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sig_procon_regular_4__1_.pdf
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ZenReKkyo
November 14, 2025
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sig_procon_regular_4__1_.pdf
ZenReKkyo
November 14, 2025
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Transcript
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ڝϓϩՊձ Regular ୈ 4 ճ Rho17 October 28, 2025 Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 1 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ 1 ͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 2 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ 1 ͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 3 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣗݾհ Rho17 • ཧֶ෦ใՊֶՊ B4(2025 ݱࡏ) • AtCoder ԫ (2009) • AGC ԿΘ͔Γ·ͤΜ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 4 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ 1 ͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 5 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching ೋ෦Ϛονϯά ೋ෦άϥϑ G(U ∪ V, E) ͕͋ΔɻબΜͩͲͷ 2 ลΛڞ༗͠ͳ͍Α͏ʹล Λ͍͔ͭ͘બͿɻ࠷େͰԿຊબΔʁ ͜ͷΑ͏ʹΛڞ༗͠ͳ͍ลͷ෦ू߹ M ⊆ E ΛϚονϯάͱ͍͍ɺลͷ ຊ͕࠷େͳͷΛ࠷େϚονϯάͱ͍͏ɻ·ͨɺ2|M| = |U| + |V | Λຬͨ͢ ͱ͖શϚονϯάͱ͍͏ɻ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 6 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching ྫ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 7 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching શͯͷลΛ༰ྔ 1 ͱͨ͠ϑϩʔΛྲྀ͢ → ࠷େϑϩʔͱͯ͠ղ͚ͨ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 8 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श Push Simultaneously (2022) N ਓͷߴڮ͘Μͱ N ݸͷϘλϯ͕͋Δɻߴڮ͘ΜͨͪͦΕͧΕඵ 1 ͷ͞ Ͱಈ͘͜ͱ͕Ͱ͖Δɻߴڮ͘ΜͱϘλϯͷ࠲ඪ͕༩͑ΒΕΔͱ͖ɺ࣌ؒ t ͷҠ ಈʹΑͬͯશͯͷߴڮ͘Μ͕ಉ࣌ʹϘλϯΛԡͤΔΑ͏ͳ t ͷ࠷খΛٻΊΑɻ • 1 ≤ N ≤ 300 ग़య: ABC401-G Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 9 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • ࣌ؒ t ʹ͍ͭͯͷఆΛղ͘ • ߴڮ͘ΜʢҎ߱ɺਓʣͱϘλϯͷؒͷదͳରԠ͚ΛͱΔ • ਓͱϘλϯΛ {ui}, {vi} ͩͱࢥ͍ɺڑ t ҎԼͰ͋ΔΑ͏ͳਓͱϘλ ϯͷϖΞʹลΛுΔ • ਓͱϘλϯͷೋ෦άϥϑͳͷͰ࠷େϚονϯάʹؼணͰ͖Δ • t ೋ୳ࡧ͍͍ͯ͠͠ɺt ͷީิลͷڑ N2 ௨Γ͔Β୳͍͍ͤͷ Ͱڑ͕খ͍͞ํ͔ΒॱʹϚονϯάͷลΛՃ͠ɺͦͷͨͼʹ S → T ϑϩʔΛྲྀ͠ॳΊͯ N ྲྀΕͨλΠϛϯάΛݟΔ͜ͱͰՄ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 10 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श Maxflow N × M ͷϚε͕͋Γɺো͕ஔ͔Ε͍ͯΔϚεͱۭ͖ϚεͷͲͪΒ͔Ͱ ͋Δɻ͜͜ʹ 1 × 2 ͷλΠϧΛॎɾԣͷ͖ͳํͰஔ͘ɻ࠷େͰ͍ͭ͘ஔ͚ Δ͔ʁ • 1 ≤ N ≤ 100 • 1 ≤ M ≤ 100 ग़య: ACL Practice Contest-D Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 11 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • λΠϧΛஔ͘ૢ࡞ΛάϦου্ͷ 2 ΛϚονͤ͞Δૢ࡞ͩͱΈΔ • άϦουάϥϑͦΕࣗମ͕ೋ෦άϥϑʂ • ࢢদͷృΓ͚Λߟ͑Δͱɺೋ෦Ϛονϯάͷղ๏͕͑Δ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 12 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ 1 ͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 13 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ શϚονϯά͕ଘࡏ͢Δ͜ͱΛಛ͚͍ͮͨ Hall ͷ݁ࠗఆཧ ೋ෦άϥϑ G = (U ∪ V, E) ʹରͯ͠ҎԼಉɻ • U ͷΛશͯΧόʔ͢ΔϚονϯά͕ଘࡏ͢Δɻ • ∀X ⊆ U ʹ͍ͭͯɺ|X| ≤ |Γ(X)|ɻͨͩ͠ Γ(X) G ্Ͱ X ʹؚ·ΕΔ ͱลͰଓ͞Ε͍ͯΔΑ͏ͳͷू߹ɻ |U| = |V | ͳΒ্ͷ݅શϚονϯάʹؔ͢Δ݅ʹͳΔɻ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 14 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ ਖ਼ੑͷུ֓ (⇒) શϚονϯάͷଘࡏΑΓ໌Β͔ɻ (⇐) ରۮΛࣔ͢ɻ࠷େϚονϯάͷαΠζ͕ |M| ͩͱͯ͠ɺ࠷খΧόʔ C ΛͱΔɻ͜͜Ͱɺ|M| = |C| ͕ΓཱͭɻU ଆͱͯ͠ U \ C, V ଆͱͯ͠ V ∩ C ΛऔΔͱ |X| > |Γ(X)| ͱͳ͍ͬͯΔΑ͏ͳྫͱͳΔɻ ʢU \ C ͱ V \ C ͷؒʹ ลແ͍ɻ͋ͬͨͱԾఆ͢Δͱ࠷খΧόʔʹ͢Δɻ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 15 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ ิ ू߹ C ⊆ V ͰɺG ͷͲͷล C ʹؚ·ΕΔʹଓ͍ͯ͠ΔΑ͏ͳͷ ΛΧόʔͱݺͿɻೋ෦άϥϑʹ͓͍ͯɺ࠷খΧόʔͱ࠷େϚονϯάͷα ΠζҰக͢Δɻ ʢ༨άϥϑ্Ͱ S ͔Β౸ୡෆՄೳͳ U ͷͱ S ͔Β౸ୡ Մೳͳ V ͷͷू߹ͱͱΕΔʣ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 16 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श ਖ਼ଇೋ෦άϥϑͷશϚονϯά શͯͷͷ͕࣍ d Ͱ͋ΔΑ͏ͳάϥϑΛ d-ਖ਼ଇάϥϑͱ͍͏ɽd-ਖ਼ଇͳೋ ෦άϥϑ G = (U ∪ V, E) Ͱ |U| = |V | Λຬͨ͢Α͏ͳͷશϚονϯάΛ ࣋ͭ͜ͱΛࣔͤɽ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 17 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श S ⊆ U ΛͱΔɽS ͔Βग़Δลͷຊ Sd Ͱ͋Δɽ͍· |S| > |Γ(S)| ͩͱԾఆ ͢Δɽ|Γ(S)| ʹೖΔลͷຊ Sd ະຬͱͳΓɼໃ६ɽ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 18 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श ABC Identity (1348) ͞ 3N ͷ A,B,C ͔ΒͳΔจࣈྻ͕༩͑ΒΕΔɽA,B,C ͦΕͧΕ N จࣈͣ ͭଘࡏ͢Δɽ͜ͷจࣈྻΛҎԼͷ݅Λຬͨ͢Α͏ͳ෦ྻ 6 ͭʢҎԼʣʹ ׂͤΑɽ • T ͷ͞ 3m • T1 = · · · = Tm • Tm+1 = · · · = T2m • T2m+1 = · · · = T3m • T1, Tm+1, T2m+1 ͦΕͧΕҟͳΔ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 19 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श ABC Identity (1348) ͞ 3N ͷ A,B,C ͔ΒͳΔจࣈྻ͕༩͑ΒΕΔɽA,B,C ͦΕͧΕ N จࣈͣ ͭଘࡏ͢Δɽ͜ͷจࣈྻΛҎԼͷ݅Λຬͨ͢Α͏ͳ෦ྻ 6 ͭʢҎԼʣʹ ׂͤΑɽ • 1 ≤ N ≤ 2 × 105 ग़య: AGC055-A Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 20 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श 3 ͣͭͷೋ෦άϥϑͰยଆʹ 1,2,3ʢલɾதɾޙʣ ͏ยଆʹ A,B,C ͱ ॻ͔Ε͍ͯΔͷΛߟ͑Δɽ ࣍ N ͷਖ਼ଇೋ෦άϥϑͳͷͰɼશϚονϯά͕ଘࡏ͢Δɽ͜ͷϚονϯ άΛऔΓআ͘ͱ࣍ N − 1 ͷਖ਼ଇೋ෦άϥϑ͕ΓɼҎԼಉ༷ʹϚονϯάΛ N ճऔΕΔɽ͋ͱϚονϯάͷछྨ͝ͱʹ 1 6 ΛదʹׂΓৼΔɽ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 21 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ 1 ͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 22 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࠷খॏΈೋ෦Ϛονϯά ଟ͘ͷ࠷େྲྀ࠷খඅ༻ྲྀʹݴ͍͑Δ͜ͱ͕Ͱ͖Δ ࠷খॏΈ࠷େϚονϯάͳͲϋϯΨϦΞϯ๏Ͱղ͚Δ߹͋Δʢࠓճলུʣ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 23 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श Domino Covering SUM (2190) H × W ͷϚε্ʹ͕ॻ͔Ε͍ͯΔɽϚε্ʹ 1 × 2 ͷυϛϊΛॏͳΒ ͳ͍Α͏ʹஔ͘͜ͱ͕ग़དྷΔɽυϛϊ͕ஔ͔Ε͍ͯͳ͍Ϛεʹॻ͔Εͨͷ ૯ͷ࠷େΛٻΊΑɽ • 1 ≤ HW ≤ 2000 • −1012 ≤ Aij ≤ 1012 ग़య: ABC407-G Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 24 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • ೋ෦άϥϑ্ͷॏΈ͖ͭϚονϯάͱߟ͑Δ • ࠾༻ͨ͠ลͷॏΈͷ࠷খΛٻΊ͍ͨ • ೋ෦άϥϑͱͯ͠Λ 2 छྨʹ͚ɼU → V ʹ༰ྔ 1 අ༻ −Aij − Ai′j′ ͷล͕͋Δͱߟ͑Δ • ෛลআڈͷͨΊɼඅ༻Λਹ্͛ͯ͠ M − Aij − Ai′j′ ʹ͠ɼྲྀྔͷ߹ܭ͔ Β Mk Ҿ͘ • ྲྀྔࣗ༝ͳͷͰɼ0 ͔Β HW/2 ·Ͱ 1 ͣͭ૿͠ͳ͕ΒௐΕྑ͍ ʢACL ͳΒ min-cost-slope ͕ศརʣ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 25 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श Minimum Coloring (2681) H × W ͷϚε্ʹന͍ۨ i ͕ஔ͔Ε͍ͯΔɽWi ԁ͏ͱۨΛࠇ͍ۨʹมߋ ͢Δ͜ͱ͕Ͱ͖ΔɽͲͷߦͲͷྻʹࠇ͍͕ۨ 1 ͭҎ্͋ΔΑ͏ʹ͢Δ࠷খί ετʁ • 1 ≤ H, W ≤ 1000 • 1 ≤ Wi ≤ 109 ग़య: ABC231-H Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 26 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • ߦͱྻΛͦΕͧΕͱͨ͠ೋ෦άϥϑͱΈͳ͢ • (i, j) ʹஔ͔ΕͨۨΛ Ri → Cj ͷίετ Wi ͷลͩͱΈͳ͢ • ٻΊΔͷ Ri, Cj શͯʹ͍ͭͯྲྀྔ 1 Ҏ্ྲྀΕΔ݅ԼͰͷ࠷খඅ༻ྲྀ • S → Ri ͕࠷খྲྀྔ 1 ίετ 0ɼRi → Cj ͕༰ྔ 1 ίετ Wij, Rj → T ͕ ࠷খྲྀྔ 1 ίετ 0 • ͋Β͔͡Ίඞཁͳ෦ʹ 1 ྲྀ͓͍ͯͨ͜͠ͱΛදݱˠάϥϑͷϙςϯγϟ ϧΛલͬͯߋ৽ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 27 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • ͋Β͔͡Ίඞཁͳ෦ʹ 1 ྲྀ͓͍ͯͨ͜͠ͱΛදݱˠάϥϑͷϙςϯγϟ ϧΛલͬͯߋ৽ • S ͕ H ෆɼRi ͕ 1 աɼCj ͕ 1 ෆɼT ͕ W ա • ॥ྲྀʹ͢ΔͨΊʹ S → T ʹίετ 0 ༰ྔແݶͷลΛுΔ • S′, T′ ΛՃ͠ɼS′ → Ri ͱ Cj → T′ ʹίετ 0 ༰ྔ 1 ͷล • S → T′, T → S′ ʹͦΕͧΕ༰ྔ H, W ͷลΛுΔ • S′ → T′ ʹྲྀͤΔ͚ͩྲྀ͢ Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 28 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श • ABC317-G (2649) • AGC037-D (2689) • ARC076-F (2819) • ARC106-E (2825) • AGC025-D (2868) • ABC215-H (?) Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 29 / 30
ڝϓϩՊձ Regular ୈ 4 ճ Rho17 ͡Ίʹ Bipartite Matching Hall
ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ճ༧ࠂ 11/7(ۚ)ʁ ҰൠϚονϯάɼύεඃ෴ͳͲɹ͋ͱର Rho17 ڝϓϩՊձ Regular ୈ 4 ճ October 28, 2025 30 / 30