Grundy - Speaker Deck

Grundy

by yosupo

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Grundy਺ @yosupot

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ҎԼͷ໰୊Λߟ͑·͢ N௖఺MลͷDAG͕༩͑ΒΕΔɻ௖఺1ʹ͸͕ۨஔ͍ͯ͋Δ A͞ΜͱB͞Μ͕ަޓʹۨΛಈ͔͢ɻ ۨ͸ลͰͭͳ͕ͬͨ௖఺ʹಈ͔ͤͯɺಈ͔ͤͳ͘ͳͬͨΒ ෛ͚ Ͳ͕ͬͪউͭʁ

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ղ๏ dfsͰղ͘ ಈ͔ͤΔ௖఺ʹʮޙʯͱॻ͔Εͨ௖఺͕͋Ε͹ʮઌʯ  ͳ͚Ε͹ʮޙʯͱॻ͘ ௖఺0ʹʮઌʯͱॻ͔ΕΕ͹ઌखඞউ  ʮޙʯͱॻ͔ΕΕ͹ޙखඞউ

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ྫ ޙ ޙ ઌ ઌ ௖఺1

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ҎԼͷ໰୊Λߟ͑·͢ N௖఺MลͷDAG͕Kछྨ༩͑ΒΕΔɻͦΕͧΕͷάϥϑͷ ௖఺1ʹ͸͕ۨஔ͍ͯ͋Δ A͞ΜͱB͞Μ͕ަޓʹɺͲΕ͔ͷάϥϑͷۨΛಈ͔͢ɻ ۨ͸ลͰͭͳ͕ͬͨ௖఺ʹಈ͔ͤͯɺಈ͔ͤͳ͘ͳͬͨΒ ෛ͚ Ͳ͕ͬͪউͭʁ

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ղ๏ ࣮͸ෳ਺ͷάϥϑ͸1ͭͷάϥϑʹ·ͱΊΒΕΔ (άϥϑ1ͷۨͷҐஔ, άϥϑ2ͷۨͷҐஔ, …, άϥϑKͷۨͷ Ґஔ)Λ௖఺ͱͯ͠ద੾ʹลΛషΔ ͜ͷάϥϑͰಉ͡Α͏ʹdfs͕Ͱ͖Δ ܭࢉྔ͸O((N+M)^K)

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O((N+M)^K)ʂʁ N=M=5000ͱͯ͠(N+M)^K͸ʁ K=1 → 10000 K=2 → 1,000,000,000 K=3 → 10,000,000,000,000 K=100 → 10^400

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΋ͬͱޮ཰ྑ͘ܭࢉͰ͖ͳ͍͔ʁ ·ͣ͸K=2ͷ৔߹Λߟ͑Δ

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ߟ࡯ 2ͭͷάϥϑ͕ͲͪΒ΋ޙखউͪͳΒ͹ɺશମͰ΋ޙखউͪ 2ͭͷάϥϑ͕(ઌखউͪ + ޙखউͪ)ͳΒ͹ɺશମͰ͸ઌख উͪ ͩͬͨΒɺ(ઌखউͪ + ઌखউͪ)ͷ৔߹͸ʁ ࣮͸શମͰઌखউͪʹͳΔྫ΋ޙखউͪʹͳΔྫ΋࡞ΕΔ

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Ұҙʹఆ·Βͳ͍ ʴ ʴ ʹ ʹ ઌखউͪ ޙखউͪ

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Ͳ͏͢Δʁ άϥϑΛʮઌखউͪʯͱʮޙखউͪʯʹ·ͱΊΔ͜ͱ͸Ͱ ͖ͳ͍ Ͱ΋2छྨ͸ఘΊΔͱͯ͠΋ɺԿछྨ͔ʹάϥϑΛ෼ྨ͢Δ ͜ͱ͸Ͱ͖ͳ͍ͷ͔ʁ

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(άϥϑA + άϥϑX)ͷউഊͱ(άϥϑB + άϥϑX)ͷউഊ ͕ɺXʹ͔͔ΘΒͣ౳͍࣌͠ɺάϥϑA,BΛʮຊ࣭తʹ౳͠ ͍ʯͱ͢Δ (ޙखউͪ+ޙखউͪ)ͱ(ޙखউͪ+ઌखউͪ)͸Ұҙʹఆ·Δ ͷͰɺޙखউͪͲ͏͠͸ʮຊ࣭తʹ౳͍͠ʯ ͳͷͰɺઌखউͪͲ͏͠΋ʮຊ࣭తʹ౳͍͠ʯͷͰ͸ʁͱ Ծఆࣦͯ͠ഊͨ͠ͷ͕ࠓͷঢ়ଶ ઌखউͪͷάϥϑ͸1छྨʹ·ͱΊΔ͜ͱ͸Ͱ͖ͳ͍ʂ

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ͱ͜ΖͰ ͖ͬࣔͨ͞͠Α͏ʹɺҎԼͷ2ͭͷάϥϑ͸౳͘͠ͳ͍ ͳͥʁ

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ͱ͜ΖͰ ࠨͷάϥϑ͸ޙखউͪʹ͔͠ߦ͚ͳ͍͕ɺӈͷάϥϑ͸ޙखউͪͱઌखউͪͷ ྆ํʹߦ͚Δ ޙखউͪͱઌखউ͕ͪ྆ํඋΘΓ࠷ڧʹݟ͑Δ(ӈͷάϥϑͷํ͕ࣗ༝౓͕ߴͦ ͏)

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͡Ό͋ ͜ͷάϥϑ͸ߋʹ࠷ڧͳͷͰ͸ʁ

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͡Ό͋ ͜ͷάϥϑ͸ߋʹ௒࠷ڧͳͷͰ͸ʁ

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࣮͸ ׬શDAGͲ͏͠͸ޓ͍ʹ౳͘͠ͳ͍͜ͱ͕ࣔͤΔ(TODO) ͳͷͰগͳ͘ͱ΋༗ݶछྨͰ͸ͳ͍

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΄΅׬શDAG͸ʁ ׬શDAGಉ࢜͸ҟͳΔ͜ͱ͕Θ͔ͬͨ ͡Ό͋׬શDAGʹ1ݸ௖఺ΛՃ͑ͨ΋ͷ͸ʁ ׬શDAGͷ͏͍͔ͪͭ͘ͷ௖఺ʹɺ௖఺1͔Βล͕ੜ͍͑ͯ ΔɺΑ͏ͳάϥϑ

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ྫ͑͹ ͜Ε͸௖఺1͔Β௖఺਺1, 3ͷDAGʹߦ͚ΔΑ͏ͳάϥϑ

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ྫ͑͹ ͜Ε͸௖఺1͔Β௖఺਺1, 3ͷ׬શDAGʹߦ͚ΔΑ͏ͳάϥϑ ͜Ε͸௖఺਺2ͷ׬શDAGͱ౳͍͠

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ͳͥʁ ௖఺਺3ͷ׬શDAGʹߦͬͯɺ༗རʹͳΔ͜ͱ͸ͳ͍ ͳͥͳΒ͹ɺ૬ख͸࣍ͷखͰ௖఺਺2ͷ׬શDAG΁ҠಈͰ͖ΔͨΊ ݁ہɺ௖఺਺1ͷ׬શDAG΁ߦ͔͘͠ͳ͍ͨΊɺ௖఺਺2ͷ׬શDAGͱຊ࣭తʹ౳͍͠ ͜ͷลΛ࢖ͬͯ༗རʹͳΔ͜ͱ͸ͳ͍

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΄΅׬શDAG ௖఺਺{a, b, c, …}ʹߦ͚Δͱ͢Δͱɺ௖఺਺=ʮ{a, b, c, …}ʹ ग़ͯ͜ͳ͍࠷খͷਖ਼੔਺ʯͷ׬શDAGͱຊ࣭తʹ౳͍͠ ྫ͑͹{1, 3, 4, 5}ʹߦ͚ΔͳΒ͹௖఺਺=2 ͳ͔ͥͱ͍͏ͱɺ3,4,5ͷͲΕʹߦ͘ͷ΋ɺ࣍ͷखͰ૬खʹ2 ʹ໭͞ΕΔͷͰɺ༗རʹͳΕͳ͍

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Ұൠʹ ઌड़ͷنଇΑΓؼೲతʹɺશͯͷάϥϑ͸׬શDAGͷͲΕ͔ͱ ຊ࣭తʹ౳͍͠ ௖఺਺=kͷ׬શDAGͱຊ࣭తʹ౳͍͠άϥϑΛlevel (k-1)ͷάϥ ϑͱݺͿ (0-indexedͰͳ͍ͱ৭ʑ໘౗) άϥϑͷlevel͸࠶ؼతʹܭࢉͰ͖ΔͨΊɺO(N+M)ͰٻΊΒΕΔ

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݁ہ (௖఺਺aͷ׬શDAG + ௖఺਺bͷ׬શDAG)ͱมܗͰ͖Δ ͜Εͷඞউ൑ఆ͸؆୯(a = b͔൱͔) O(N+M)

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K=3 K=3ͷ৔߹͸ʁ ࠷ॳͷ۪௚ղΛࢥ͍ग़͢ͱɺෳ਺ݸͷάϥϑ͸1ݸͷάϥϑʹ·ͱΊΒΕΔ ͦͯ͠ɺk=2ͳΒ͹શͯͷάϥϑ͸׬શDAG΁มܗͰ͖Δ͜ͱΛ࢖͏ (άϥϑA + άϥϑB + άϥϑC)=  {άϥϑA + (άϥϑBͱάϥϑCΛ·ͱΊͨ΋ͷ)}=  (׬શDAG + άϥϑB + άϥϑC) Αͬͯk=2ͷ࣌ͱಉ͡نଇͰ׬શDAG΁มܗͯ͠Α͍

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K=4 K=4ͷ৔߹͸ʁ (άϥϑA + άϥϑB + άϥϑC + άϥϑD)=  {άϥϑA+(άϥϑBͱάϥϑCͱάϥϑDΛ·ͱΊͨ΋ͷ)}=  (׬શDAG + άϥϑB + άϥϑC + άϥϑD)

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݁ہ (׬શDAG + ׬શDAG + … + ׬શDAG)ʹมܗͰ͖Δ ͜ͷඞউ൑ఆ͸ʁ(࣮͸NimήʔϜͱ͍͏༗໊ͳܗͷήʔϜʹͳ͍ͬͯΔ…) (׬શDAG + ׬શDAG + … + ׬શDAG)=  {(׬શDAGͱ׬શDAGΛ·ͱΊͨ΋ͷ) + (K-2ݸͷ׬શDAGΛ·ͱΊͨ΋ͷ)}=  ʹมܗͰ͖Δ ͱ͜ΖͰɺK=2ͳͷͰ(׬શDAGͱ׬શDAGΛ·ͱΊͨ΋ͷ)͸׬શDAG΁มܗͰ͖Δ ͭ·Γɺ(׬શDAGͱ׬શDAGΛ·ͱΊͨ΋ͷ)Λ׬શDAGʹมܗͰ͖ΔͱɺK-1ʹؼ ணͰ͖Δ

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׬શDAG + ׬શDAG (׬શDAGͱ׬શDAGΛ·ͱΊͨ΋ͷ)ͱ౳͍͠׬શDAGͷ ௖఺਺͸ʁ ࣮͸ɺ௖఺਺(a+1)ͱ(b+1)Λ·ͱΊΔͱɺ௖఺਺(a xor b)+1 ͱ౳͘͠ͳΔ(TODOઆ໌) ͭ·Γɺlevel aͱlevel bΛ߹੒͢Δͱɺlevel (a xor b)ʹͳΔ (͜Ε͕0-indexedʹͨ͠ཧ༝)

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ղ๏ ͱ͍͏͜ͱ͸ɺlevel a + b + …ͳΒ͹ɺlevel (a xor b xor …) ͱ౳͍͠ ͜Ε͕0͔Ͳ͏͔൑ผ͢Ε͹౴͑ O(K(N+M))ʂ