sig_procon_regular_5.pdf

Transcript

1. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 November 7, 2025 Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 1 / 35
2. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ໨࣍ 1 ͸͡Ίʹ 2 ઢܕܭը๏ 3 ڇήʔ 4 ૒ରੑ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 2 / 35
3. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ໨࣍ 1 ͸͡Ίʹ 2 ઢܕܭը๏ 3 ڇήʔ 4 ૒ରੑ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 3 / 35
4. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࣗݾ঺հ Rho17 • ཧֶ෦৘ใՊֶՊ B4(2025 ೥ ݱࡏ) • AtCoder ԫ (2009) • ࣍Կ΍Δ͔ωλ੾Εؾຯ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 4 / 35
5. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ໨࣍ 1 ͸͡Ίʹ 2 ઢܕܭը๏ 3 ڇήʔ 4 ૒ରੑ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 5 / 35
6. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࠷దԽ໰୊ ͋Δ੍໿ͷԼͰ͋Δؔ਺ʢ໨తؔ਺ʣΛ࠷େ/࠷খʹ͢Δ΋ͷΛٻΊΔ໰୊Λ࠷ దԽ໰୊ͱݺͿɻಛʹ੍໿ͱ໨తؔ਺ͷ྆ํ͕ઢܕؔ਺Ͱ͋ΔΑ͏ͳ΋ͷΛઢ ܗܭը໰୊ʢLPʣͱ͍͏ɻ max cT x s.t. Ax ≤ b x ≥ 0 Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 6 / 35
7. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ઢܕܭը๏ ྫɿ࠷େϑϩʔ໰୊Λઢܕܭը໰୊ͱͯ͠ఆࣜԽ͢Δɻ max ∑ e∈δ+(s) xe − ∑ e∈δ−(s) xe s.t. ∑ e∈δ+(v) xe = ∑ e∈δ−(v) xe (v ∈ V \ {s, t}) 0 ≤ xe ≤ ce (e ∈ E) Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 7 / 35
8. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ઢܕܭը๏ ઢܕܭը໰୊ʹର͠ɺ ʮશͯͷม਺͸੔਺஋ΛͱΔʯͱ͍͏৚݅Λ෇͚଍ͨ͠΋ ͷΛ੔਺ܭը໰୊ʢIPʣͱ͍͏ɻ IP ໰୊ͷ੔਺৚݅Λ֎ͯ͠ LP ໰୊ʹ͢Δ͜ͱΛ LP ؇࿨ͱݺͿɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 8 / 35
9. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ઢܕܭը๏ ྫɿ࠷େಠཱू߹Λઢܕܭը໰୊ͱͯ͠ఆࣜԽ͢Δɻ max ∑ xi s.t. xu + xv ≤ 1 forall (u, v) ∈ E 0 ≤ xi ≤ 1 ೋ෦άϥϑͷ৔߹ɺLP ؇࿨ͨ͠৔߹ͱղ͕Ұக͢Δɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 9 / 35
10. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ઢܕܭը๏ • ੔਺ܭը໰୊͸ NP-Hard ͳͷͰҰൠʹ͸ଟ߲ࣜ࣌ؒͰղ͚ͳ͍ • ઢܕܭը໰୊͸ଟ߲ࣜ࣌ؒͰղ͚Δ • γϯϓϨοΫε๏ͳͲղ͘ํ๏͸͋Δ͕ɺڝϓϩͰ͸ઢܕܭը໰୊Λͦͷ ··ղ͘͜ͱ͸΄΅ͳ͍ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 10 / 35
11. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ໨࣍ 1 ͸͡Ίʹ 2 ઢܕܭը๏ 3 ڇήʔ 4 ૒ରੑ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 11 / 35
12. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ڇήʔ ҎԼͷΑ͏ͳܗͷઢܕܭը໰୊͕༩͑ΒΕͨͱ͢Δɽ maximize xn − x1 s.t. xui + di ≥ xvi ੍໿ͱ໨తؔ਺͕ 2 ม਺ͷࠩʹؔ͢Δෆ౳ࣜͷ࣌ɼ࠷୹ܦ࿏໰୊ͱಉ༷ʹղ͘ ͜ͱ͕Ͱ͖Δɽ͜ͷ৔߹ɼui → vi ʹίετ di ͷลΛுͬͨͷͪʹ 1 → N ͷ ࠷୹ܦ࿏ΛٻΊΕ͹ྑ͍ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 12 / 35
13. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ڇήʔ ౸ୡෆೳͷ৔߹ɼղ͸ඇ༗քʢ͍͘ΒͰ΋େ͖͘Ͱ͖Δʣ ɽෛͷϧʔϓ͕ଘࡏ͠ ͨ৔߹ɼ࣮ߦෆೳʢղͳ͠ʣ ɽ ࣮͸࠷୹ܦ࿏໰୊ͷ૒ରʹͳ͍ͬͯΔ͜ͱ͕֬ೝͰ͖Δɽ ʢޙड़ʣ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 13 / 35
14. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ 01Sequence (1963) 0 ͱ 1 ͔ΒͳΔ਺ྻ {Ai} ͰɼҎԼͷ M ݸͷ৚݅Λຬͨ͢΋ͷΛߟ͑Δɽ • ∑Rj i=Lj Ai ≥ Xj ؚ·ΕΔ 1 ͷݸ਺͕࠷খͷ΋ͷΛग़ྗͤΑɽ • 1 ≤ N ≤ 2 × 105 • 1 ≤ M ≤ 2 × 105 • ৚݅Λຬͨ͢਺ྻ͸ඞͣଘࡏ͢Δ ग़య: ABC216-G Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 14 / 35
15. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ • ྦྷੵ࿨ΛऔΔͱɼSR − SL−1 ≥ Xj • ਺ྻͷཁૉ͕ 0 ͔ 1 ͳͷͰɼ0 ≤ Si+1 − Si ≤ 1 • ໨తؔ਺͸ SN − S0 ͷ࠷খԽɼͭ·Γ S0 − SN ͷ࠷େԽ SR − Xj ≥ SL−1 Si+1 ≥ Si Si + 1 ≥ Si+1 ෛลೖΓˠؒʹ߹Θͳ͍ͷͰෛลΛফ͢ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 15 / 35
16. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ 0 ͱ 1 ΛೖΕସ͑ɼ0 ͷݸ਺ʹ͍ͭͯͷ৚݅ʹͯ͠ΈΔ • ྦྷੵ࿨ΛऔΔͱɼSR − SL−1 ≤ R − L − Xj + 1 • ਺ྻͷཁૉ͕ 0 ͔ 1 ͳͷͰɼ0 ≤ Si+1 − Si ≤ 1 • ໨తؔ਺͸ SN − S0 ͷ࠷େԽ SL−1 + R − L − Xj + 1 ≥ SR Si+1 ≥ Si Si + 1 ≥ Si+1 ͜ΕͰμΠΫετϥͰ͖Δܗʂ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 16 / 35
17. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ໨࣍ 1 ͸͡Ίʹ 2 ઢܕܭը๏ 3 ڇήʔ 4 ૒ରੑ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 17 / 35
18. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ Α͋͘Δઆ໌ɿઢܕܭը໰୊ͷ৚݅ࣜΛ͍͔ͭ͘଍͠Ҿ͖͢Δͱղͷ্ք͕ಘ ΒΕΔɽ͜ͷ܎਺ΛٻΊΔ໰୊͕ݩͷ໰୊ͷ૒ର໰୊ɽ ˠఱ࠽తͳൃ૝Ͱ܎਺Λಋ͖ग़͍ͯ͠ΔΑ͏ʹฉ͑ͯ͜͠·͍ɼྑ͘ ͳ͍ɽ ɽ ɽ ɽ ɽ ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 18 / 35
19. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ max cT x s.t. Ax ≤ b (P) ʹର͠ɼ૒ର໰୊͸ min yT b s.t. yT A = cT y ≥ 0 (D) Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 19 / 35
20. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ max cT x s.t. Ax ≤ b x ≥ 0 (P) ʹର͠ɼ૒ର໰୊͸ min yT b s.t. yT A ≥ cT y ≥ 0 (D) Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 20 / 35
21. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ max cT x s.t. Ax = b x ≥ 0 (P) ʹର͠ɼ૒ର໰୊͸ min yT b s.t. yT A ≥ cT (D) Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 21 / 35
22. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ ڧ૒ରఆཧ P,D ͕ڞʹ࣮ߦՄೳͳΒ͹ɼ2 ͭͷղ͸Ұக͢Δɽi.e. cT x = yT b Λຬͨ͢ x, y ͕ଘࡏ͢Δɽ • min-max ఆཧ͸͜ͷܗʂ • ࠷େϑϩʔɾ࠷খΧοτ • ࠷େೋ෦Ϛονϯάɾ࠷খ௖఺Χόʔ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 22 / 35
23. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ ূ໌ͷུ֓ ऑ૒ରఆཧ cT x ≤ yT b ͸͙͢ʹݴ͑Δɽ൓ରଆΛݴ͏ͨΊʹ Farkas ͷิ୊Λ ༻͍Δɽ Farkas ͷิ୊ Ax = b, x ≥ 0 ͱ AT y ≥ 0, yT b < 0 ͷ͏͍ͪͣΕ͔ҰํͷΈ͕੒ཱ͢Δɽ ಉ༷ʹɼAx ≤ b ͱ AT y = 0, yT b < 0, y ≥ 0 ͷ͍ͣΕ͔ҰํͷΈ͕੒ཱ͢Δɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 23 / 35
24. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ D ଆͷղΛ γ ͱͯ͠ɼ cT x ≥ γ ͱͳΒͳ͍ͱԾఆ͢Δɽ͜ͷ࣌ɼ   A −cT   x ≤   b −γ   ͸੒ཱ͠ͳ͍ͷͰɼFarkas ͷิ୊ΑΓ   A −cT   T   y λ   = 0,   b −γ   T   y λ   < 0,   y λ   ≥ 0 Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 24 / 35
25. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ૒ରੑ λ = 0 ͱ͢Δͱ A ʹ͍ͭͯ Farkas ͷิ୊ͷޙ൒͕੒ཱ͢ΔͷͰ Ax ≰ b ͜Ε͸ x ͕ A ͷղͰ͋Δ͜ͱʹໃ६ɽ λ > 0 ͷ࣌ɼ y/λ ΋࣮ߦՄೳղʹͳΔ͕ɼy ͷ࠷దੑʹ൓͢ΔͷͰໃ६ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 25 / 35
26. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ Infinite Knapsack (2522) ॏ͞ Ai, ମੵ Bi, Ձ஋ Ci ͷՙ෺͕ͦΕͧΕແݶݸ͋Δɽॏ͞ͷ߹ܭ͕ X ҎԼ ͔ͭମੵͷ߹ܭ͕ X ҎԼʹͳΔΑ͏ʹ඼෺ΛબΜͩ࣌ͷՁ஋ͷ࿨ͷ࠷େ஋Λ f(X) ͱ͢Δɽ limX→∞ f(X)/X ΛٻΊΑɽ • 1 ≤ N ≤ 2 × 105 • 108 ≤ Ai, Bi, Ci ≤ 109 ग़య: ABC275-G Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 26 / 35
27. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ • ͦΕͧΕͷ඼෺Λ࣮਺ݸબͿͱͯ͠΋໰୊ͳ͍ɽ • ∑ Aix′ i ≤ X, ∑ Bix′ i ≤ X ͷ΋ͱͰ f(X) = ∑ Cix′ i Λ maximize • X ͰׂΓɼ ∑ Aixi ≤ 1, ∑ Bixi ≤ 1 ͷ΋ͱͰ f(X)/X = ∑ Cixi Λ maximize • ૒ରΛऔΔɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 27 / 35
28. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ max cT x s.t. Ax ≤ b x ≥ 0 (P) ʹର͠ɼ૒ର໰୊͸ min yT b s.t. yT A ≥ cT y ≥ 0 (D) max cT x s.t.   A1 · · · AN B1 · · · BN   x ≤   1 1   min y + z s.t. Aiy + Biz ≥ Ci y, z ≥ 0 Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 28 / 35
29. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ min y + z s.t. Aiy + Biz ≥ Ci y, z ≥ 0 Λղ͖͍ͨɽ y + z = k Ͱೋ෼୳ࡧ͢ΔɽAiy + Bi(k − y) ≥ Ci Λຬͨ͢ y ͷൣғͷڞ௨෦ ෼͕ 0 ≤ y ʹ͋Δ͔൑ఆ͢Ε͹ྑ͍ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 29 / 35
30. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ Security Camera 2 (2887) ೋ෦άϥϑ (L, R) ͕༩͑ΒΕΔɽࠨଆͷ௖఺ i ʹΧϝϥΛஔ͘ίετ͸ Ai, ӈଆͷ௖఺ j ʹΧϝϥΛஔ͘ίετ͸ Bj. Ұͭͷ௖఺ʹΧϝϥ͸͍͘ΒͰ΋ஔ ͚Δɽશͯͷ૊ (i, j) ʹର͠ɼ i ͱ j ʹஔ͔Ε͍ͯΔΧϝϥͷ߹ܭݸ਺͕ Ci,j Ҏ্ʹͳΔΑ͏ʹ͍ͨ͠ɽͦͷΑ͏ͳ࠷খίετ͸ʁ • 1 ≤ L, R ≤ 100 • 1 ≤ Ai, Bj ≤ 10 • 0 ≤ Ci,j ≤ 100 ग़య: ABC224-H Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 30 / 35
31. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ • minimize ∑ AiLi + ∑ BjRj • Ai + Bj ≥ Ci,j max cT x s.t. Ax ≤ b x ≥ 0 (P) min yT b s.t. yT A ≥ cT y ≥ 0 (D) Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 31 / 35
32. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ max cT x s.t. Mx ≤   A B   x ≥ 0 (P) min   L R   T   A B   s.t.   L R   T M ≥ cT   L R   ≥ 0 (D) M ͸ (L + R) × LR ߦྻɼ iR + j ྻ໨͸ i ߦ໨ͱ L + j ߦ໨ͷΈ͕ 1 Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 32 / 35
33. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ ล (i, j) ʹରͯ͠஋ xi,j ͕ఆ·͍ͬͯΔɽL ଆͷ௖఺ i′ ʹ͍ͭͯɼ ∑ j xi′,j ≤ Ai′ . R ଆͷ௖఺ j′ ʹ͍ͭͯɼ ∑ i xi,j′ ≤ Bj′ . ∑ Ci,jxi,j Λ࠷େԽ ͤΑɽ S → Li ͷ༰ྔ͕ Ai, Ri → T ͷ༰ྔ͕ Bi Ͱ Li → Rj ʹඅ༻ Ci,j ͷล͕͋Δ ࣌ͷ࠷େඅ༻ྲྀΛٻΊΒΕΕ͹ྑ͍ɽίετΛ-1 ഒͯ͠ਹ্͛͢Ε͹࠷খඅ༻ ྲྀͱͯ͠ٻΊΒΕΔɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 33 / 35
34. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࿅श໰୊ • ABC404-G (1894) • ARC031-D (2825?) • AGC347-G (2876) • ABC356-G (3022) ͦͷଞ AGC ͷޙΖͷํͱ͔ʹ͍͔ͭ͘...... Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 34 / 35
35. ڝϓϩ෼Պձ Regular ୈ 5 ճ Rho17 ͸͡Ίʹ ઢܕܭը๏ ڇήʔ ૒ରੑ ࣍ճ༧ࠂ ͍ͭʹ͠Α͏ & Կ͠Α͏...... Rho17 ڝϓϩ෼Պձ Regular ୈ 5 ճ November 7, 2025 35 / 35