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sig_procon_regular_4__1_.pdf

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November 14, 2025
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 sig_procon_regular_4__1_.pdf

Avatar for ZenReKkyo

ZenReKkyo

November 14, 2025

Transcript

  1. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 October 28, 2025 Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 1 / 30
  2. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ໨࣍ 1 ͸͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 2 / 30
  3. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ໨࣍ 1 ͸͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 3 / 30
  4. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣗݾ঺հ Rho17 • ཧֶ෦৘ใՊֶՊ B4(2025 ೥ ݱࡏ) • AtCoder ԫ (2009) • AGC Կ΋Θ͔Γ·ͤΜ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 4 / 30
  5. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ໨࣍ 1 ͸͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 5 / 30
  6. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching ೋ෦Ϛονϯά ೋ෦άϥϑ G(U ∪ V, E) ͕͋ΔɻબΜͩͲͷ 2 ล΋୺఺Λڞ༗͠ͳ͍Α͏ʹล Λ͍͔ͭ͘બͿɻ࠷େͰԿຊબ΂Δʁ ͜ͷΑ͏ʹ୺఺Λڞ༗͠ͳ͍ลͷ෦෼ू߹ M ⊆ E ΛϚονϯάͱ͍͍ɺลͷ ຊ਺͕࠷େͳ΋ͷΛ࠷େϚονϯάͱ͍͏ɻ·ͨɺ2|M| = |U| + |V | Λຬͨ͢ ͱ͖׬શϚονϯάͱ͍͏ɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 6 / 30
  7. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching ྫ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 7 / 30
  8. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Bipartite Matching શͯͷลΛ༰ྔ 1 ͱͨ͠ϑϩʔΛྲྀ͢ → ࠷େϑϩʔͱͯ͠ղ͚ͨ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 8 / 30
  9. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ Push Simultaneously (2022) N ਓͷߴڮ͘Μͱ N ݸͷϘλϯ͕͋Δɻߴڮ͘Μͨͪ͸ͦΕͧΕඵ଎ 1 ͷ଎͞ Ͱಈ͘͜ͱ͕Ͱ͖Δɻߴڮ͘ΜͱϘλϯͷ࠲ඪ͕༩͑ΒΕΔͱ͖ɺ࣌ؒ t ͷҠ ಈʹΑͬͯશͯͷߴڮ͘Μ͕ಉ࣌ʹϘλϯΛԡͤΔΑ͏ͳ t ͷ࠷খ஋ΛٻΊΑɻ • 1 ≤ N ≤ 300 ग़య: ABC401-G Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 9 / 30
  10. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ • ࣌ؒ t ʹ͍ͭͯͷ൑ఆ໰୊Λղ͘ • ߴڮ͘ΜʢҎ߱ɺਓʣͱϘλϯͷؒͷద੾ͳରԠ෇͚ΛͱΔ໰୊ • ਓͱϘλϯΛ௖఺ {ui}, {vi} ͩͱࢥ͍ɺڑ཭ t ҎԼͰ͋ΔΑ͏ͳਓͱϘλ ϯͷϖΞʹลΛுΔ • ਓͱϘλϯͷೋ෦άϥϑͳͷͰ࠷େϚονϯάʹؼணͰ͖Δ • t ͸ೋ෼୳ࡧͯ͠΋͍͍͠ɺt ͷީิ͸ลͷڑ཭ N2 ௨Γ͔Β୳ͤ͹͍͍ͷ Ͱڑ཭͕খ͍͞ํ͔ΒॱʹϚονϯάͷลΛ௥Ճ͠ɺͦͷͨͼʹ S → T ϑϩʔΛྲྀ͠ॳΊͯ N ྲྀΕͨλΠϛϯάΛݟΔ͜ͱͰ΋Մ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 10 / 30
  11. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ Maxflow N × M ͷϚε໨͕͋Γɺো֐෺͕ஔ͔Ε͍ͯΔϚεͱۭ͖ϚεͷͲͪΒ͔Ͱ ͋Δɻ͜͜ʹ 1 × 2 ͷλΠϧΛॎɾԣͷ޷͖ͳํ޲Ͱஔ͘ɻ࠷େͰ͍ͭ͘ஔ͚ Δ͔ʁ • 1 ≤ N ≤ 100 • 1 ≤ M ≤ 100 ग़య: ACL Practice Contest-D Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 11 / 30
  12. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ • λΠϧΛஔ͘ૢ࡞ΛάϦου্ͷ 2 ௖఺ΛϚονͤ͞Δૢ࡞ͩͱΈΔ • άϦουάϥϑ͸ͦΕࣗମ͕ೋ෦άϥϑʂ • ࢢদͷృΓ෼͚Λߟ͑Δͱɺೋ෦Ϛονϯάͷղ๏͕࢖͑Δ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 12 / 30
  13. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ໨࣍ 1 ͸͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 13 / 30
  14. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ ׬શϚονϯά͕ଘࡏ͢Δ͜ͱΛಛ௃͚͍ͮͨ Hall ͷ݁ࠗఆཧ ೋ෦άϥϑ G = (U ∪ V, E) ʹରͯ͠ҎԼ͸ಉ஋ɻ • U ͷ௖఺ΛશͯΧόʔ͢ΔϚονϯά͕ଘࡏ͢Δɻ • ∀X ⊆ U ʹ͍ͭͯɺ|X| ≤ |Γ(X)|ɻͨͩ͠ Γ(X) ͸ G ্Ͱ X ʹؚ·ΕΔ ௖఺ͱลͰ઀ଓ͞Ε͍ͯΔΑ͏ͳ௖఺ͷू߹ɻ |U| = |V | ͳΒ͹্ͷ৚݅͸׬શϚονϯάʹؔ͢Δ৚݅ʹͳΔɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 14 / 30
  15. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ ਖ਼౰ੑͷུ֓ (⇒) ׬શϚονϯάͷଘࡏΑΓ໌Β͔ɻ (⇐) ରۮΛࣔ͢ɻ࠷େϚονϯάͷαΠζ͕ |M| ͩͱͯ͠ɺ࠷খ఺Χόʔ C ΛͱΔɻ͜͜Ͱɺ|M| = |C| ͕੒ΓཱͭɻU ଆͱͯ͠ U \ C, V ଆͱͯ͠ V ∩ C ΛऔΔͱ |X| > |Γ(X)| ͱͳ͍ͬͯΔΑ͏ͳྫͱͳΔɻ ʢU \ C ͱ V \ C ͷؒʹ ล͸ແ͍ɻ͋ͬͨͱԾఆ͢Δͱ࠷খ఺Χόʔʹ൓͢Δɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 15 / 30
  16. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά Hall ͷ݁ࠗఆཧ ิ଍ ఺ू߹ C ⊆ V ͰɺG ͷͲͷล΋ C ʹؚ·ΕΔ௖఺ʹ઀ଓ͍ͯ͠ΔΑ͏ͳ΋ͷ Λ఺ΧόʔͱݺͿɻೋ෦άϥϑʹ͓͍ͯɺ࠷খ఺Χόʔͱ࠷େϚονϯάͷα Πζ͸Ұக͢Δɻ ʢ࢒༨άϥϑ্Ͱ S ͔Β౸ୡෆՄೳͳ U ͷ௖఺ͱ S ͔Β౸ୡ Մೳͳ V ͷ௖఺ͷ࿨ू߹ͱͱΕΔʣ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 16 / 30
  17. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ ਖ਼ଇೋ෦άϥϑͷ׬શϚονϯά શͯͷ௖఺ͷ࣍਺͕ d Ͱ͋ΔΑ͏ͳάϥϑΛ d-ਖ਼ଇάϥϑͱ͍͏ɽd-ਖ਼ଇͳೋ ෦άϥϑ G = (U ∪ V, E) Ͱ |U| = |V | Λຬͨ͢Α͏ͳ΋ͷ͸׬શϚονϯάΛ ࣋ͭ͜ͱΛࣔͤɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 17 / 30
  18. ڝϓϩ෼Պձ 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 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 18 / 30
  19. ڝϓϩ෼Պձ 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 ͭଘࡏ͢Δɽ͜ͷจࣈྻΛҎԼͷ৚݅Λຬͨ͢Α͏ͳ෦෼ྻ 6 ͭʢҎԼʣʹ෼ ׂͤΑɽ • T ͷ௕͞͸ 3m • T1 = · · · = Tm • Tm+1 = · · · = T2m • T2m+1 = · · · = T3m • T1, Tm+1, T2m+1 ͸ͦΕͧΕҟͳΔ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 19 / 30
  20. ڝϓϩ෼Պձ 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 ͭଘࡏ͢Δɽ͜ͷจࣈྻΛҎԼͷ৚݅Λຬͨ͢Α͏ͳ෦෼ྻ 6 ͭʢҎԼʣʹ෼ ׂͤΑɽ • 1 ≤ N ≤ 2 × 105 ग़య: AGC055-A Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 20 / 30
  21. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ 3 ௖఺ͣͭͷೋ෦άϥϑͰยଆʹ͸ 1,2,3ʢલɾதɾޙʣ ΋͏ยଆʹ͸ A,B,C ͱ ॻ͔Ε͍ͯΔ΋ͷΛߟ͑Δɽ ࣍਺ N ͷਖ਼ଇೋ෦άϥϑͳͷͰɼ׬શϚονϯά͕ଘࡏ͢Δɽ͜ͷϚονϯ άΛऔΓআ͘ͱ࣍਺ N − 1 ͷਖ਼ଇೋ෦άϥϑ͕࢒ΓɼҎԼಉ༷ʹϚονϯάΛ N ճऔΕΔɽ͋ͱ͸Ϛονϯάͷछྨ͝ͱʹ 1 6 Λద౰ʹׂΓৼΔɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 21 / 30
  22. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ໨࣍ 1 ͸͡Ίʹ 2 Bipartite Matching 3 Hall ͷ݁ࠗఆཧ 4 ॏΈ͖ͭϚονϯά Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 22 / 30
  23. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࠷খॏΈೋ෦Ϛονϯά ଟ͘ͷ໰୊͸࠷େྲྀ΍࠷খඅ༻ྲྀʹݴ͍׵͑Δ͜ͱ͕Ͱ͖Δ ࠷খॏΈ࠷େϚονϯάͳͲϋϯΨϦΞϯ๏Ͱղ͚Δ৔߹΋͋Δʢࠓճ͸লུʣ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 23 / 30
  24. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ Domino Covering SUM (2190) H × W ͷϚε໨্ʹ੔਺͕ॻ͔Ε͍ͯΔɽϚε໨্ʹ 1 × 2 ͷυϛϊΛॏͳΒ ͳ͍Α͏ʹஔ͘͜ͱ͕ग़དྷΔɽυϛϊ͕ஔ͔Ε͍ͯͳ͍Ϛεʹॻ͔Εͨ੔਺ͷ ૯࿨ͷ࠷େ஋ΛٻΊΑɽ • 1 ≤ HW ≤ 2000 • −1012 ≤ Aij ≤ 1012 ग़య: ABC407-G Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 24 / 30
  25. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ • ೋ෦άϥϑ্ͷॏΈ͖ͭϚονϯάͱߟ͑Δ • ࠾༻ͨ͠ลͷॏΈͷ࠷খ஋ΛٻΊ͍ͨ • ೋ෦άϥϑͱͯ͠௖఺Λ 2 छྨʹ෼͚ɼU → 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 ͣͭ૿΍͠ͳ͕Βௐ΂Ε͹ྑ͍ ʢACL ͳΒ min-cost-slope ͕ศརʣ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 25 / 30
  26. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ Minimum Coloring (2681) H × W ͷϚε໨্ʹന͍ۨ i ͕ஔ͔Ε͍ͯΔɽWi ԁ෷͏ͱۨΛࠇ͍ۨʹมߋ ͢Δ͜ͱ͕Ͱ͖ΔɽͲͷߦͲͷྻʹ΋ࠇ͍͕ۨ 1 ͭҎ্͋ΔΑ͏ʹ͢Δ࠷খί ετ͸ʁ • 1 ≤ H, W ≤ 1000 • 1 ≤ Wi ≤ 109 ग़య: ABC231-H Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 26 / 30
  27. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࿅श໰୊ • ߦͱྻΛͦΕͧΕ௖఺ͱͨ͠ೋ෦άϥϑͱΈͳ͢ • (i, j) ʹஔ͔ΕͨۨΛ Ri → Cj ͷίετ Wi ͷลͩͱΈͳ͢ • ٻΊΔ΋ͷ͸ Ri, Cj શͯʹ͍ͭͯྲྀྔ 1 Ҏ্ྲྀΕΔ৚݅ԼͰͷ࠷খඅ༻ྲྀ • S → Ri ͕࠷খྲྀྔ 1 ίετ 0ɼRi → Cj ͕༰ྔ 1 ίετ Wij, Rj → T ͕ ࠷খྲྀྔ 1 ίετ 0 • ͋Β͔͡Ίඞཁͳ෦෼ʹ 1 ྲྀ͓͍ͯͨ͜͠ͱΛදݱˠάϥϑͷϙςϯγϟ ϧΛલ΋ͬͯߋ৽ Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 27 / 30
  28. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

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  29. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

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  30. ڝϓϩ෼Պձ Regular ୈ 4 ճ Rho17 ͸͡Ίʹ Bipartite Matching Hall

    ͷ݁ࠗఆཧ ॏΈ͖ͭϚονϯά ࣍ճ༧ࠂ 11/7(ۚ)ʁ ҰൠϚονϯάɼύεඃ෴ͳͲɹ͋ͱ૒ର Rho17 ڝϓϩ෼Պձ Regular ୈ 4 ճ October 28, 2025 30 / 30