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ਖ਼͘͠࡞ΔϥϯμϜέʔε UV Study: ϏΞόογϡʂ ڝٕϓϩάϥϛϯά LT ձ tsutaj July 11, 2023 tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 1 / 24

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ࣗݾ঺հ ▶ Twitter: @tsutaj ▶ ࠷ۙ͸ AHC ʹग़ͨΓ ICPC OB/OG ձʹ͍·͢ ▶ ֶੜͷͱ͖͸େֶ༗ࢤͰ࡞໰ΛΑ͘΍ͬͯ·ͨ͠ ▶ ໋ཱ߹॓ɺձ௡߹॓ɺ๺େ߹॓ɺetc... ▶ ࠷ۙ PAST ্ڃʙΤΩεύʔτຊΛࣥච͠·ͨ͠ ▶ ࠓճ͸ڝϓϩ࡞໰ʹ࢖͑ͦ͏ͳςΫΛ঺հ͠·͢ ▶ աڈͷൃද (ࢀߟ): ڝϓϩ࡞໰Λࢧ͑Δٕज़ Link ▶ ͳ͚ͳ͠ͷ࡞໰ϊ΢ϋ΢Λগͣͭ͠์ྲྀ͍ͨ͠ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 2 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 3 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 4 / 24

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ϥϯμϜέʔεΛ࢖͏৔໘ ϥϯμϜʹέʔεΛ࡞Γͨ͘ͳΔ৔໘͸͍͔ͭ͋͘Δ ▶ ڝϓϩͰ໰୊Λղ͍͍ͯͯɺόάΛಛఆ͍ͨ͠ͱ͖ ▶ ڝϓϩͷςετέʔεΛ࡞Γ͍ͨͱ͖ ▶ ࣮ߦ͢Δ͝ͱʹҟͳΔग़ྗ݁Ռʹ͍ͨ͠ͱ͖ ▶ ਓһͷׂΓ౰ͯ ▶ நબ ͜ͷεϥΠυͰ ʮਖ਼͍͠ʯ ϥϯμϜέʔεΛ࡞ΕΔΑ͏ʹͳΓ·͠ΐ͏ʂ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 5 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 6 / 24

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͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ ▶ ͜ͷൃදʹ͓͍ͯɺ ʮਖ਼͍͠ϥϯμϜέʔεʯ = ภΓ͕ͳ͍ ▶ ΞϧΰϦζϜΛؒҧ͑ΔͱɺҰ༷Ͱͳ͘ͳΔ͜ͱ΋ଟʑ͋Δ ▶ ςετέʔε͕ภΔͱɺಛఆͷཁૉ͕ͨ͘͞Μબ͹ΕͯࠔΔ͜ͱ΋ ▶ ྫ: ͦͷγϟοϑϧɺຊ౰ʹγϟοϑϧͰ͔͢ʁ Կؾͳ͍མͱ݀͠ʹϋ Ϛͬͨ࿩ Link ▶ ಉ͡ϖΞ͕සൟʹબ͹Εͯ͠·͍ͬͯΔ ▶ ͜ͷൃදͰ͸ҎԼͷؔ਺ͷଘࡏΛೝΊɺසൟʹ࢖͍·͢ ▶ rand(a, b) := a Ҏ্ b ະຬͷ੔਺ΛҰ༷ϥϯμϜʹฦ͢ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 7 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 8 / 24

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ॱྻΛϥϯμϜʹ࡞Δ ▶ ௕͞ N ͷॱྻ P ͸ N! ௨Γ͋Δ ▶ Ұ༷ϥϯμϜʹॱྻΛ࡞Δʹ͸ɺN! ௨Γͷॱྻ͢΂͕ͯಉ֬͡཰Ͱग़ Ε͹Α͍ ▶ ࣍ϖʔδͰϥϯμϜʹ࡞Δํ๏Λ঺հ Figure: Ҿ༻ݩ: GeeksforGeeks Link tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 9 / 24

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ॱྻΛϥϯμϜʹ࡞Δ ௕͞ N ͷॱྻ P Λ࡞Δʹ͸ɺྻ A = (1, . . . , N) Λγϟοϑϧ͢Ε͹Α͍ ϑΟογϟʔɾΠΣʔπͷγϟοϑϧ Link 1: P ← A 2: for i = 0, . . . , N − 2 : 3: j ← rand(i, N) 4: Pi ͱ Pj Λަ׵͢Δ 5: end for 6: return P ▶ ಛ௃: ܭࢉྔ͕ O(N) Ͱɺ͔ͭภΓ͕ͳ͍ʂ ▶ A Ͱ i ൪໨ (0-indexed) ͩͬͨཁૉ͸ɺP Ͱ N − i ௨Γͷ৔ॴʹೖΓ ಘΔ ▶ ͦͷͨΊɺ͜ͷΞϧΰϦζϜͰಘΒΕΔग़ྗ͸ N! ௨Γ͋Δ ▶ ॱྻͷ૯਺ʹҰக ▶ ௕͞ N ͷ͢΂ͯͷॱྻ͕͜ͷํ๏Ͱੜ੒͞Ε͏Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 10 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 11 / 24

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N ݸͷத͔Β 2 ݸબͿํ๏ ▶ (0, 1, . . . , N − 1) ͷத͔Β 2 ݸબͿ͜ͱΛߟ͑Δ ▶ ͸͡ΊʹɺΑ͋͘Δؒҧ͍Λ঺հ ▶ ࢲ΋࣮ࡍʹ΍ͬͯ͠·ͬͨ͜ͱͷ͋Δख๏ N ݸͷத͔Β 2 ݸબͿํ๏ʢؒҧ͍ʣ 1: a ← rand(0, N − 1) 2: b ← rand(a + 1, N) 3: return (a, b) ▶ ͜Ε͸ಉ༷ʹ͔֬Β͘͠ͳΒͳ͍ͷͰ஫ҙʂ ▶ ۩ମྫΛग़ͯ͠ߟ͑ͯΈ·͠ΐ͏ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 12 / 24

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N ݸͷத͔Β 2 ݸબͿํ๏ N ݸͷத͔Β 2 ݸબͿํ๏ʢؒҧ͍ʣ 1: a ← rand(0, N − 1) 2: b ← rand(a + 1, N) 3: return (a, b) ▶ ภ͍ͬͯΔ͜ͱΛ͔֬ΊͯΈΑ͏ ▶ a = 0, b = 1 ͱͳΔ֬཰: 1 (N−1)2 ▶ a ͕ 0 ʹͳΔ֬཰΋ɺb ͕ 1 ʹͳΔʢ৚݅෇͖ʣ֬཰΋ 1 N−1 ▶ a = N − 2, b = N − 1 ͱͳΔ֬཰: 1 N−1 ▶ a ͕ N − 2 ʹͳΔ֬཰͕ 1 N−1 Ͱɺͦͷͱ͖ b ͸ඞͣ N − 1 ʹͳΔ ▶ ྆ऀ͸ຊདྷಉ֬͡཰Ͱग़ͯ΄͍͕͠ɺޙऀͷ΄͏͕ N − 1 ഒग़΍͘͢ɺ ภ͍ͬͯΔ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 13 / 24

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N ݸͷத͔Β 2 ݸબͿํ๏ Ͱ͸ɺͲͷΑ͏ʹ͢Ε͹ภΓͳ͘औΕΔ͔ʁ N ݸͷத͔Β 2 ݸબͿํ๏ 1: repeat 2: a ← rand(0, N) 3: b ← rand(0, N) 4: until a ̸= b ▶ a = b ͷέʔεΛഉআͭͭ͠ a, b ΛϥϯμϜʹܾΊΔͷΛ܁Γฦ͢͜ ͱͰɺҰ༷ϥϯμϜʹϖΞΛܾΊΔ͜ͱ͕Ͱ͖Δ ▶ a < b ʹ͍ͨ͠৔߹͸ɺ্هͷํ๏Ͱ a, b ΛબΜͩޙʹඞཁʹԠͯ͡ swap(a, b) ͢Ε͹Α͍ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 14 / 24

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N ݸͷத͔Β K ݸબͿํ๏: expected O(K) ▶ K ≤ N 2 ͷͱ͖ɺ࣍ͷํ๏Ͱ expected O(K) Λୡ੒Ͱ͖Δ ▶ K ≤ N 2 ΑΓɺཁૉΛ 1 ݸબͿͨΊʹඞཁͳࢼߦճ਺ͷظ଴஋͸ 2 ҎԼ ▶ S ͸ unordered_set ͳͲͰ͓࣋ͬͯ͘ͱΑ͍ N ݸͷத͔Β K ݸબͿํ๏ (K ≤ N 2 ͷͱ͖) 1: S = () 2: for i = 0, . . . , K − 1 : 3: repeat 4: v ← rand(0, N) 5: until v not in S 6: S ʹ v Λ௥Ճ͢Δ 7: end for 8: return S ▶ K ≥ N 2 ͷͱ͖͸ɺબ͹ͳ͍ཁૉΛ্هͷํ๏Ͱܾఆ͢Δ͜ͱͰ expected O(K) Λୡ੒Ͱ͖Δ ▶ ิू߹Λߟ͍͑ͯΔ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 15 / 24

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N ݸͷத͔Β K ݸબͿํ๏: deterministic O(K log K) ▶ ࣍ͷํ๏Ͱ deterministic O(K log K) Λୡ੒Ͱ͖Δ N ݸͷத͔Β K ݸબͿํ๏ 1: t ← N − K + 1, d ← ௕͞ K ͷ഑ྻ 2: for i = 0, . . . , K − 1 : 3: r ← rand(0, t + i) 4: if r < t : d[i] ← r 5: else d[i] ← d[r − t] 6: end if 7: end for 8: d Λঢॱιʔτ͢Δ 9: for i = 0, . . . , K − 1 : 10: d[i] ← d[i] + i 11: end for 12: return d ▶ ͜ͷΞϧΰϦζϜͷΞΠσΞ͸ͪ͜ΒΛࢀরͷ͜ͱ Link tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 16 / 24

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1 ͸͡Ίʹ ϥϯμϜέʔεΛ࢖͏৔໘ ͜ͷൃදʹ͓͚Δ ʮਖ਼͍͠ʯ ϥϯμϜέʔεͱ͸ 2 ϥϯμϜέʔεΛ࡞ͬͯΈΑ͏ ॱྻΛϥϯμϜʹ࡞Δ N ݸͷத͔Β K ݸબͿ K = 2 ͩͬͨΒʁ ֬཰తख๏ ܾఆతख๏ άϥϑΛϥϯμϜʹ࡞Δ άϥϑΛ࡞Δ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 17 / 24

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άϥϑΛ࡞Δํ๏ ௖఺Λ۠ผͨ͠ʢ௖఺ʹ൪߸͕͍ͭͨʣάϥϑΛ࡞Δ͜ͱΛߟ͑Δ ▶ ϥϯμϜάϥϑΛ࡞Δ ▶ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 18 / 24

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ϥϯμϜάϥϑΛ࡞Δํ๏ ▶ N ௖఺ M ลͷϥϯμϜάϥϑΛ࡞Δ͜ͱΛߟ͑Δ ▶ ͋Γ͑Δลͷ૯਺: E = N(N−1) 2 N ௖఺ M ลͷϥϯμϜάϥϑΛ࡞Δํ๏ ઌ΄ͲͷʮN ݸͷத͔Β K ݸબͿํ๏ʯΛԠ༻ ▶ M < E 2 ͷͱ͖ ▶ ลΛϥϯμϜʹ M ݸબͼɺN ௖఺ M ล͔ΒͳΔάϥϑ G Λฦ͢ ▶ M ≥ E 2 ͷͱ͖ ▶ ลΛϥϯμϜʹ E − M ݸબͿ ▶ N ௖఺ E − M ล͔ΒͳΔάϥϑ G Λ࡞Δ ▶ G ͷิάϥϑ G Λฦ͢ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 19 / 24

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ϥϯμϜάϥϑΛ࡞Δํ๏ʢ͓·͚ʣ N ௖఺ M ลͷʮ࿈݁ͳʯϥϯμϜάϥϑΛ࡞Δ৔߹͸Ͳ͏͢Δ͔ʁ Α͘ݟΒΕΔɺ࿈݁ͳϥϯμϜάϥϑΛ࡞Δख๏ ▶ N ௖఺͔ΒͳΔ໦Λ࡞Δ ▶ ໦ΛϥϯμϜʹ࡞Δํ๏͸ޙड़ ▶ ࢒Γ M − N + 1 ลΛϥϯμϜʹ௥Ճ ͔͠͠ɺݫີʹ͸͜Ε͸Ұ༷ϥϯμϜͰͳ͍ ▶ ݫີʹҰ༷ϥϯμϜɺ͔ͭߴ଎ʹ࡞Δํ๏͕͋Ε͹ڭ͍͑ͯͩ͘͞ ▶ Codeforces Ͱͷٞ࿦ Link tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 20 / 24

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άϥϑΛ࡞Δํ๏ ௖఺Λ۠ผͨ͠ʢ௖఺ʹ൪߸͕͍ͭͨʣάϥϑΛ࡞Δ͜ͱΛߟ͑Δ ▶ ϥϯμϜάϥϑΛ࡞Δ ▶ ໦Λ࡞Δ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 21 / 24

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໦Λ࡞Δํ๏ ▶ N ௖఺ͷ໦͔Βɺ1 ͔Β N ͷ੔਺͔ΒͳΔ௕͞ N − 2 ͷ਺ྻ (Prüfer sequence) ͕Ұҙʹ࡞ΕΔ ▶ Prüfer sequence ͔Β໦Λ࡞Δ͜ͱ͕Ͱ͖Δ͠ɺͦͷٯ΋Ͱ͖Δ ▶ ໦ΛҰ༷ϥϯμϜʹ࡞Δʹ͸ɺ1 ͔Β N ͷ੔਺͔ΒͳΔ௕͞ N − 2 ͷ ਺ྻΛϥϯμϜʹ࡞Γɺ໦ʹม׵͢Ε͹Α͍ ▶ ໦͔Β Prüfer sequence Λ࡞Δํ๏͸ Appendix ʹهࡌ ҎԼͷઆ໌͸ 1-indexed Ͱ͋Δ͜ͱʹ஫ҙ Prüfer sequence ͔Β໦Λ࡞Δ: O(N log N) Link 1: function prufer_sequence_to_tree(A) 2: (d1 , d2 , . . . , dN ) ← A Ͱͷ i ͷग़ݱճ਺ + 1 // ௖఺ͷ࣍਺ʹରԠ 3: for i = 1, . . . , N − 2 : 4: dv = 1 ͔ͭ௖఺൪߸͕࠷খͷ௖఺ v ΛٻΊΔ 5: ௖఺ v ͱ Ai ͷؒʹลΛՃ͑Δ 6: dv ͱ di ͷ஋Λ 1 ݮΒ͢ 7: end for 8: du ̸= 0, dv ̸= 0 ͱͳΔ௖఺ u ͱ v ͷؒʹลΛՃ͑Δ 9: end function tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 22 / 24

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·ͱΊ ▶ ࠓճ͸ɺҎԼΛҰ༷ϥϯμϜʹ࡞Δํ๏Λ঺հͨ͠ ▶ গ͠Ͱ΋όάΒͤΔͱҰ༷Ͱͳ͘ͳΔͷͰɺόάʹ஫ҙ ▶ ΑΓྑ͍ϥϯμϜέʔεΛ࡞ΕΔΑ͏ʹͳΓ·͠ΐ͏ʂ ର৅෺ ܭࢉྔ ॱྻ O(N) N ݸͷத͔Β૬ҟͳΔ K ݸΛબͿ expected O(K) deterministic O(K log K) ϥϯμϜάϥϑ expected O(M) ໦ O(N log N) (࿈݁ϥϯμϜάϥϑ) expected O(N log N + (M − N)) ˞Ұ༷ϥϯμϜͰͳ͍ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 23 / 24

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Appendix: ໦͔Β Prüfer sequence Λ࡞Δ ໦͔Β Prüfer sequence Λ࡞Δ: O(N log N) ▶ 2 ͭͷ௖఺͕࢒Δ·ͰɺҎԼΛ܁Γฦ͢ ▶ ͦͷ࣌఺Ͱͷ໦ͷ༿ͷ͏ͪɺ൪߸͕࠷΋খ͍͞΋ͷΛબͿ ▶ ྡ઀͢Δ௖఺Λ Prüfer sequence ʹ௥Ճ͠ɺબΜͩ༿Λ໦͔ΒऔΓআ͘ tsutaj ਖ਼͘͠࡞ΔϥϯμϜέʔε July 11, 2023 24 / 24