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*33FBEJOHळ࿦จ঺հ
 "DDFMFSBUFE2VFSZ1SPDFTTJOH7JB
 4JNJMBSJUZ4DPSF1SFEJDUJPO 4*(*3 
 'BTU"QQSPYJNBUF'JMUFSJOHPG4FBSDI3FTVMUT 4PSUFECZ"UUSJCVUF 4*(*3 גࣜձࣾϝϧΧϦ
 ઙ໺୎໵

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"DDFMFSBUFE2VFSZ1SPDFTTJOH7JB 4JNJMBSJUZ4DPSF1SFEJDUJPO 4*(*3 w 63- w IUUQTEMBDNPSHDJUBUJPODGN JE w ݚڀͷ໨త w సஔΠϯσοΫεܕͷݕࡧΤϯδϯͰUPQLͷΫΤϦॲཧΛ
 ߴ଎Խ͢Δ͜ͱ w ํ๏ w ΫΤϦॲཧதͷಈతͳจॻͷࢬמΓ QSVOJOH ʹண໨͠ɺ
 ػցֶशΛԉ༻ͯ͠ΞϧΰϦζϜΛվળ

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લఏ w ҰൠతͳసஔΠϯσοΫεΛ૝ఆ w λʔϜͱϙεςΟϯάϦετ͕อଘ͞Ε͍ͯΔ w ϙεςΟϯάϦετʹ͸จॻ*%ʢιʔτࡁΈʣͷू߹͕ೖ͍ͬͯΔ w λʔϜ͝ͱͷ࠷େείΞͳͲͷิॿతͳσʔλ΋֨ೲͰ͖Δͱ͢Δ w ͜ͷݚڀͰԾఆ͢ΔείΞؔ਺ w جຊతʹ͸#.Λ૝ఆ w ΫΤϦʹؚ·ΕΔ֤λʔϜͷߩݙͷ଍͠߹Θͤ

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ΫΤϦॲཧ w EPDVNFOUBUBUJNF %""5 ʹΑΔΫΤϦॲཧΛ૝ఆ w λʔϜ͝ͱʹϙεςΟϯάϦετΛॲཧ͢ΔͷͰ͸ͳ ͘ɺෳ਺ͷϙεςΟϯάϦετΛಉ࣌ʹΧʔιϧͰॲཧ w ͢΂ͯͷΧʔιϧΛҰ൪લʹηοτ͠ɺͦͷจॻͷΫΤ Ϧॲཧ͕ऴΘͬͨΒޙΖʹͣΒ͢ w ͢΂ͯͷΧʔιϧ͕࠷ޙ·Ͱ͍ͬͨΒऴྃɺUPQLͷจ ॻू߹Λฦ͢

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ಈతࢬמΓʹΑΔΫΤϦॲཧ w ΫΤϦॲཧதʹɺͦΕ·Ͱʹॲཧͨ͠จॻͷUPQLͷείΞ Λܭࢉɾอ࣋͢Δʢ੨ઢʣ w ࠷ऴతʹ ʹऩଋ͢Δ w จॻͷείΞͷVQQFSCPVOEΛܭࢉ͢Δ w λʔϜ͝ͱͷ࠷େείΞ NBYTDPSF ΛΠϯσοΫεʹอଘ͓ͯ͘͠ͳͲ w VQQFSCPVOE͕ᮢ஋ ΑΓখ͍͞είΞͷจॻ͸҆શʹεΩοϓͰ͖Δ w ͜ͷํ๏͸ϩεϨεɺͭ·Γਖ਼֬ͳUPQLͷจॻ͕ಘΒΕΔ θ Θk θ

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طଘͷಈతࢬמΓΞϧΰϦζϜ w ޮ཰తͳΫΤϦॲཧͷํ๏ͱͯ͠.BY4DPSF 8"/% #.8 CMPDLNBY8"/% ͕͋Δ w ͢΂ͯجຊతͳઓུ͸લड़ͷಈతࢬמΓͱಉ͡ w 8"/%͸ݡ͍ΞϧΰϦζϜͰ.BY4DPSFΑΓͨ͘͞Μ εΩοϓͰ͖ΔΑ͏ʹ͍ͯ͠Δ w #.8 CMPDLNBY8"/% ͸8"/%ΛϒϩοΫԽ͠ɺ ֤ϒϩοΫʹ౷ܭ৘ใΛ΋ͭΑ͏ʹ͞Βʹվྑͨ͠΋ͷ

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͜ͷݚڀͷఏҊख๏ w ͢΂ͯͷจॻʹ͍ͭͯΫΤϦॲཧ͢Δͱɺ ͸ɺUPQLείΞ ʹऩଋ͢Δ w ػցֶशΛ࢖ͬͯΫΤϦͷಛ௃ྔ͔Β࠷ऴతͳᮢ஋ Λ༧ଌ͢Δʢ੺ઢʣ w ͜ΕʹΑΓɺଟ͘ͷจॻΛεΩοϓͰ͖ΔΑ͏ʹͳΔߴ଎ԽͰ͖Δ θ Θk (D, Q) Θk (D, Q)

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3FTFBSDI2VFTUJPOT w 32 w సஔΠϯσοΫεΛॲཧ࢝͠ΊΔલʹ ΛਪఆͰ͖Δ͔ʁ w 32 w ΋͠લ΋ͬͯ ΛਪఆͰ͖ΔͳΒɺͦͷࣄલ஌ࣝʹΑͬͯ
 Ͳͷఔ౓ΫΤϦॲཧΛ଎͘Ͱ͖Δ͔ʁ w 32 w ͷࣄલ஌ࣝΛ࢖ͬͯ.BY4DPSFɺ8"/%ɺ#.8ͷΑ͏ͳख๏ ΛվྑͰ͖Δ͔ʁ Θk (D, Q) Θk (D, Q) Θk (D, Q)

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ΦϥΫϧʢਆୗʣΛ࢖࣮ͬͨݧ w 32΋͠લ΋ͬͯ ΛਪఆͰ͖ΔͳΒɺͦͷࣄલ஌ࣝʹΑͬͯͲͷఔ ౓ΫΤϦॲཧΛ଎͘Ͱ͖Δ͔ʁ΁ͷճ౴ w ΛΫΤϦॲཧͷલʹܭࢉ͓͖ͯ͠ɺ֤ࢬמΓΞϧΰϦζϜʹᮢ஋ͱ͠ ͯ౉ͯ͠͠·͏ w ࠷దͳᮢ஋͕෼͔͍ͬͯΔঢ়ଶͰͷɺ."9ύϑΥʔϚϯεΛܭଌ w ࣮ݧ৚݅ w (PWσʔληοτɺ ΫΤϦͰ࣮ߦ࣌ؒΛϛϦඵ୯ҐͰܭଌ w ࢬמΓΞϧΰϦζϜͱͯ͠.BY4DPSF 8"/% #.8ͷͭΛ࢖ͬͯൺֱ Θk (D, Q) Θk (D, Q)

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ΦϥΫϧʢਆୗʣΛ࢖࣮ͬͨݧ w ΦϥΫϧΛ࢖͏ͱϕʔεϥΠϯ ΑΓ΋ʙ΄Ͳߴ଎Խ w Λਖ਼֬ʹਪఆ͢Δ͜ͱͰɺ ߴ଎Խͷ༨஍͕͋Δʂ ∇ Θk (D, Q)

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֬཰Ϟσϧϕʔεͷᮢ஋༧ଌ w ֬཰ϞσϧΛ࢖ͬͨ༧ଌ w ༧ଌ࣌ؒࠐΈͰશମͷΫΤϦॲཧΛ଎͘͠ͳ͍ͱ͍͚ͳ͍ͷͰɺγϯϓϧͳϞσϧͰ w ͷ෼෍͕Ψ΢ε෼෍Ͱ͋ΔͱԾఆ͠ɺύϥϝʔλΛֶशͯ͠༧ଌ෼෍ͱ͢Δ w ͦͷ෼෍ͷ্Ͱ࠷దͳ Λਪఆ͢Δʢ ʣ Θk ̂ Θk q = ?

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w PWFSFTUJNBUFͨ͠ͱ͖ʹΑΓϖφϧςΟΛ༩͍͑ͨ w ֬཰Ϟσϧʹଛࣦؔ਺Λ૊Έ߹ΘͤΔ w ࠓճ͸PWFSFTUJNBUFͨ͠ͱ͖ͷଛࣦΛ ഒͱ ͢ΔΑ͏ͳઢܗͷଛࣦؔ਺ʢ੨ઢʣΛߟ͑Δ α ֬཰Ϟσϧϕʔεͷᮢ஋༧ଌ w .JOJNVN#BZFT3JTL .#3 w ༧ଌ෼෍ͷ্Ͱɺଛࣦͷظ଴஋͕࠷খͱ ͳΔΑ͏ͳ Λબ୒Ͱ͖Δ w ࠷దͳ ͸ ͷRVBOUJMFͱͳΔ ͜ͱ͕ূ໌͞Ε͍ͯΔ<> w ͨͱ͑͹ɺ ͷͱ͖RVBOUJMF ࢛෼Ґ఺ ͕࠷ద w Ϟσϧʹ͸#BZFTJBO-JOFBS3FHSFTTJPO #-3 Λ࢖ͬͨ w ղੳతʹʢดܗࣜͰʣղ͕ٻΊΒΕΔͷ Ͱߴ଎ʹܭࢉͰ͖Δ ̂ Θk ̂ Θk 1/(α + 1) α = 3 <>1'$ISJTUP⒎FSTFOBOE'9%JFCPME0QUJNBMQSFEJDUJPOVOEFSBTZNNFUSJDMPTT&DPOPNFUSJD5IFPSZ r

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ଛࣦϕʔεͷᮢ஋༧ଌ w ଛࣦؔ਺Λ࢖ͬͨػցֶशϞσϧ΋ߟ͑Δ w ༩͑ΒΕͨଛࣦؔ਺ʹରͯ͠௚઀ֶश w ଛࣦؔ਺ʹ͸ඇରশʹͨ͠)VCFSMPTTΛ࠾༻ w ඍ෼Մೳ w ϚϧνϨΠϠʔύʔηϓτϩϯ .-1 w .-1-
 ૚ʢӅΕ૚͕ʣͷ.-1 w .-1-
 ૚ʢӅΕ૚͕ʣͷ.-1 w ͍ͣΕ΋$16Ͱޮ཰తʹܭࢉՄೳ

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Ϟσϧͷֶश w Ϟσϧͷֶशʹ࢖ͬͨΫΤϦϩά w .JDSPTPGU2VFSZMPH w ΫΤϦ w σʔληοτ w (PWສ݅ w $MVF8FC#ສ݅ w ֶशࡁΈϞσϧͷ༧ଌ଎౓ w NT .-1- w NT .-1- w NT #-3 w ͍ͣΕ΋NTҎ಺ͳͷͰे෼଎͍ ֶशʹ࢖ͬͨಛ௃ྔ

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ਪఆͨ͠ᮢ஋Λ࢖ͬͨ
 ΫΤϦॲཧ PWFSFTUJNBUFͯͨ͠Β࠶ܭࢉ ώʔϓαΠζ͕༰ྔʢLݸʣΛ௒͑ͨͱ͖ͷॲཧ ώʔϓͷ࠷খείΞΑΓେ͖͚Ε͹ώʔϓʹೖΕΔ είΞͷVQQFSCPVOE͕ Λ௒͑Δ
 จॻ͚ͩείΞܭࢉ θ ᮢ஋ͷ༧ଌ஋ Λ ͷॳظ஋ͱ͢Δ ̂ Θk θ

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"MHPSJUINʹΑΔ
 ΫΤϦॲཧͷ࣮ߦ଎౓ ϕʔε
 ϥΠϯ ϕʔε
 ϥΠϯ ύʔηϓτϩϯ #BZFTJBO-JOFBS3FHSFTTJPO গ͚ͩ͠଎͘ͳͬͨ

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"MHPSJUIN͕஗͍ཧ༝ w PWFSFTUJNBUFͨ͠ͱ͖ʹ ࠶ܭࢉ͕૸Δ w ͜Ε͕ॏ͍

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"MHPSJUINʢվྑ൛ʣ w ໰୊ʹͳ͍ͬͯΔλʔϜͷ૊Έ߹Θͤ ʢαϒΫΤϦʣΛܭࢉ w αϒΫΤϦʹϚον͢Δจॻ͚ͩ
 ݟΕ͹Α͍ w จॻ͕͔ͳΓߜΕΔͷͰߴ଎ʹ
 ࠶ܭࢉՄೳ

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"MHPSJUINʹΑΔ
 ΫΤϦॲཧͷ࣮ߦ଎౓ w தԝ஋Ͱ࠷େͰ଎͘ͳͬͨ w ͨͩ͠ಉ͡ Ͱൺ΂ͨͱ͖ w ࠷ద஋ͱൺ΂Δͱ͘Β͍ w "MHPSJUIN͸RVBOUJMF͕େ͖͍
 ʢΞάϨογϒͳʣ΄͏͕଎͍ q = 0.5

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3FTFBSDI2VFTUJPOT w 32 w సஔΠϯσοΫεΛॲཧ࢝͠ΊΔલʹ ΛਪఆͰ͖Δ͔ʁ w ˠώʔϓͷᮢ஋͕༧ଌՄೳͰ͋Δ͜ͱΛࣔͨ͠ w 32 w ΋͠લ΋ͬͯ ΛਪఆͰ͖ΔͳΒɺͦͷࣄલ஌ࣝʹΑͬͯ
 Ͳͷఔ౓ΫΤϦॲཧΛ଎͘Ͱ͖Δ͔ʁ w ˠಈతࢬמΓΞϧΰϦζϜͷ଎౓͕Ҏ্վળ͢Δ͜ͱΛࣔͨ͠ w 32 w ͷࣄલ஌ࣝΛ࢖ͬͯ.BY4DPSFɺ8"/%ɺ#.8ͷΑ͏ͳख๏ΛվྑͰ͖Δ͔ʁ w ˠ͢΂ͯͷख๏ͰΞϧΰϦζϜతͳվળ͕ՄೳͰ͋Δ͜ͱΛࣔͨ͠ Θk (D, Q) Θk (D, Q) Θk (D, Q)

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ײ૝ w ܾఆతͳํ๏ʢ Λ࢖͏ϕʔεϥΠϯʣͰ΋े෼Ͱ͸ʁ w 5BCMFΛΈΔͱɺϕʔεϥΠϯ͔Βͷࠩ͸ͦΕ΄Ͳେ͖͘ͳ͍ w "MHPSJUINͰଟগվળ͢Δͱ͸͍͑ɺࠩ෼͸͍͍ͤͥఔ౓ w ࢬמΓΞϧΰϦζϜͷࠩͷ΄͏͕େ͖͍ w ࢓૊Έ͕γϯϓϧͳ ͷ΄͏͕औΓճ͠΍͍͢ʢػցֶश͍Βͳ͍͠ʣ w ΋ͬͱ͍͍ϞσϧΛ࡞ΕΔ༨஍͸͋Δ w ಛ௃ྔ΍ػցֶशΞϧΰϦζϜͷվળ͕Մೳ w ਅͷᮢ஋ʢΦϥΫϧʣʹ͚ۙͮ͹ͦΕͳΓʹ଎͘ͳΔ Qk Qk

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'BTU"QQSPYJNBUF'JMUFSJOHPG4FBSDI 3FTVMUT4PSUFECZ"UUSJCVUF 4*(*3 w 63- w IUUQTEMBDNPSHDJUBUJPODGN JE w ໰୊ઃఆ w ଐੑʹΑΔιʔτॱΛकΓͳ͕Β%$(ͷΑ͏ͳϝτϦοΫΛ࠷దԽ͢Δ໰୊ w ͜ͷݚڀͷ੒Ռ w ߴ଎ʹ΄΅࠷దͳ݁ՌΛฦ͢ΞϧΰϦζϜΛఏҊ w ۙࣅʹΑΔΤϥʔ཰ͱ࠷ѱ࣌ؒܭࢉྔͷཧ࿦อূ͋Γ w ࣮ݧతʹ΋ߴ଎ੑͱΤϥʔ཰ͷόϥϯε͕Α͍͜ͱ͕ࣔ͞Εͨ

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໰୊ઃఆ'JMUFSJOH!L w ௨ৗɺϢʔβ͕ݕࡧͰཉ͍͠ͷ͸ SFMFWBODFΛ࠷େԽͨ͠4&31 w 8FCݕࡧɺ4/4ɺετϦʔϛϯά αʔϏεɺ&$αΠτͳͲ w ଟ͘ͷαʔϏεͰ͸ɺଐੑʢ࣌ؒ΍Ձ֨ ͳͲʣʹΑΔιʔτʹରԠ͍ͯ͠Δ w ͨͱ͑͹"NB[POͰJQIPOFͷݕࡧ݁ ՌΛՁ֨ͷ͍҆ॱʹ͢ΔͱSFMFWBODFͷ
 ௿͍ΞΠςϜ্͕ʹདྷΔͨΊɺϢʔβମ ݧΛଛͳ͏ BNB[PODPKQͰՁ֨ͷ͍҆ॱͰJQIPOFΛݕࡧ
 ʢՈిˍΧϝϥΧςΰϦʣ

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໰୊ઃఆ'JMUFSJOH!L w 'JMUFSJOH!L w SFMFWBODFͷϦετ ʢॱং͖ͭʣ͕
 ༩͑ΒΕΔ w ϝτϦοΫ2Λ࠷େԽ͢Δɺ࠷େL݅ͷ෦෼Ϧετ
 Λฦ͢ R = < r1 , . . . , rn >

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0QUJNBM'JMUFSJOH 015 w ࣌ؒܭࢉྔ Ͱ࠷దͳ݁Ռʢۙࣅͳ͠ʣΛܭࢉͰ͖Δ w ಈతܭը๏ʢϝϞԽʣΛ࢖ͬͯܭࢉ w O L͕େ͖͘ͳΔͱ஗͍ Θ(nk)

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ώϡʔϦεςΟΫεʹΑΔํ๏ w $VUP⒎015 w ͋Β͔͡ΊܾΊͨSFMFWBODFͷᮢ஋ͰϑΟϧλ͔ͯ͠Β015Ͱ͞ΒʹϑΟϧλ w ࠷ѱέʔεͰ͸Τϥʔ཰ʢਫ਼౓ʣͷอূ͕ͳ͍ w ᮢ஋ͷઃఆʹΑΔͨΊ w ࣌ؒܭࢉྔ͸վળͤͣ࠷ѱέʔεͰ w 5PQL015 w UPQLͷϦετΛٻΊɺͦΕΛ͞Βʹ015ͰϑΟϧλ͢Δ w LݸະຬͷϦετʹ͢Δ͜ͱͰϝτϦοΫΛ࠷େԽͰ͖Δ͜ͱ΋͋ΔͨΊ w PQUJNBMʢۙࣅʹΑΔΤϥʔ཰͕ʣͰ͋Δ͜ͱ͕ূ໌Ͱ͖Δ w ࣌ؒܭࢉྔ͸࠷ѱͰ Θ(nk) Θ(n log k + k2)

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Џ'JMUFSJOH w $VUP⒎΍5PQLͳͲͷώϡʔϦεςΟΫεͰ͸Τϥʔ཰ΛίϯτϩʔϧͰ͖ͳ͍ w Џ'JMUFSJOH͸Τϥʔ཰ΛЏҎԼʹอূͨۙ͠ࣅղΛɺ
 ࣌ؒܭࢉྔ Ͱݟ͚ͭΔʢ ʣ w Џ'JMUFSJOH͸ҎԼͷεςοϓ͔ΒͳΔ w 3JHIUQSVOJOH w %JTDSFUJ[BUJPO w 5ISFTIPMEJOH w ֤ϑΣʔζͰى͜Γ͏Δ࠷ѱέʔεʹɺޙͷϑΣʔζͰରԠ͢Δ͜ͱͰ
 ܭࢉ࣌ؒΛ࡟͍ͬͯ͘ Θ(n + k2 log(1−ϵ) (ϵ/k)) 0 < ϵ < 1

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࣮ݧ݁Ռ w Џ'JMUFSJOH͸5PQL015ͱݞΛฒ΂Δ଎͞Ͱɺখ͍͞Τϥʔ཰Λୡ੒͍ͯ͠Δ w Џ'JMUFSJOHͷ ɺ Ͱ͸Τϥʔ཰Λ΄΅ʹ཈͑ͳ͕Β015ΑΓ΋அવ଎͍ ϵ = 0.01 ϵ = 0.001