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一般グラフの最大マッチング
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yosupo
March 26, 2017
3
2k
一般グラフの最大マッチング
yosupo
March 26, 2017
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Transcript
Ұൠάϥϑͷ ࠷େϚονϯά ాɹߊฏ
Ұൠ࠷େϚονϯάͱͳΜͧ w άϥϑ͕༩͑ΒΕΔͷͰɺล͕࠷େͷϚονϯά ΛٻΊ͍ͯͩ͘͞ w ϚονϯάͱɺάϥϑͷลΛɺͲͷลಉ࢜ Λڞ༗͠ͳ͍Α͏ʹ͍͔ͭ͘બΜͩͷͰ͢ w ͳ͓ɺ͜ͷߨٛͰجຊతʹάϥϑ࿈͔݁ͭ୯७ ͳͷͱ͠·͢
Ϛονϯάͷྫ
ϚονϯάͰͳ͍ྫ
lҰൠz࠷େϚονϯά w ͳΜͰzҰൠzͱ͍͍ͭͯΔͷʁ w ࣮ɺ༩͑ΒΕΔάϥϑΛೋ෦άϥϑʹݶఆ͢Δͱ ͬͱָͳʹͳΓ·͢ ڈͷߨٛͰͬͨʁ w ͜Ε͕ೋ෦࠷େϚονϯάͱݺΕ͍ͯͯɺ༩͑Β
ΕΔάϥϑ͕ҰൠάϥϑʹͳΔͱҰൠ࠷େϚονϯ άͰ͢
ܭࢉྔ w 0 7? ۪ͳ࣮ ΘΓ͔͠؆୯ w 0 7?
಄ྑ࣮͘ &ENPOETͷՖղΞϧΰϦ ζϜ ͜͜·ͰΓ·͢ w 0 㲋7 & 46(0* ࠓճΓ·ͤΜ
ࢀߟจݙ w ߹ͤ࠷దԽୈೋ൛ #ίϧς+ϑΟʔήϯஶ ઙઙହ ਔখஉฏా༁ ؙળग़൛ฏ
࣍ ᶃ .૿Ճύε ᶄ ด࿏Λ௵͢ ᶅ 0 7? ᶆ
0 7? લ ᶇ 0 7? ޙ
ᶃ.૿Ճύε
ᶃͷॏཁϙΠϯτ άϥϑͱϚονϯάʹ͍ͭͯ .૿Ճύε͕͋Δ 㱻࠷େϚονϯάͰͳ͍
.૿Ճύε w άϥϑͱϚονϯά͕༩͑ΒΕΔ w ҎԼͷ݅Λຬͨ͢ͷ͕.૿Ճύε w ͕͞ح w ྆ͷ͕ϚονϯάʹΘΕ͍ͯͳ͍ w
ύεͷล͕Ϛονϯάʹɺ ؚ·Εͳ͍ ؚΉ ؚ·Εͳ͍ ؚΉ ʜ ؚ·Εͳ͍
.૿Ճύεͷྫ
ॏཁϙΠϯτ .૿Ճύε͕͋Δ 㱻࠷େϚονϯάͰͳ͍ ͜ΕΛࣔ͢
ॏཁϙΠϯτ w .૿Ճύε͕͋Δ 㱺࠷େϚονϯάͰͳ͍ w ࠷େϚονϯάͰͳ͍ 㱺.૿Ճύε͕͋Δ
ॏཁϙΠϯτ w .૿Ճύε͕͋Δ 㱺࠷େϚονϯάͰͳ͍ w ࠷େϚονϯάͰͳ͍ 㱺.૿Ճύε͕͋Δ
.૿Ճύε͕͋Δ㱺࠷େϚονϯάͰͳ͍ w ͬͪ͜؆୯ w .૿Ճύε ؚ·Εͳ͍ ؚΉ ؚ·Εͳ͍ ؚΉ ʜ
ؚ·Εͳ͍ ͳͷͰɺ͜ΕΛ ؚΉ ؚ·Εͳ͍ ؚΉ ؚ·Εͳ͍ ʜ ؚΉ ʹ͢Δ w ͜ΕͰϚονϯά͕૿ͤͨͷͰ0,
ྫ
ॏཁϙΠϯτ w .૿Ճύε͕͋Δ 㱺࠷େϚονϯάͰͳ͍ w ࠷େϚονϯάͰͳ͍ 㱺.૿Ճύε͕͋Δ
࠷େϚονϯάͰͳ͍㱺.૿Ճύε͕͋Δ w ࣮ͬͪ͜؆୯ w άϥϑͱ࠷େͰͳ͍Ϛονϯά.Λߟ͑ɺ. ʹ૿Ճύε͕ଘࡏ͢Δ͜ͱΛࣔ͢ɻ͜͜Ͱɺͦͷά ϥϑͷ࠷େϚονϯάΛ ͳΜͰ͍͍ͷͰબͼ . ͱ͢Δɻ
w &.YPS.Λߟ͑Δ &ɺ.͔.ͷͲͬͪ ͔ʹؚ͚ͩ·ΕΔลͷू߹
ྫ . .
ྫ . . .?.
w &ʹͲΜͳੑ࣭͕͋Δ͔ʁ w ͯ͢ͷͷ࣍d 㲎. .ͱʹ͔࣍ w ͭ·Γύε͔αΠΫϧ͔͠ͳ͍ w
࣍ͷ͔Βݩ.ͱݩ.ͷล͕ຊͣͭ w ߋʹɺݩ.ͷลݩ.ͷลΑΓଟ͍ w ͭ·Γɺඞͣ. . . ʜ .ͱ͍͏ύε͕ଘࡏ͢Δ ˠ͜Εɺ.ͷ.૿Ճύε
ॏཁϙΠϯτ ࠶ܝ .૿Ճύε͕͋Δ 㱻࠷େϚονϯάͰͳ͍
ᶄด࿏Λ௵͢
ͦͦ w ᶃͰɺάϥϑͷ.૿ՃύεΛൃݟͰ͖ΕϚον ϯά͕૿ͤͯϋοϐʔͱ͍͏͜ͱ͕Θ͔ͬͨ w దʹൃݟ͠Α͏ͱ͢ΔͱͲ͏ͳΔͩΖ͏ʁ
؆୯ͳ w άϥϑͱT U͕༩͑ΒΕ·͢ w T͔ΒUͷύεΛൃݟ͍ͯͩ͘͠͞
ղ๏ w EGT όοΫτϥοΫ w ୳ࡧͨ͠Λ͏୳ࡧ͠ͳ͍Α͏ʹͨ͠ Β 0 7
w άϥϑͱϚονϯά͕༩͑ΒΕ·͢ w .૿ՃύεΛݟ͚͍ͭͯͩ͘͞
ղ๏ w EGT όοΫτϥοΫ w ελʔτɺΰʔϧϚονϯάʹͬͯͳ͍ w ୳ࡧͨ͠ Ϛονϯάͷล͔
Λ͏୳ࡧ ͠ͳ͍Α͏ʹͨ͠Β 0 7 w Λ લͬͨͷϚονϯά ͱ લ ͬͨͷϚονϯάͰͳ͍ ʹഒՃ͢Δͱߟ͑ ΒΕΔ
ྫ
ྫ ͳΜͩ͜Εʜͨ·͛ͨͳ͋
ղ๏JTӕ
Կ͕͍͚ͳ͔ͬͨͷ͔ w ղ๏ͰɺΛछྨʹഒՃͯ͠EGT͍ͯ͠Δ w ͜ͷछྨͷΛ྆ํ͍ͬͯΔύεͱ͍͏ͷΛ ݟ͚ͭͯ͠·͏ɻ w ͜ΕΛഒՃͨ͠ޙͳΒύεʹͳ͍ͬͯΔ͕ɺ ݩͷάϥϑʹ͢ͱด࿏͕ੜ·Εͯ͠·͏
w ͜ͷEGTͩͱɺ.૿Ճύε͚ͩΛਖ਼֬ʹݟ͚ͭΒΕ ΔΘ͚Ͱແ͍ w ͰɺଟগͷใಘΒΕΔ w ͠ɺݟ͚ͭͨͷ͕ύεͰແ͍ ด࿏͕͋ͬ ͨ ͱͨ͠ΒҎԼͷܗ͕ಘΒΕΔ
ۮຊ ด࿏ͷ͞ح
w ด࿏ͷ͞حˠΛഒՃͨ݁͠Ռύεʹͳ Δ͔Β w ࠷ॳͷχϣϩͷ͕͞ۮˠด࿏ͷҰ൪࠷ޙͷล ϚονϯάʹΘΕ͍ͯͳ͍ล ۮຊ ด࿏ͷ͞ح
͜ΕΛͲ͏͢Δʁ
None
ʁʁʁ w ࣮͜͏͍͏ด࿏௵ͯ͠ɺz࠷େϚονϯά͔ Ͳ͏͔zมΘΒͳ͍ w ด࿏Λݟ͚ͭͨΒɺ௵͢ˠϚονϯάΛ૿͢ˠ ͢ɺͱݴ͏͜ͱΛߦ͑Α͍ w ௵ͨ݁͠ՌͷάϥϑͰϚονϯά͕૿ͤͨΒඞͣ ݩͷάϥϑͰϚονϯά͕૿ͤΔ
w ˢ͜͏͍͏ܗͷด࿏Λ௵ͯ͠ɺz࠷େϚονϯά ͔Ͳ͏͔zมΘΒͳ͍ w ͜ΕΛࣔ͢ w ۮຊɺด࿏ͷ͞حɹΊͪΌͪ͘Όॏཁ ۮຊ ด࿏ͷ͞ح
લ४උ ͜ΕΛ
લ४උ ͜͏
݁Ռ ح ্ਤͷΑ͏ʹɺ ࠷ॳͷχϣϩͷ͕͞ͷ߹͚ͩߟ͑Εྑ͍
௵͢લ͕࠷େϚονϯά 㱻௵͢ޙ͕࠷େϚονϯά ͜ΕΛࣔ͢
௵͢લ͕࠷େϚονϯάͰͳ͍ 㱻௵͢ޙ͕࠷େϚονϯάͰͳ͍
௵͢લ͕࠷େϚονϯάͰͳ͍ 㱺௵͢ޙ͕࠷େϚονϯάͰͳ͍ ௵͢ޙ͕࠷େϚονϯάͰͳ͍ 㱺௵͢લ͕࠷େϚονϯάͰͳ͍
௵͢લ͕࠷େϚονϯάͰͳ͍ 㱺௵͢ޙ͕࠷େϚονϯάͰͳ͍ ௵͢ޙ͕࠷େϚονϯάͰͳ͍ 㱺௵͢લ͕࠷େϚονϯάͰͳ͍
ᶃΛ͔ͭ͏ ௵͢લʹ.૿Ճύε͕͋Δ 㱺௵͢ޙʹ.૿Ճύε͕͋Δ
w ௵͢લͷάϥϑʹ.૿Ճύε͕͋ΔͱԾఆ͢Δ w ͜Ε͕௵ͨͪ͢Λͬͯͳ͔ͬͨΒવ௵ͨ͠ޙʹ ͋Δ w ௵ͨ͠ͷ͏ͪͲΕ͔ΛͬͯͨΒʁ w ·ͣɺ௵ͨͨͪ͠ͷ͏ͪɺϚονϯάʹͬͯͳ͍ ݸ͚ͩͳͷͰɺύεͷͷ͏ͪͲ͔ͬͪ௵͢
Ͱແ͍ w ͔ͦ͜ΒύεΛݟ͍ͯ͘ͱɺͲ͔͜Ͱ௵͢ʹͿ͔ͭΔ
w ύεΛݟͯߦ͘ͱ͍͔ͭ௵͢ͷ͏ͪͲΕ͔͕ग़ͯ͘ Δ w ͚ͩͲ௵ͨͪ͢ɺͯ͢௵͢ಉ࢜ͰϚονϯ άͯ͠ΔPSϚονϯάͯ͠ͳ͍ w Αͬͯɺ௵͢ʹzϚονϯάͯ͠ͳ͍ลzͰೖΓࠐΉ w ͱ͜ΖͰ௵ͨ͠ޙϚονϯάʹͬͯͳ͍ʹͳΔ
w Αͬͯɺͦ͜ͰύεΛଧͪΕɺ௵ͨ͠ޙͷάϥϑͰ .૿ՃύεͱͳΔ
ࣔͤͨ ௵͢લʹ.૿Ճύε͕͋Δ 㱺௵͢ޙʹ.૿Ճύε͕͋Δ ͭ·Γ ௵͢લ͕࠷େϚονϯάͰͳ͍ 㱺௵͢ޙ͕࠷େϚονϯάͰͳ͍
௵͢લ͕࠷େϚονϯάͰͳ͍ 㱺௵͢ޙ͕࠷େϚονϯάͰͳ͍ ௵͢ޙ͕࠷େϚονϯάͰͳ͍ 㱺௵͢લ͕࠷େϚονϯάͰͳ͍
w ·ͣɺ௵͢ͱ͖ʹඞͣϚονϯάͷຊ ด࿏ͷ ͞ ݸݮΔ w ٯʹɺ௵ͨ͠ޙͷάϥϑͷϚονϯά͕༩͑ΒΕΕ ɺ ด࿏ͷ͞ ݸ૿͑ͨϚονϯάΛݩͷά
ϥϑͰߏஙͰ͖Δ w ௵ͨ͠Λͬͯͳ͔ͬͨΒͦͷ··͢ w ͬͯͯɺݩͷάϥϑͰͷͦͷϚονϯάʹͬ ͯΔҎ֎Λ͚ͬͭ͘Δ
͜͏ͳͬͨͱͨ͠Β
͜͏
͜͏ͳͬͨͱͨ͠Β
͜͏
w ௵ͨ݁͠Ռ͕࠷େϚονϯά͡Όແ͍ͱ͢Δ w ௵ͨ݁͠ՌͰͷ࠷େϚονϯάΛߟ͑ͯɺͦΕΛ ͢͜ͱʹΑΓɺݩͷάϥϑͰͷݩʑͷϚονϯάΑ Γຊͷଟ͍Ϛονϯά͕ಘΒΕΔ w ͭ·ΓݩͷάϥϑͰ࠷େϚονϯάͰແ͍
ࣔͤͨ ௵͢ޙ͕࠷େϚονϯάͰͳ͍ 㱺௵͢લ͕࠷େϚονϯάͰͳ͍
ࣔͤͨ ௵͢લ͕࠷େϚονϯάͰͳ͍ 㱻௵͢ޙ͕࠷େϚονϯάͰͳ͍
ࣔͤͨ ௵͢લ͕࠷େϚονϯά 㱻௵͢ޙ͕࠷େϚονϯά
0 7? ΞϧΰϦζϜ
0 7? w طʹ͔͍ͬͯΔਓ͍Δ͔ʁ w άϥϑͱϚονϯάΛ༩͑ͯɺϚονϯάΛ૿͠ ͯฦ͕ؔ͢࡞ΕΕྑ͍ ͜ΕΛ૿͑ͳ͘ͳΔ· Ͱݺͼଓ͚Δ
0 7? w ؔͷதɺ w దʹ.૿ՃύεΛ୳ͦ͏ͱ͢Δ w ͳ͔ͬͨΒ͏࠷େϚονϯάͳͷͰ0, w ݟ͚ͭͨͷ͕ύεͩͬͨΒϚονϯάΛͦΕͰ૿ͤ0,
w ด࿏͕͋ͬͨΒʁ w ด࿏Λ௵ͨ͋͠ͱͷάϥϑͱϚονϯάΛҾʹɺ࠶ؼ తʹ͜ͷؔΛݺͿˠ෮ݩͯ͠SFUVSO
ܭࢉྔ w ·ͣɺҰ൪֎ଆͰؔΛݺͿճ͕0 7 w ؔͷதͰ࠶ؼతʹ0 7 ճ͙Β͍ؔΛݺͿ ด
࿏Λ௵͢ʹݸͣͭݮΔ w ͦͷதͰɺ.૿ՃύεΛݟ͚ͭΑ͏ͱ͢Δͷ ʹ0 7 & 0 7? w ݁Ռͱͯ͠ɺ0 7?
0 7? લ
લJTԿ w FENPOETͷՖղΞϧΰϦζϜͰɺ w άϥϑͱϚονϯά͕༩͑ΒΕͨ࣌ʹϚονϯά Λ૿ͯ͠ฦؔ͢Λ࡞Γɺ0 7 ճݺͿ w ·ͰมΘΓ·ͤΜ
w ͜ͷؔΛɺ࠶ؼతʹؔΛݺͿͷΛΊΔ͜ͱ ʹΑΓ0 7? ʹ͠·͢
લJTԿ w ͔͠͠ɺ͍͖ͳΓϚονϯάΛ0 7? Ͱ૿͢ͷ ͍͠ w ͱΓ͋͑ͣ࠷େϚονϯά͔Ͳ͏͔͚ͩΛ0 7? Ͱఆ͢Δ͜ͱΛߟ͑·͢
w .૿ՃύεΛ෮ݩ͢Δύʔτ͕͔ͳΓ͍͠ʜ
0 7? w 0 7? ൛ʹແବͳͱ͜Ζͳ͍ͩΖ͏͔ʁ w .૿ՃύεΛൃݟ͠Α͏ͱͯ͠ɺحͷαΠΫϧ͕ ͋ͬͨΒ௵͢ˡ͜͜·ͰΑ͍ w
ͦͷ͋ͱɺ·͔ͨΒEGTΛ࢝ΊΔͷ͕ແବ w EGTΛ͠ͳ͕ΒɺحͷαΠΫϧ͕͋ͬͨΒద࣌௵ ͯ͘͠ɺΈ͍ͨͳײ͡ͰΕ͏·͍͜ͱճEGT ͢Δ͚ͩͰͲ͏ʹ͔ͳΒͳ͍ͩΖ͏͔
0 7? w ୳ࡧΛ࡞͍ͬͯ͘ w ࠜϚονϯάʹΘΕ͍ͯͳ͍ͨͪ
w ϚονϯάʹΘΕ͍ͯͳ͍ͨͪਂ͞ w ਂ͕͞ۮͷΛۮɺحͷΛحͱݺͿ w ۮͱࢠΛͭͳ͙ลɺϚονϯάʹΘΕ͍ͯͳ ͍ล w حͱࢠΛͭͳ͙ลɺϚονϯάʹΘΕ͍ͯΔ
ล
͜͏͍͏ײ͡ͷΛ࡞͍ͬͯ͘ ۮ ح ۮ ح ۮ
୳ࡧ w ۮ͔Βੜ͑ΔลΛݟ͍ͯ͘ w ลͷ૬खͷछྨʹΑΓɺύλʔϯʹ߹͚͢Δ w ʹݱΕ͍ͯͳ͍ˠ૬खͱɺͦͷϚονϯάઌΛʹ Ճ͢Δ w حˠԿ͠ͳ͍
w ҧ͏ͷۮˠʁʁ w ಉ͡ͷۮˠʁʁ
ҧ͏ͷۮͷ߹ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
ҧ͏ͷۮͷ߹ w ลͷࠨӈΛY Zͱ͢Δͱɺ YͷͷࠜdYͷύε Y Z ZdZͷͷࠜͷύε
ɺ.૿Ճύεʹͳͬ ͍ͯΔ w Αͬͯɺҧ͏ͷۮಉ࢜Λͭͳ͙ล͕͋ͬͨΒͦ Ε࠷େϚονϯάͰແ͍
ಉ͡ͷۮͷ߹ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
͜Εͱ͔ ۮ ح ۮ ح ۮ
ಉ͡ͷͷ߹ w YͱZΛܨ͍ͰΔͱ͢Δ w ࣮ɺMDB Y Z [ͱ͢Δͱɺ Yd[dZ ͱ
Y Z ͔ΒͳΔด࿏ɺ௵ͯ͠ྑ͍ด࿏ ʹͳ͍ͬͯΔ w ͳͷͰɺಉ͡ͷۮಉ࢜Λͭͳ͙ล͕ ͋ͬͨΒ௵͢ Y Z [
௵ͯ͠ྑ͍ด࿏ ࠶ܝ ۮຊ ด࿏ͷ͞ح
w ͜ΕΒΛ܁Γฦ͢ͱɺ͍͔ͭ.૿Ճύε͕ݟ͔ͭΔ͔ɺ͘͠ ɺߋʹۮͱͭͳ͕Δลͷઌ͕ح͚ͩʹͳΔ w ۮͱͭͳ͕Δลͷઌ͕ح͚ͩͳͷͰɺۮͷ͏ͪϚονϯάʹ ͑ΔͷͷݸحͷݸҎԼ w ͨͩحશͯۮͱશͯϚονϯά͍ͯ͠Δ w ͔͠ϚονϯάʹΘΕ͍ͯͳ͍શͯۮ
w Αͬͯ͜ΕҎ্Ϛονϯά૿ͤͳ͍ w Αͬͯ͜͏ͳΔ·Ͱ܁Γฦͤྑ͍ w ͜ΕͳΒదʹΔͱ0 7? Ͱ࠷େϚονϯά͔ఆͰ͖·͢
0 7? ޙ
ޙ w ࠷େϚονϯά͔0 7? Ͱఆ͢ΔͷΘΓ͔͠ ؆୯ w ͚ͩͲ࣮ࡍ૿ͦ͏ͱ͢Δͱɺେมʜ w ૿ՃύεΛ͏·͘෮ݩͰ͖ΔΑ͏ʹ͠ͳ͍͚ͯ͘
ͳ͍
૿Ճύεͷ෮ݩ w ૿Ճύε͕ඞཁʹͳΔͷɺҧ͏ͷۮಉ࢜Λͭ ͳ͙ล͕ݟ͔ͭͬͨ࣌ w ͭ·Γɺͷۮʹ͍ͭͯͦͷ͔Β࢝·Γɺࠜ ͰऴΘΔΑ͏ͳɺzϚονϯάʹͬͯΔ ͬͯͳ ͍ ͬͯΔ
ͬͯͳ͍ ʜ ͬͯͳ͍zύε͕෮ ݩͰ͖ΕΑ͍
૿Ճύεͷ෮ݩ w ͬͯΔ ͬͯͳ͍ ʜ ͬͯͳ͍ͱ͍͏ύεͷ෮ݩ͕ ͍ͨ͠ w ϚονϯάʹͬͯΔลͱ͍͏ͷ͝ͱʹҰҙʹఆ· Δ
w Ϛονϯάʹͬͯͳ͍ลɺ͜ΕͲΕΛબ͍ ͍ΜͩΖ͏ʁ ˠٯʹɺʹ͍ͭͯ͜ͷz࣍ʹ͏͖Ϛονϯάʹͬ ͯͳ͍ลͲΕ͔zΛมʹ࣋ͬͯɺ͏·͘ཧ͢Εʁ
૿Ճύεͷ෮ݩ w ҎԼͷछྨͷมΛߟ͑Δ w NU V VͱϚονϯάͯ͠Δ NBUDI w
OY V Vͱͷؒʹล͋Δ͚ͲɺϚονϯάͯ͠ͳ ͍ͷ͏ͪͲΕ͔ͭ OFYU
NU OYͷ੍ w ҙͷۮVʹ͍ͭͯɺ w NU V OY NU V
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ະଓͷՃ w ͜Ε؆୯ w OY V ͭͳ͙Ͱ0, w W͔ΒTUBSU͢Δͱύεɺ w
W V ͭͳ͙ dͱͳΔ ۮ V W ͭͳ͙
Λ௵͢ w ͜Εେม w ·ͣۮ͕શ෦ݸͷ߹ V W
Λ௵͢ w ͜͏͢ΕΑͦ͞͏ʁ V W
Λ௵͢ w ۮͷαΠζ͕શ෦ͷͱ͖ɺ ·ͣVdMDBͱWdMDBͷύεΛٻ ΊΔ w ͦͷύε্ͷMDBҎ֎ͷۮYʹͭ ͍ͯ OY Y
ͦͷύεͷݸ༿ଆ ʹ͢Δ w ࣮ࡍۮ͕αΠζͰͳ͍ͨ Ίɺେม V W
Λ௵͢ V W -$"
͜͏ͯ͠ V W
͜͏ V W
Λ௵͢ w VͱW͔Β࢝·Δ.૿ՃύεΛ࡞Δ w -$" ͭͷύε͕௨Δɺ࠷ࠜʹ͍ۙۮ ΛٻΊΔ w -$"·ͰͷɺϚονϯάʹ௨Βͳ͍ลΛํʹҠಈͰ ͖ΔΑ͏ʹ͢Δ
V W͔ΒۮճਐΜͩYʹ͍ͭͯɺY ͷॴଐ͢Δۮ͕-$"Ҏ্Ͱͳ͚ΕɺOY Y ύεͷ ݸલ w -$"Ҏ্ͳͷɺલਤͷΑ͏ʹɺV͔ΒͷύεͱW͔Βͷ ύε-$"ͷதͰަΘΔ͜ͱ͕͋Δ͔Β
NU OYͷ੍ ࠶ܝ w ҙͷۮVʹ͍ͭͯɺ w NU V OY NU
V NU OY NU V OY NU OY NU V ʜ OY NU ʜ V ʜ w ͕ɺzV͔Β࢝·ΓɺͦͷͷࠜͰऴΘΔύεzͱͳ ΔΑ͏ʹ͍ͨ͠ w Λ௵ͨ͋͠ͱ͜ΕΛຬ͍ͨͯ͠Δ͔ʁ
NU OYͷ੍ w ·ͣɺ-$"ͷࢠ -$"আ͘ ͚ͩߟ͑Εྑ͍ w V͔ΒͱW͔Βͷύεʹؚ·ΕΔશͯ0, w ͦΕҎ֎ɺͲ͔͜ͰV͔ΒͱW͔ΒͷύεʹͿͭ
͔Δ͕ɺͿ͔ͭͬͨΒύεͷ͚ͩΛͬͯ-$" ·ͰӡΕΔͷͰ0,
.૿ՃύεΛ૿͢ w ҧ͏ͷۮΛͭͳ͙߹ w ྆ํͷ͔ΒύεΛ࡞ͬͯɺదʹ૿ͤ0, w Ҏ্ΑΓɺ0 7? ͰϚονϯά͕૿ͤΔ͜ͱ͕ Θ͔ͬͨ
͜ΕͰ0 7? ͕ग़དྷ·ͨ͠