NGT-ONNGによる近似最近傍探索

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M3 ৘ใݕࡧ࿦จಡΈձ #2 hiromu-nakamura @po3rin 3FBEJOHQBQFSl0QUJNJ[BUJPOPG*OEFYJOH#BTFEPOL /FBSFTU/FJHICPS(SBQIGPS1SPYJNJUZ4FBSDIJO)JHI EJNFOTJPOBM%BUBz /(50//(ʹΑΔۙࣅ࠷ۙ๣୳ࡧ

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தଜ߂෢ ΤϜεϦʔͷ"*ɾػցֶशνʔϜͰݕࡧج൫पΓ Λ୲౰͍ͯ͠Δɻ(P͕޷͖ɻ ֤छΞΧ΢ϯτQPSJOͰ΍ͬͯ·͢ɻ )*SPNV/BLBNVSB

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$POUFOUT *OUSPEVDUJPOdϕϯνϚʔΫ͔ΒݟΔ"//ͷݱঢ়d /(50//(3FMBUFE8PSL /(50//( ͓·͚

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*OUSPEVDUJPO dϕϯνϚʔΫ͔ΒݟΔ"//ͷݱঢ়d

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"// ߴ࣍ݩͷϕΫτϧۭؒͰσʔλͷ࠷ۙ๣Λݕࡧ͢Δ͜ͱ͸ॏཁͳλεΫͰ͋Δɻ ը૾σʔληοτ͔Βྨࣅͨ͠ը૾Λݟ͚ͭΔ ৘ใݕࡧͰ͸ɺςΩετຒΊࠐΈΛ࢖༻ͯ͠ɺྨࣅهࣄΛࢀর͢Δ ಛʹਂ૚ֶशϞσϧͷ࣍ݩ਺͸૿Ճ͢Δ܏޲͕͋ΔͨΊɺ਺ඦສΛ௒͑Δߴ࣍ݩ ϕΫτϧΛݕࡧ͢Δඞཁ͕͋Γɺۙࣅ࠷ۙ๣୳ࡧ BQQSPYJNBUFOFBSFTU OFJHICPS "// ݕࡧํ๏͸ओྗɻ

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"// ఏҊ͞Ε͍ͯΔ࠷ۙ๣ݕࡧํ๏͸ଟ਺ଘࡏ IOTX GBJTT 'BDFCPPL੡ ONTMJC લճՏ߹͞Μ͕࢖༻ qBOO 0QFO$7Ͱ΋࣮૷͞Ε͍ͯΔ &UDʜ ࠷ۙ๣୳ࡧͷϕϯνϚʔΫΛߦ͏ϓϩδΣΫτʮ#FODINBSLJOHOFBSFTUOFJHICPSTʯ <>Ͱ֤ख๏ΛλεΫ͝ͱʹൺֱՄೳɻ <>IUUQTHJUIVCDPNFSJLCFSOBOOCFODINBSLT

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"// ͳʹ΍Β:BIPP+"1"/ΒʹΑͬͯ/(5ͱ͍͏΋ͷ͕ੈքτοϓϨϕϧ଎౓ୡ੒ͨ͠Β͍͠<> <>IUUQTUFDICMPHZBIPPDPKQFOUSZ

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#FODINBSLJOHOFBSFTUOFJHICPST <>IUUQTHJUIVCDPNFSJLCFSOBOOCFODINBSLT

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#FODINBSLJOHOFBSFTUOFJHICPST <>IUUQTHJUIVCDPNFSJLCFSOBOOCFODINBSLT

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IUUQTBSYJWPSHBCT Αͬͯࠓճͷ࿦จ /(5 /FJHICPSIPPEHSBQIBOEUSFF ͷ࿦จ

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/(50//(3FMBUFE8PSL d/(50//(ొ৔·ͰͷܦҢΛ͓͑͞Δd

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3FMBUFE8PSL ࠷ۙ๣୳ࡧ͸ɺπϦʔϕʔεͱάϥϑϕʔεʹେ͖͘෼ྨ͞ΕΔɻଠࣈ͸/(5ʹ࣮૷͞Ε͍ͯΔΞϧΰϦζϜɻ ˙πϦʔϕʔε LEUSFF "// LEπϦʔʹۙࣅݕࡧΛద༻͢Δํ๏ '-"// ෳ਺ͷLEπϦʔ͕ฒߦͯ͠ݕࡧ͞ΕΔϥϯμϜԽ͞ΕͨLEπϦʔ ˙άϥϑϕʔε )/48 ۙࣅۙ๣άϥϑʹϨΠϠʔͷ֓೦Λಋೖ "//( ۙࣅ,//( 1"//( ΫΤϦ࣌ؒΛ୹ॖ͢ΔͨΊʹ"//(ͷ֤ϊʔυͷΤοδΛ࡟আ͢Δ 0//( ϕϯνϚʔΫτοϓϨϕϧ

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"//( ϊʔυʹLݸͷ࠷ۙ๣ϊʔυ΁ͷ࿈઀Λߦ͏άϥϑɻ ࿈݁άϥϑΛอূ͠ͳ͍ͷͰ෼཭͞Εͨάϥϑ͕ൃੜ͢Δ ਤ ɻ L//( "//(<>͸ϊʔυͣͭແ޲ΤοδͰஞ࣍తʹ࿈݁͢Δ͜ͱͰ࿈݁άϥϑΛอূ͢Δ ਤ ɻ "//( <>IUUQTTZJNHKQJEPDTSFTFBSDI@MBCBSUJDMFTNJXBTBLJJQTKUPEQEG

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"//(5 ϥϯμϜʹબ୒ͨ͠ϊʔυ͔ΒάϥϑΛ୳ࡧ͢ΔͷͰάϥϑ͕େྔͷϊʔυΛ༗͢Δ৔ ߹ʹ୳ࡧίετ͕૿Ճ͢Δɻಛʹ୳ࡧى఺࣍ୈͰίετ͕௓Ͷ্͕Δɻ "//(5<>͸"//(ʹ͓͍ͯ୳ࡧى఺Λܾఆ͢ΔͷʹEWQUSFF<>Λར༻͢Δ͜ͱʹ ΑΓ௿ίετͰۙ๣ϊʔυΛ୳ࡧͰ͖Δ ਤ ɻ "//( <>IUUQTTZJNHKQJEPDTSFTFBSDI@MBCBSUJDMFTNJXBTBLJJQTKKOMQEG "//(5

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"//(5 EWQUSFF EZOBNJDWBOUBHFQPJOUUSFF ͸୳ࡧى఺ΛબͿͨΊ͚ͩʹ࢖ΘΕΔɻ͜ͷྫͰ͸ۭؒશମΛ੺ͷಉ ৺ԁɺΦϨϯδͷಉ৺ԁɺ੨ͷಉ৺ԁʹΑΓॱʹ෼ׂ͢Δ͜ͱͰ෦෼ۭؒ΁ࡉ෼Խ͠ɺͦͷ෦෼ۭؒͷแؚؔ܎ Λࣔ͢πϦʔߏ଄Λੜ੒͢Δɻ͜ΕͰ୳ࡧى఺Λେ෯ʹߜΓࠐΊΔɻಈత໦ͷߏஙΞϧΰϦζϜ͸ޙ೔ௐࠪʜ

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/(50//( d/(50//(ʹΑΔΤοδௐ੔d

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.PUJWBUJPO "//( "//(5Ͱ͸Τοδ਺ʹΑͬͯ͸࣍ͷ໰୊͕ൃੜ͢Δɻ ೖྗΤοδ͕ۃ୺ʹগͳ͍ϊʔυ͸౸ୡ͢Δ֬཰͕Լ͕ΔͷͰݕࡧ࿙Ε͢ΔՄೳ Τοδ਺͕શମతʹগͳ͍ͱԕճΓʹͳΔՄೳੑ͕͋Γɺݕࡧ࿙Ε͢ΔՄೳੑ্͕͕Δ Τοδ਺͕ଟ͍ͱ୳ࡧ͕࣌ؒ૿͑Δɻ ͭ·Γ ೖྗΤοδͷ਺͸ϊʔυͰҰఆʹ͍ͨ͠ ग़ྗΤοδ਺ͷ૿ՃʹΑΔ଎౓௿Լ͕ൃੜ͠ͳ͍ൣғͰ֤ϊʔυͷೖྗΤοδ਺Λ૿΍͢ɻ ͜ΕΒΛϊʔυʹΤοδͷௐ੔ʹΑͬͯղܾ͢Δͷ͕/(50//( 0QUJNJ[FE/FBSFTU /FJHICPST(SBQI Ͱ͋Γɺ"//(5ʹΤοδௐ੔΍ύεௐ੔Λ͔͚ͨ΋ͷͰ͋Δɻ

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EFHSFFBEKVTUNFOUNFUIPET QBUIBEKVTUNFOUNFUIPE /(50//(Ͱ͸ͭΤοδௐ੔๏ͱݕࡧ࣌ͷύεௐ੔๏ΛఏҊ͍ͯ͠Δɻ ੩త࣍਺ௐ੔,//(ͷΤοδͱ൓సΤοδ͔Β࡞ͬͨάϥϑͰೖग़࣍਺Λେ·͔ʹௐ੔͢Δ ੍໿෇͖੩త࣍਺ௐ੔ೖग़࣍਺ΛΑΓਖ਼֬ʹௐ੔͢Δ ಈత࣍਺ௐ੔ݕࡧ࣌ʹཁٻ͞ΕΔਫ਼౓͔Βग़࣍਺Λಈతʹܾఆ͢Δ γϣʔτΧοτΤοδআڈʹΑΔύεௐ੔ ϊʔυͷग़࣍਺͕ߴ͘ͳΓɺΫΤϦ͕࣌ؒ௕͘ͳΔͷΛͳΜͱ͔͍ͨ͠

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੩త࣍਺ௐ੔ d,//(ͷΤοδͱ൓సΤοδ͔Β࡞ͬͨάϥϑͰೖग़࣍਺Λେ·͔ʹௐ੔d ·ͣ͸ϊʔυͷೖग़࣍਺Λେࡶ೺ʹἧ͍͑ͨ F@P F@J͕༩͑ΒΕͨ࣌ɺ֤ϊʔυ͔Βڑ཭͕୹͍ॱʹ F@Pݸͷग़ྗΤοδ͕બ୒͞ΕΔɻF@Jݸͷग़ྗΤοδ ΋୹͍ॱʹબ୒͞ΕΔɻ બ͹ΕͨF@Pݸͷग़ྗΤοδ܈͸ͦͷ··࠾༻͠ɺF@J ݸͷग़ྗΤοδ܈͸શͯ൓సͯ͠ೖྗΤοδͱͯ͠࠾ ༻͢Δɻ ࠨਤ͸L F@P F@Jͷ࣌Λࣔ͢ɻେ·͔ʹFP FJʹ࣍਺͕ἧ͑ΒΕ͍ͯΔɻ ࿦จͰ͸͜ΕΛௐ੔άϥϑͱ໋໊͍ͯ͠Δɻ ௐ੔ ݸͷΤοδ eo ݸͷ൓సΤοδ ei

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੍໿෇͖࣍਺ௐ੔ தଜ͕·ͩཧղͰ͖ͯͳ͍આ dೖग़࣍਺ΛΑΓਖ਼֬ʹௐ੔͢ΔͨΊͷ੍໿෇͖ͷ࣍਺ௐ੔d ઌ΄Ͳͷํ๏ͩͱେࡶ೺ʹ࣍਺Λௐ੔Ͱ͖Δ͕ɺҰ෦ͷϊʔυ͸ߴ ͍ग़࣍਺Λ࣋ͭ܏޲͕͋Δɻͦ͜Ͱɺग़ࣗ਺Λ૿΍͞ͳ͍Α͏ʹೖ ग़࣍਺Λௐ੔͢Δํ๏͕௥ՃͰఏҊ͞Ε͍ͯΔɻ ઌ΄ͲͷΞϧΰϦζϜͰF@PͰੜ੒͞ΕΔௐ੔άϥϑ(@UΛੜ੒ ͢Δɻ(@JΛߏஙதͷάϥϑ(@Fͷసஔάϥϑͱ͢Δɻ (@U಺ͷϊʔυͷग़࣍਺ͷঢॱͰͭͷϊʔυ͕બ୒͞Εɺͦͷ ϊʔυͷྡ઀ϊʔυͷ͍ۙॱͦΕͧΕʹ͍ͭͯɺ(@Jͷྡ઀ϊʔυ ͷग़࣍਺͕F@JΑΓ௿͘ɺ(@Fͷϊʔυͷग़࣍਺͕F@PΑΓ௿͍৔ ߹ɺྡ઀ϊʔυͱͷΤοδ͕(@Fʹ௥Ճ͞ΕΔɻ (@FϊʔυʹF@PݸͷΤοδؚ͕·ΕΔ·ͰɺݩͷάϥϑͷΤοδ ͕ϊʔυʹ௥Ճ͞Ε·͢ɻ

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ௐ੔ Gt Ge Gi Gt Ge Gi F@J F@P L Ge TUBHF݁Ռ ˔ ˔ ˔ ࠷ޙͷϊʔυ·Ͱॲཧ ੍ݶ෇͖੩త࣍਺ௐ੔TUBHF ඞͣ͠΋࿈઀άϥϑʹͳΒͳ͍ʁ

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Ge Ge G 4UBHF ੍ݶ෇͖੩త࣍਺ௐ੔TUBHF F@J F@P L ඞͣ͠΋࿈઀άϥϑʹͳΒͳ͍ʁΜʁʁ ࠓޙཁௐࠪ

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γϣʔτΧοτΤοδআڈʹΑΔ ύεௐ੔ EWQUSFFΛ࢖༻ͯ͠ΫΤϦʹ΄΅ྡ઀͢ΔϊʔυΛ୳ ࡧى఺ͱ͍ͯ͠ΔͨΊɺγϣʔτΧοτΤοδ͸ແବ ʹग़࣍਺Λ૿΍͍ͯ͠Δ৔߹͕͋Δɻ ͜ͷΑ͏ͳΤοδΛ࡟আ͢ΔͱΫΤϦ࣌ؒΛ୹ॖͰ͖ Δɻ ୅ସύε͕ͭҎ্ͷϊʔυͰߏ੒͞Ε͍ͯΔ৔߹ ࠨ ਤB Λ࡟আ͠Α͏ͱ͢ΔͱΫΤϦ࣌ؒͷํ͕େ͖͘ͳ ΔͨΊͭͷϊʔυͰߏ੒͞Ε͍ͯΔύεͷΈ͕࡟আ͞ ΕΔ ࠨਤCͷ৔߹ ɻ γϣʔτΧοτΤοδ͕୅ସύε಺ͷΤοδΑΓ୹͍ ৔߹ ࠨਤD ͷΑ͏ͳ৔߹͸࡟আ͞Εͳ͍ɻ

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ಈత࣍਺ௐ੔ dݕࡧ࣌ʹཁٻ͞ΕΔਫ਼౓͔Βग़࣍਺Λಈతʹܾఆ͢Δd ߴਫ਼౓͕ඞཁͳ৔߹͸ߴ͍ग़࣍਺͕ඞཁɻҰํͰ଎౓ ͕༏ઌ͞ΕΔ৔߹͸ߴ͍ग़࣍਺͸ෆཁɻ͕ͨͬͯ͠ɺ ग़࣍਺͸ݕࡧ࣌ʹಈతʹௐ੔Ͱ͖Δͱخ͍͠ɻ ୳ࡧ࣌ʹߟྀ͢Δग़࣍਺͸ਫ਼౓ௐ੔ύϥϝʔλЏͷؔ ਺Ͱ͋ΔFQΛೖྗͱͯ͠ௐ੔͢Δɻ ਫ਼౓ௐ੔ύϥϝʔλͷЏ͸୳ࡧ൒ܘʹ΋ӨڹΛ༩͑ Δɻͭ·Γ,OO4FBSDIͷਫ਼౓Λೋͭͷ؍఺Ͱܾఆ͢ Δɻ

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74NFUIPET %"͕࠷΋ڑ཭ܭࢉ͕গͳ͘ࡁΉɻ σʔληοτʹΑͬͯ͸4"Ͱ͑͞΄΅࠷ྑͷάϥϑߏ଄ΛߏஙͰ͖Δɻ 4"$͸ɺ(*45ͷΑ͏ͳߴ࣍ݩͷσʔληοτʹޮՌత͔΋͠Εͳ͍ɻ 4"ͱ4"$ͷ࣍਺͸%"ͷ࣍਺ΑΓ΋গͳ͍ͨΊϝϞϦ࢖༻ྔͷ࡟ݮΛ༏ઌ͢Δඞཁ͕͋Δ৔߹͸ɺ4"·ͨ͸ 4"$͕બ୒ࢶͱͳΔɾ

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3FTVMUT σʔληοτ͝ͱͷ֤छ࠷ۙ๣୳ࡧख๏ൺֱ

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ग़࣍਺ΛݮΒ͢ɻೖ࣍਺͸Ұఆʹ͢ΔΤοδௐ੔ͱ͍͏ΞΠσΞ͸໘ന͍ɻ ͨͩಈతͳߋ৽ɺ࡟আΛߟྀͯ͠ͳ͍ͷͰɺΞϓϦέʔγϣϯͰ࢖͏ͷ͸ෆ޲ ͖ɻ੩తͳݕࡧʹ͸࢖͑Δ͚Ͳͦ͏ͳΔͱ࢖͍Ͳ͜Ζ͕ɻɻ 7BME΋/(50//(͸αϙʔτ͠ͳ͍ͱݴ͍ͬͯΔɻͦΓΌͦ͏ɻ ΍͸Γੈ͸)/48࣌୅ͳͷ͔ ॴײ

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͓·͚ͱิ଍

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<>IUUQTHJUIVCDPNWEBBTWBME 7BME<>͸/(5ϥΠϒϥϦΛ࢖ͬͨ෼ࢄϕΫτϧݕࡧΤϯδϯɻLTGSJFOEMZͰɺ )FMNͰҰܸͰσϓϩΠ͕Մೳɻ௨৴͸H31$Λ࢖͍ͬͯΔɻ͔͠΋(P੡ΧοέΣɻ ΫϥΠΞϯτϥΠϒϥϦ΋ఏڙ͞Ε͓ͯΓ1ZUIPO (P /PEFKTͳͲʹରԠ͍ͯ͠Δɻ

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ͨͩ7BMEͩͱ/(5"0//(͔͠ରԠͯ͠ͳ͍͍͘͞ɻɻɻ ࠶ߏஙͱ෼ࢄΤϯδϯ͕೉ͦ͠͏ͩ͠ɻɻɻ OHUίϚϯυ΋Τοδௐ੔ܥΞϧΰϦζϜ͸໌ࣔతʹ࠶ߏஙΛୟ͘ܗࣜʹͳ͍ͬͯΔɻ ϕϯνϚʔΫݟΔͱΤοδௐ੔͕ಈతʹͰ͖ͳ͍ͱ·ͩ)/48ܥʹ܉ഋ্͕͕Γͦ͏ɻ

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ͻΌʔʔʔ

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͓·͚ େ·͔ͳ੩తͳ࣍਺ௐ੔ํ๏ͷग़ॴ ͳͥ͜ͷΞΠσΞ͕ग़ͯ͘Δ͔ɻ