i(e\013ects)e(in)h(nearest-neigh)o(b)q (or)h(searc)o(hing.)1100 127 y Fb(Discr)n(ete)f(and)g(Computational)e (Ge)n(ometry)p Fu(,)g(16)j(\(1996\),)g(pp.)1100 172 y(155{176.)1039 233 y([2])20 b(S.)i(Ary)o(a,)h(D.M.)e(Moun)o(t,)i(N.S.)e(Netan)o(y)o (ah)o(u,)j(R.)d(Silv)o(er-)1100 278 y(man,)16 b(and)g(A.)e(W)m(u.)23 b(An)15 b(optimal)i(algorithm)g(for)e(appro)o(x-)1100 324 y(imate)j(nearest)f(neigh)o(b)q(or)i(searc)o(hing,)31 b(In)16 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)1100 370 y(the)i(5th)f(A)o(nnual)f(A)o(CM-SIAM)i(Symp)n(osium)e(on)i(Discr)n (ete)1100 415 y(A)o(lgorithms)p Fu(,)11 b(1994,)i(pp.)26 b(573{582.)1039 476 y([3])20 b(J.L.)f(Ben)o(tley)m(.)36 b(Multidimen)q(sio)q(nal)23 b(binary)e(searc)o(h)f(trees)1100 522 y(used)12 b(for)e(asso)q(ciativ)o(e)j(searc)o(hing.)i Fb(Communic)n(ations)9 b(of)i(the)1100 567 y(A)o(CM)p Fu(,)i(18)g(\(1975\),)h(pp.)f(509{517.)1039 628 y([4])20 b(S.)12 b(Berc)o(h)o(told)i(and)e(D.A.)f(Keim.)16 b(High-dimension)q (al)f(Index)1100 673 y(Structures.)i(In)12 b(Pro)q(ceedings)i(of)d (SIGMOD,)i(1998,)f(p.)g(501.)1100 719 y(See)36 b Fa(http://ww)o(w.i)o (nf)o(orm)o(at)o(ik.)o(uni)o(-h)o(all)o(e.)o(de/)16 b(keim/)1100 765 y(SIGMOD98Tut)o(or)o(ial)o(.ps)o(.g)o(z)1039 825 y Fu([5])k(E.)11 b(Cohen.)g(M.)g(Datar,)h(S.)f(F)m(ujiw)o(ara,)g(A.)g (Gionis,)i(P)m(.)d(Indyk,)1100 871 y(R.)h(Mot)o(w)o(ani,)h(J.)e(D.)h (Ullman.)h(C.)f(Y)m(ang.)i(Finding)h(In)o(terest-)1100 916 y(ing)g(Asso)q(ciations)h(without)f(Supp)q(ort)g(Pruning.)k(T)m(ec) o(hnical)1100 962 y(Rep)q(ort,)11 b(Computer)f(Science)g(Departmen)o (t,)h(Stanford)f(Uni-)1100 1008 y(v)o(ersit)o(y)m(.)1039 1068 y([6])20 b(T.M.)15 b(Chan.)25 b(Appro)o(ximate)16 b(Nearest)g(Neigh)o(b)q(or)h(Queries)1100 1114 y(Revisited.)76 b(In)31 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)h(the)g(13th)f(A)o(nnual) 1100 1160 y(A)o(CM)15 b(Symp)n(osium)d(on)i(Computational)d(Ge)n (ometry)p Fu(,)h(1997,)1100 1205 y(pp.)i(352-358.)1039 1266 y([7])20 b(S.)12 b(Cost)g(and)h(S.)f(Salzb)q(erg.)17 b(A)11 b(w)o(eigh)o(ted)i(nearest)g(neigh)o(b)q(or)1100 1311 y(algorithm)20 b(for)e(learning)j(with)d(sym)o(b)q(olic)i (features.)32 b Fb(Ma-)1100 1357 y(chine)12 b(L)n(e)n(arning)p Fu(,)f(10)i(\(1993\),)g(pp.)g(57{67.)1039 1418 y([8])25 b(S.)13 b(Chaudh)o(uri,)j(R.)d(Mot)o(w)o(ani)h(and)g(V.)e(Narasa)o(yy)o (a.)19 b(\\Ran-)1100 1463 y(dom)26 b(Sampling)i(for)d(Histogram)i (Construction:)43 b(Ho)o(w)1100 1509 y(m)o(uc)o(h)20 b(is)g(enough?".)38 b(In)20 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)i (SIGMOD'98)p Fu(,)1100 1555 y(pp.)14 b(436{447.)1039 1615 y([9])20 b(T.M.)f(Co)o(v)o(er)g(and)h(P)m(.E.)f(Hart.)36 b(Nearest)19 b(neigh)o(b)q(or)j(pat-)1100 1661 y(tern)11 b(classi\014cation.)k Fb(IEEE)c(T)m(r)n(ansactions)d(on)i(Information) 1100 1706 y(The)n(ory)p Fu(,)i(13)h(\(1967\),)g(pp.)h(21{27.)1020 1767 y([10])20 b(P)m(.)c(Ciaccia,)i(M.)e(P)o(atella)h(and)g(P)m(.)e (Zezula,)29 b(A)15 b(cost)i(mo)q(del)1100 1812 y(for)d(similarit)o(y)j (queries)f(in)f(metric)f(spaces)22 b(In)14 b Fb(Pr)n(o)n(c)n(e)n(e)n (dings)1100 1858 y(of)g(PODS'98)p Fu(,)d(pp.)i(59{68.)1020 1919 y([11])20 b(S.)34 b(Deerw)o(ester,)k(S.)c(T.)f(Dumais,)40 b(T.K.)32 b(Landauer,)1100 1964 y(G.W.)10 b(F)m(urnas,)i(and)f(R.A.)e (Harshman.)k(Indexing)f(b)o(y)f(laten)o(t)1100 2010 y(seman)o(tic)i (analysis.)i Fb(Journal)c(of)g(the)g(So)n(ciety)e(for)i(Informa-)1100 2056 y(tion)i(Scienc)n(es,)d Fu(41)j(\(1990\),)g(pp.)g(391{407.)1020 2116 y([12])20 b(L.)12 b(Devro)o(y)o(e)g(and)h(T.J.)d(W)m(agner.)15 b(Nearest)d(neigh)o(b)q(or)i(meth-)1100 2162 y(o)q(ds)i(in)g (discrimination)q(.)25 b Fb(Handb)n(o)n(ok)12 b(of)j(Statistics)p Fu(,)d(v)o(ol.)j(2,)1100 2207 y(P)m(.R.)10 b(Krishnaiah)j(and)e(L.N.)f (Kanal,)h(eds.,)g(North-Holland,)1100 2253 y(1982.)1020 2314 y([13])20 b(R.O.)12 b(Duda)i(and)f(P)m(.E.)f(Hart.)k Fb(Pattern)c(Classi\014c)n(ation)e(and)1100 2359 y(Sc)n(ene)i(A)o (nalysis.)i Fu(John)g(Wiley)h(&)d(Sons,)i(NY,)e(1973.)1020 2420 y([14])20 b(H.)10 b(Edelsbrunner.)16 b Fb(A)o(lgorithms)9 b(in)i(Combinatorial)e(Ge)n(om-)1100 2465 y(etry.)16 b Fu(Springer-V)m(erlag,)f(1987.)1020 2526 y([15])20 b(C.)g(F)m(aloutsos,)i(R.)e(Barb)q(er,)i(M.)d(Flic)o(kner,)24 b(W.)19 b(Niblac)o(k,)1100 2571 y(D.)f(P)o(etk)o(o)o(vic,)h(and)f(W.)f (Equitz.)30 b(E\016cien)o(t)19 b(and)e(e\013ectiv)o(e)1100 2617 y(querying)f(b)o(y)d(image)h(con)o(ten)o(t.)k Fb(Journal)12 b(of)h(Intel)r(ligent)e(In-)1100 2663 y(formation)h(Systems)p Fu(,)f(3)i(\(1994\),)g(pp.)g(231{262.)p eop %%Page: 12 12 12 11 bop -30 -33 a Fu([16])20 b(C.)15 b(F)m(aloutsos)i(and)g(D.W.)e (Oard.)25 b(A)15 b(Surv)o(ey)i(of)e(Informa-)50 12 y(tion)i(Retriev)n (al)g(and)f(Filtering)h(Metho)q(ds.)25 b(T)m(ec)o(hnical)17 b(Re-)50 58 y(p)q(ort)e(CS-TR-3514,)g(Departmen)o(t)h(of)e(Computer)h (Science,)50 104 y(Univ)o(ersit)o(y)g(of)e(Maryland,)i(College)f(P)o (ark,)f(1995.)-30 165 y([17])20 b(M.)j(Flic)o(kner,)28 b(H.)22 b(Sa)o(whney)m(,)k(W.)e(Niblac)o(k,)j(J.)22 b(Ashley)m(,)50 210 y(Q.)16 b(Huang,)i(B.)e(Dom,)i(M.)e(Gork)n(ani,)j(J.)d(Hafner,)h (D.)f(Lee,)50 256 y(D.)e(P)o(etk)o(o)o(vic,)h(D.)f(Steele,)h(and)f(P)m (.)f(Y)m(ank)o(er.)20 b(Query)14 b(b)o(y)h(im-)50 302 y(age)c(and)h(video)f(con)o(ten)o(t:)17 b(the)10 b(QBIC)g(system.)k Fb(IEEE)d(Com-)50 347 y(puter)p Fu(,)h(28)h(\(1995\),)g(pp.)g(23{32.) -30 408 y([18])20 b(J.K.)12 b(F)m(riedman,)i(J.L.)e(Ben)o(tley)m(,)i (and)f(R.A.)g(Fink)o(el.)18 b(An)13 b(al-)50 454 y(gorithm)19 b(for)f(\014nding)i(b)q(est)e(matc)o(hes)h(in)f(logarithmic)j(ex-)50 500 y(p)q(ected)i(time.)43 b Fb(A)o(CM)23 b(T)m(r)n(ansactions)c(on)h (Mathematic)n(al)50 545 y(Softwar)n(e)p Fu(,)12 b(3)h(\(1977\),)g(pp.)g (209{226.)-30 606 y([19])20 b(T.)d(Figiel,)j(J.)c(Lindenstrauss,)k(V.)c (D.)h(Milman.)31 b(The)17 b(di-)50 652 y(mension)c(of)e(almost)h (spherical)h(sections)g(of)e(con)o(v)o(ex)g(b)q(o)q(dies.)50 698 y Fb(A)n(cta)i(Math.)f Fu(139)h(\(1977\),)g(no.)g(1-2,)g(53{94.)-30 759 y([20])20 b(A.)15 b(Gersho)h(and)g(R.M.)e(Gra)o(y)m(.)24 b Fb(V)m(e)n(ctor)14 b(Quantization)e(and)50 804 y(Data)h(Compr)n (ession.)i Fu(Klu)o(w)o(er,)e(1991.)-30 865 y([21])20 b(T.)c(Hastie)h(and)g(R.)e(Tibshirani)q(.)29 b(Discriminan)o(t)19 b(adaptiv)o(e)50 911 y(nearest)d(neigh)o(b)q(or)h(classi\014catio)q(n.) 26 b(In)15 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)i(the)50 957 y(First)e(International)d(Confer)n(enc)n(e)h(on)h(Know)r(le)n(dge)g (Disc)n(ov-)50 1002 y(ery)i(&)g(Data)e(Mining)p Fu(,)g(1995,)h(pp.)g (142{149.)-30 1063 y([22])20 b(H.)11 b(Hotelling.)17 b(Analysis)d(of)d(a)g(complex)i(of)e(statistical)j(v)n(ari-)50 1109 y(ables)19 b(in)o(to)f(principal)i(comp)q(onen)o(ts.)31 b Fb(Journal)16 b(of)h(Educ)n(a-)50 1155 y(tional)12 b(Psycholo)n(gy)p Fu(,)e(27)j(\(1933\).)g(pp.)g(417{441.)-30 1216 y([23])20 b(P)m(.)11 b(Indyk,)h(G.)e(Iy)o(engar,)i(N.)e(Shiv)n (akumar.)16 b(Finding)d(pirated)50 1261 y(video)20 b(sequences)f(on)f (the)g(In)o(ternet.)32 b(T)m(ec)o(hnical)19 b(Rep)q(ort,)50 1307 y(Computer)14 b(Science)g(Departmen)o(t,)g(Stanford)g(Univ)o (ersit)o(y)m(.)-30 1368 y([24])20 b(P)m(.)j(Indyk)h(and)f(R.)g(Mot)o(w) o(ani.)47 b(Appro)o(ximate)25 b(Nearest)50 1414 y(Neigh)o(b)q(or)18 b({)e(T)m(o)o(w)o(ards)g(Remo)o(ving)i(the)f(Curse)f(of)g(Dimen-)50 1459 y(sionalit)o(y)m(.)34 b(In)17 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e (of)i(the)g(30th)g(Symp)n(osium)f(on)50 1505 y(The)n(ory)d(of)g (Computing)p Fu(,)e(1998,)i(pp.)h(604{613.)-30 1566 y([25])20 b(K.V.)10 b(Ra)o(vi)j(Kan)o(th,)e(D.)g(Agra)o(w)o(al,)h(A.)e(Singh.)15 b(\\Dimension-)50 1612 y(alit)o(y)j(Reduction)g(for)e(Similarit)o(y)j (Searc)o(hing)f(in)f(Dynamic)50 1657 y(Databases".)c(In)d Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)i(SIGMOD'98)p Fu(,)e(166)i({)g(176.) -30 1718 y([26])20 b(K.)31 b(Karh)o(unen.)376 1709 y(\177)371 1718 y(Ub)q(er)h(lineare)h(Metho)q(den)g(in)g(der)50 1764 y(W)m(ahrsc)o(heinl)q(ic)o(h)q(k)o(eits)q(rec)o(hn)o(u)q(ng.)j Fb(A)o(nn.)17 b(A)n(c)n(ad.)g(Sci.)h(F)m(en-)50 1810 y(nic)n(ae)p Fu(,)11 b(Ser.)i(A137,)g(1947.)-30 1871 y([27])20 b(V.)15 b(Koivune)i(and)f(S.)f(Kassam.)23 b(Nearest)16 b(neigh)o(b)q(or)h(\014lters)50 1916 y(for)e(m)o(ultiv)n(ariate)i (data.)22 b Fb(IEEE)15 b(Workshop)e(on)i(Nonline)n(ar)50 1962 y(Signal)d(and)h(Image)f(Pr)n(o)n(c)n(essing)p Fu(,)e(1995.)-30 2023 y([28])20 b(N.)g(Kata)o(y)o(ama)h(and)h(S.)e(Satoh.)40 b(The)21 b(SR-tree:)32 b(an)21 b(in-)50 2069 y(dex)e(structure)g(for)e (high-dimensi)q(on)q(al)k(nearest)e(neigh)o(b)q(or)50 2114 y(queries.)40 b(In)21 b Fb(Pr)n(o)n(c.)e(SIGMOD'97)p Fu(,)i(pp.)f(369{380.)40 b(The)50 2160 y(co)q(de)11 b(is)g(a)o(v)n (ailable)i(from)d Fa(http://www.)o(rd)o(.na)o(cs)o(is.)o(ac.)o(jp)o(/) 50 2206 y(~katayama/h)o(om)o(epa)o(ge/)o(re)o(sea)o(rc)o(h/s)o(rtr)o (ee)o(/En)o(gl)o(ish)o(.h)o(tml)-30 2267 y Fu([29])20 b(N.)11 b(Linial,)i(E.)e(London,)h(and)g(Y.)f(Rabino)o(vic)o(h.)17 b(The)11 b(geome-)50 2312 y(try)g(of)g(graphs)g(and)h(some)f(of)f(its)h (algorithmic)j(applications.)50 2358 y(In)k Fb(Pr)n(o)n(c)n(e)n(e)n (dings)c(of)j(35th)f(A)o(nnual)g(IEEE)h(Symp)n(osium)f(on)50 2404 y(F)m(oundations)10 b(of)i(Computer)f(Scienc)n(e)p Fu(,)f(1994,)i(pp.)g(577{591.)-30 2465 y([30])20 b(M.)h(Lo)o(\023)-18 b(ev)o(e.)40 b(F)m(onctions)22 b(aleastoires)h(de)e(second)h(ordere.)50 2510 y Fb(Pr)n(o)n(c)n(essus)8 b(Sto)n(chastiqu)o(es)f(et)i(mouvement)f (Br)n(ownian)p Fu(,)g(Her-)50 2556 y(mann,)14 b(P)o(aris,)f(1948.)-30 2617 y([31])20 b(B.)10 b(S.)f(Manjunath.)k(Airphoto)e(dataset.)h Fa(http://vi)o(val)o(di)o(.)50 2663 y(ece.ucsb.ed)o(u/)o(Man)o(jun)o (at)o(h/r)o(es)o(ear)o(ch.)o(ht)o(m)1020 -33 y Fu([32])20 b(B.)f(S.)f(Manjunath)i(and)f(W.)f(Y.)g(Ma.)33 b(T)m(exture)18 b(features)1100 12 y(for)c(bro)o(wsing)h(and)f(retriev)n(al)h(of)e (large)i(image)f(data.)19 b Fb(IEEE)1100 58 y(T)m(r)n(ansactions)11 b(on)h(Pattern)g(A)o(nalysis)e(and)i(Machine)g(Intel-)1100 104 y(ligenc)n(e,)18 b(\(Sp)n(e)n(cial)e(Issue)g(on)i(Digital)f(Libr)n (aries\))p Fu(,)g(18)h(\(8\),)1100 149 y(pp.)c(837-842.)1020 212 y([33])20 b(G.S.)14 b(Manku,)g(S.)f(Ra)r(jagopalan,)j(and)e(B.G.)f (Lindsa)o(y)m(.)20 b(Ap-)1100 257 y(pro)o(ximate)d(Medians)f(and)g (other)f(Quan)o(tiles)i(in)f(One)e(P)o(ass)1100 303 y(and)20 b(with)e(Limited)i(Memory)m(.)34 b(In)18 b Fb(Pr)n(o)n(c)n(e)n(e)n (dings)e(of)i(SIG-)1100 349 y(MOD'98)p Fu(,)13 b(pp.)g(426{435.)1020 411 y([34])20 b(Y.)13 b(Matias,)i(J.S.)e(Vitter,)h(and)g(M.)g(W)m(ang.) 20 b(W)m(a)o(v)o(elet-based)1100 457 y(Histograms)11 b(for)f(Selectivit)o(y)i(Estimations.)h(In)d Fb(Pr)n(o)n(c)n(e)n(e)n (dings)1100 502 y(of)k(SIGMOD'98)p Fu(,)d(pp.)i(448{459.)1020 565 y([35])20 b(R.)i(Mot)o(w)o(ani)g(and)h(P)m(.)e(Ragha)o(v)n(an.)44 b Fb(R)n(andomize)n(d)19 b(A)o(lgo-)1100 610 y(rithms)p Fu(.)d(Cam)o(bridge)f(Univ)o(ersit)o(y)g(Press,)e(1995.)1020 672 y([36])20 b(M.)d(Otterman.)27 b(Appro)o(ximate)18 b(matc)o(hing)g(with)e(high)i(di-)1100 718 y(mensionali)q(t)o(y)d (R-trees.)g(M.Sc.)c(Sc)o(holarly)k(pap)q(er,)d(Dept.)g(of)1100 764 y(Computer)j(Science,)20 b(Univ.)14 b(of)f(Maryland,)j(College)f(P) o(ark,)1100 809 y(MD,)e(1992.)1020 872 y([37])20 b(A.)e(P)o(en)o (tland,)j(R.W.)d(Picard,)i(and)f(S.)f(Sclaro\013.)33 b(Photo-)1100 917 y(b)q(o)q(ok:)22 b(to)q(ols)16 b(for)e(con)o(ten)o (t-based)j(manipulation)h(of)d(image)1100 963 y(databases.)k(In)13 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)i(the)g(SPIE)g(Confer)n(enc)n(e)f (on)1100 1009 y(Stor)n(age)i(and)g(R)n(etrieval)g(of)g(Image)h(and)f (Vide)n(o)g(Datab)n(ases)1100 1054 y(II)p Fu(,)f(1994.)1020 1117 y([38])20 b(G.)c(Salton)h(and)g(M.J.)d(McGill.)28 b Fb(Intr)n(o)n(duction)12 b(to)k(Mo)n(dern)1100 1162 y(Information)d(R)n(etrieval.)k Fu(McGra)o(w-Hill)f(Bo)q(ok)f(Compan)o (y)m(,)1100 1208 y(New)e(Y)m(ork,)f(NY,)g(1983.)1020 1270 y([39])20 b(H.)15 b(Samet.)25 b Fb(The)16 b(Design)e(and)h(A)o (nalysis)e(of)i(Sp)n(atial)f(Data)1100 1316 y(Structur)n(es.)h Fu(Addison-W)m(esley)m(,)g(Reading,)g(MA,)d(1989.)1020 1378 y([40])20 b(J.)i(R.)g(Smith.)45 b(In)o(tegrated)23 b(Spatial)i(and)e(F)m(eature)f(Im-)1100 1424 y(age)d(Systems:)29 b(Retriev)n(al,)22 b(Analysis)e(and)f(Compression.)1100 1469 y(Ph.D.)13 b(thesis,)g(Colum)o(bia)h(Univ)o(ersit)o(y)m(,)g(1997.) f(Av)n(ailable)i(at)1100 1515 y Fa(ftp://ftp.c)o(tr)o(.co)o(lum)o(bi)o (a.e)o(du)o(/CT)o(R-R)o(es)o(ear)o(ch)o(/ad)o(ve)o(nt/)1100 1561 y(public/publ)o(ic)o(/jr)o(smi)o(th)o(/th)o(es)o(is)1020 1623 y Fu([41])20 b(T.)14 b(Sellis,)k(N.)c(Roussop)q(oulos,)j(and)f(C.) e(F)m(aloutsos.)23 b(Multi-)1100 1669 y(dimensional)18 b(Access)13 b(Metho)q(ds:)20 b(T)m(rees)13 b(Ha)o(v)o(e)h(Gro)o(wn)g (Ev-)1100 1714 y(erywhere.)37 b(In)19 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)e (of)i(the)f(23r)n(d)h(Internationa)o(l)1100 1760 y(Confer)n(enc)n(e)12 b(on)h(V)m(ery)g(L)n(ar)n(ge)f(Data)h(Bases)p Fu(,)e(1997,)i(13{15.) 1020 1822 y([42])20 b(A.W.M.)10 b(Smeulders)i(and)f(R.)f(Jain,)i(eds.)g Fb(Image)f(Datab)n(ases)1100 1868 y(and)21 b(Multi-me)n(dia)e(Se)n(ar)n (ch.)39 b Fu(Pro)q(ceedings)24 b(of)d(the)g(First)1100 1914 y(In)o(ternational)k(W)m(orkshop,)f(IDB-MMS'96,)h(Amsterdam)1100 1959 y(Univ)o(ersit)o(y)15 b(Press,)e(Amsterdam,)g(1996.)1020 2022 y([43])20 b(J.R.)25 b(Smith)h(and)g(S.F.)e(Chang.)54 b(Visually)28 b(Searc)o(hing)1100 2067 y(the)21 b(W)m(eb)g(for)e(Con)o (ten)o(t.)39 b Fb(IEEE)20 b(Multime)n(dia)e Fu(4)j(\(1997\):)1100 2113 y(pp.)h(12{20.)g(See)g(also)g Fa(http://disn)o(ey)o(.ct)o(r.)o (col)o(umb)o(ia)o(.)1100 2159 y(edu/WebSEEk)1020 2221 y Fu([44])e(G.)11 b(Wyszec)o(ki)i(and)e(W.S.)g(St)o(yles.)j Fb(Color)e(scienc)n(e:)i(c)n(onc)n(epts)1100 2267 y(and)j(metho)n(ds,)g (quantitativ)o(e)e(data)h(and)h(formulae)p Fu(.)28 b(John)1100 2312 y(Wiley)15 b(and)f(Sons,)g(New)e(Y)m(ork,)g(NY,)h(1982.)1020 2374 y([45])20 b(R.)f(W)m(eb)q(er,)i(H.)d(Sc)o(hek,)j(and)e(S.)g (Blott.)35 b(A)18 b(quan)o(titativ)o(e)1100 2420 y(analysis)f(and)d(p)q (erformance)g(study)h(for)e(Similarit)o(y)k(Searc)o(h)1100 2466 y(Metho)q(ds)e(in)f(High)g(Dimensional)i(Spaces.)j(In)13 b Fb(Pr)n(o)n(c)n(e)n(e)n(dings)1100 2511 y(of)18 b(the)f(24th)g (Internationa)o(l)e(Confer)n(enc)n(e)h(on)i(V)m(ery)f(L)n(ar)n(ge)1100 2557 y(Data)c(Bases)f(\(VLDB\))p Fu(,)g(1998,)h(pp.)g(194-205.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF