響け！動的計画法 (DP) 入門〜個人的まとめ
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༷ʑͳ%1 w͏%1 wΔ%1 w֬%1 wCJU%1 wܻ%1 wظ%1 w۠ؒ%1 w%1 πϦʔ%1
wήʔϜ%1 w෦ྻ%1 wૠೖ%1 wൈڈ%1 w͢%1 w࿈݁%1 ϑϩϯςΟΞ๏ w5SJF%1 wશํҐ%1 wΠϯϥΠϯ%1 w"MJFO%1 ࠓ͜͜·Ͱʂ ࢀরઌ"U$PEFS ղ63-Λ֤εϥΠυʹهࡌ͍ͯ͠·͢ -
φοϓαοΫ ༰ྔa ΓΜ͝ ͪ͘Θ Έ͔Μ ͠
Ձ ॏ͞ ΓΜ͝ ԁ ͪ͘Θ ԁ Έ͔Μ ԁ ͠ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ԁ ྫ C = 7 N = 4 φοϓαοΫʹೖͬͨͷՁςʔϒϧ ԁ ԁ ԁ ԁ ԁ ԁ ԁ 2 Ձॏ͞ ॱͰᩦཉʹ ͠߹ΘͤΕ͍͍ͷͰ "࠷దղʹͳΒͳ͍ ༰ྔ ͷφοϓαοΫͱ छྨͷ Ձ ॏ͞ ͕༩͑ΒΕͨ࣌ Λ͑ͳ͍ൣғͰՁ૯Λ࠷େԽ͢ΔΑ͏ͳͷબͼํΛ͑Α C N C 1 ≤ N ≤ 100 1 ≤ C ≤ 105 1 ≤ wi ≤ C 1 ≤ vi ≤ 109 ඵҎ vi wi &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ%,OBQTBDLIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@E -
ೋछྨͷه๏ w Խࣜ w ϝϞԽ࠶ؼ 1SPT
ؔݺͼग़͠ʹΑΔελοΫΦʔόʔϑϩʔ̈ൃੜ͠ͳ͍ ΠϯϥΠϯ%1 ηάͰՃࢉ ͷΑ͏ͳߴԽςΫ͕ద༻Ͱ͖Δ ίʔφʔέʔεߟྀͷͨΊͷJGจ͕૿͑ΔIUUQJEFPOFDPN-J;R ̈࠶ؼͱελοΫɾΦʔόʔϑϩʔ IUUQXXXGPTLVJTLZPUPVBDKQdJHBSBTIJDMBTTQMSFDJUFSIUNM 1SPT τϙϩδΧϧιʔτ̉ෆཁ ίʔφʔέʔεͰίʔυ͕؆ૉ ̉τϙϩδΧϧιʔτ IUUQTFOXJLJQFEJBPSHXJLJ5PQPMPHJDBM@TPSUJOH dp[i][c] = dp[i − 1][c] JGΛબͳ͍ JGΛબͿ͔ͭ c − wi ≥ 0 max(dp[i − 1][c − wi ] + vi , dp[i − 1][c]) { ·Ͱͷͷத͔Βॏ͕͞ʹͳΔΑ͏ʹબΜͩ࣌ͷՁͷ૯ͷ࠷େ dp[i][c] i c Long long solve(long long i, long long c){ if(dp[i][c] != -1) return dp[i][c]; if(i == 0) return dp[i][c] = 0; if((c-w[i]) >= 0) return dp[i][c] = max(solve(i-1,c-w[i])+v[i], solve(i-1,c)); else return dp[i][c] = solve(i-1,c); } IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT -
͏%1Δ%1 w ͏%1 dp[i][c] = dp[i − 1][c]
JGΛબͳ͍ JGΛબͿ͔ͭ c − wi ≥ 0 max(dp[i − 1][c − wi ] + vi , dp[i − 1][c]) { &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ%,OBQTBDLIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@E dp[i + 1][c + wi ] = max(dp[i][c] + vi , dp[i + 1][c + wi ]) JG Γઌ͕༰ྔҎ c + wi ≤ C dp[i + 1][c] = max(dp[i + 1][c], dp[i][c]) IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT ʜ J J J ʜ DXJ EQ<J><DX@J> D EQ<J><D> EQ<J><D> EQ<J ><D> D XJ EQ<J ><D X@J> w Δ%1 ঢ়ଶΛભҠ dp[i][c] ͔Βঢ়ଶΛભҠ dp[i][c] ·Ͱͷͷத͔Βॏ͕͞ʹͳΔΑ͏ʹબΜͩ࣌ͷՁͷ૯ͷ࠷େ dp[i][c] i c -
֬%1 w ֬Ͱঢ়ଶΛද͢%1 ྫEQ<J> ʙͱͳΔ֬ Λਖ਼ͷحͱ͠·͢ɻ ຕͷίΠϯ͕͋Γ·͢ɻίΠϯʹ
ͱ൪߸͕ৼΒΕ͍ͯ·͢ɻ ֤ ʹ͍ͭͯɺίΠϯΛ͛Δͱɺ֬ Ͱද͕ग़ͯɺ֬ Ͱཪ͕ग़·͢ɻ ଠ܅ ຕͷίΠϯΛͯ͛͢·ͨ͠ɻ ͜ͷͱ͖ɺදͷݸ͕ཪͷݸΛ্ճΔ֬ΛٻΊ͍ͯͩ͘͞ɻ ೖྗͯ͢Ͱ͋Δɻ N N 1,2,…, N i (1 ≤ i ≤ N) i pi 1 − pi N 1 ≤ N ≤ 102 1 ≤ a1 < a2 < ⋯ < aN ≤ K &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ*$PJOTIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@J EQ<><> GPS JOUJ J O J GPS JOUK K J K \ EQ<J ><K > EQ<J><K> Q<J> EQ<J ><K> EQ<J><K> Q<J> ^ EPVCMFBOT GPS JOUJ O J O J BOT EQ<O><J> DPVU TFUQSFDJTJPO BOT FOEM IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT EQ<J><K> JຕͷίΠϯ·Ͱ͛ͨͱ͖ ද͕KຕͰΔ֬ w ํͱղྫ ঢ়ଶͷભҠΛߟ͑Δͱ ҎԼͷࣜʹͳΔ ίΠϯJ͕දͷ߹ ίΠϯ͕දͷ֬ + = dp[i][j] * i ίΠϯ͕ཪͷ֬ + = dp[i][j] * i dp[i + 1][j + 1] dp[i + 1][j] ίΠϯJ͕ཪͷ߹ -
CJU%1 w CJUͰঢ়ଶΛද͢%1 ྫEQ<CJU><J> w ੍͕/ ͳͲͱখ͍࣌͞େ͜Ε
ඖͷ͏͗͞ ʙ/ ͕͍·͢ɻ͜ΕΒ ඖͷ͏͕͗͞ెڝΛ͠·ͨ͠ɻ ಉண͍·ͤΜͰͨ͠ɻߴڮ܅ ਓͷ؍٬͔ΒใΛूΊ·ͨ͠ɻ ൪ͷ؍٬ʹΑΔͱɺ͏͗͞ ͏͗͞ ΑΓઌʹΰʔϧͨͦ͠͏Ͱ͢ɻ ͯ͢ͷ؍٬ͷใʹ߹க͢ΔΑ͏ͳணॱ͕Կ௨Γߟ͑ΒΕΔ͔ٻΊ͍ͯͩ͘͞ɻ ͷͯ͢૬ҟͳΔɻ ͯ͢ͷ؍٬ͷใʹ߹க͢ΔΑ͏ͳணॱ͕গͳ͘ͱͻͱͭଘࡏ͢Δɻ N N M i xi yi 2 ≤ N ≤ 16 1 ≤ M ≤ N(N − 1)/2 1 ≤ xi , yi ≤ N xi ≠ yi (xi , yi ) "U$PEFS#FHJOOFS$POUFTU%ెڝIUUQTBUDPEFSKQDPOUFTUTBCDUBTLTBCD@E ྫ N = 4 M = 3 w ΑΓઌʹΰʔϧ w ΑΓઌʹΰʔϧ w ΑΓઌʹΰʔϧ ύλʔϯaணॱ Ґ Ґ Ґ Ґ ͏͗͞ શ෦Ͱ௨Γ -
CJU%1ίʔυ IUUQTBUDPEFSKQDPOUFTUTBCDTVCNJTTJPOT EQ<CJU> ౸ணॱͷ௨Γ EQ<> GPS JOUCJU CJU O CJU
\CJU ͔ΒCJU ·Ͱ GPS JOUK K O K \ JG CJU K DPOUJOVF طʹ౸ணͨ͠Kύεɹ CPPMG USVF H<K><L> KLΑΓઌʹ౸ண L͕طʹ౸ண͍ͯͯ͠K͕ະ౸ணͳΒGBMTF GPS JOUL L O L JG CJU L H<K><L> G GBMTF JG G EQ<CJUc K > EQ<CJU> ྫ͑ CJU ʹCJU Λ͠߹ΘͤΔ ^ ^ DPVU EQ< O > FOEM "U$PEFS#FHJOOFS$POUFTU%ెڝIUUQTBUDPEFSKQDPOUFTUTBCDUBTLTBCD@E w ํͱղྫ w CJUͷ֤ܻ͕Λද͠ ౸ண ະ౸ண ྫCJU ʜ ͕౸ண ະ౸ண w CJUঢ়ଶͷભҠΛߟ͑Δ CJU CJU CJU -
ܻ%1 w ݅Λຬ͔ͨ͢ ্ݶ͔൱͔ ͷঢ়ଶΛද͢%1 ྫEQ<ܻ><݅><ະຬPS͍͠ͷϑϥά> ߹ͷ w
TNBMMFSϑϥάΛઃ͚ͯ ্݅ݶͷཧΛߦ͏ w ੍͕, ?ͳͲͱେ͖͍࣌େ͜Ε Ҏ্ ҎԼͷͷ͏ͪɺेਐදهʹ͓͚Δ֤ܻͷࣈͷ૯͕ ͷഒͰ͋ΔΑ ͏ͳͷԿݸͰ͠ΐ͏͔ Ͱׂͬͨ༨ΓΛٻΊ͍ͯͩ͘͞ɻ 1 K D 1e9 + 7 1 ≤ K < 1010000 1 ≤ D ≤ 100 &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ4%JHJU4VNIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@T ྫ, % EHU㱡 ෦ ͕ҎԼ ,ҎԼ ͳΒՃࢉ͢Δ EHU㱡͕ҎԼ͔ͭEHU㱢 ෦ ͕ҎԼͳΒՃࢉ EHU㱡͕ҎԼ ,ະຬ ͳΒՃࢉ EHU ͷ࣌ ʜ ʜ w ํͱղྫ w ,ҎԼͷΒΛઌ಄͔ΒܻͣͭࣈΛݟΔ w ֤ܻͷࣈͷ૯ʹ͍ͭͯɺ%Ͱ༨ΛͱΔ w ͋Δͷ͕,ҎԼͳΒɺՃࢉ͢Δ -
ܻ%1ίʔυ &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ4%JHJU4VNIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@T IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT EQ<EHU><E><JTMFTT> ઌ಄͔Β֬ఆܻͨ͠EHU ֤ܻͷࣈͷΛ%ͰׂͬͨË ,ະຬஸͷϑϥάJTMFTTΛ࣋ͭͷ EQ<><><>
GPS JOUEHU EHU / EHU GPS JOUE E % E GPS JOUJTMFTT JTMFTT JTMFTT \ GPS JOUOYU OYU OYU \ JG OYU ,<EHU> ͋ΔͷܻEHU͕,ͷEHUΑΓখ͍͞߹ʜJTMFTT ʹՃࢉ EQ<EHU >< E OYU %><> EQ<EHU><E><JTMFTT>.0% .0% FMTFJG OYU ,<EHU> ͋ΔͷܻEHU͕,ͷEHUͱಉ͡߹ʜ֤JTMFTTʹՃࢉ EQ<EHU >< E OYU %><JTMFTT> EQ<EHU><E><JTMFTT>.0% .0% FMTF͋ΔͷܻEHU͕,ͷEHUΑΓେ͖͍߹ʜJTMFTTͳΒՃࢉ JG JTMFTT EQ<EHU >< E OYU %><JTMFTT> EQ<EHU><E><JTMFTT>.0% .0% ^ ,ͱ͍͠ ,ະຬ DPVU EQ</><><> EQ</><><> .0% .0% .0% FOEM ̈ " # $"$ #$ͱՁ -
͜͜·Ͱͷ·ͱΊ w ಈతܭը๏ʮ͋ΔΛখ͍͞෦ʹׂ͠ ෦ ͷղΛར༻͢Δ͜ͱͰݩͷΛղ͘ΞϧΰϦζϜʯ w ه๏ʮԽࣜʯͱʮϝϞԽ࠶ؼʯ͕͋ΔΑ w छྨछྨҎ্ɺྫͱͯͭ͠ڍ͛ΔΑ w
CJU%1CJUͰঢ়ଶΛද͢%1 w ܻ%1ܻϑϥάͰঢ়ଶΛද͢%1 w ֬%1֬Ͱঢ়ଶΛද͢%1 -
༷ʑͳ%1 w͏%1 wΔ%1 w֬%1 wCJU%1 wܻ%1 wظ%1 w۠ؒ%1 w%1 πϦʔ%1
wήʔϜ%1 w෦ྻ%1 wૠೖ%1 wൈڈ%1 w͢%1 w࿈݁%1 ϑϩϯςΟΞ๏ w5SJF%1 wશํҐ%1 wΠϯϥΠϯ%1 w"MJFO%1 ࢀরઌ"U$PEFS ղ63-Λ֤εϥΠυʹهࡌ͍ͯ͠·͢ -
ظ%1 &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ+4VTIJIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@K w ظͰঢ়ଶΛද͢%1 ྫEQ<J> ʙͱͳΔظ /ຕͷࡼ͕͋Γ·͢ɻࡼʹ ͱ൪߸͕ৼΒΕ͍ͯ·͢ɻ ࠷ॳɺ֤
ʹ͍ͭͯɺࡼJʹ ݸͷण͕࢘ஔ͔Ε͍ͯ·͢ɻ ͯ͢ͷण͕࢘ແ͘ͳΔ·Ͱɺଠ܅࣍ͷૢ࡞Λ܁Γฦ͠ߦ͍·͢ɻ ͷ͕֬Ͱग़ΔαΠίϩΛৼΓɺग़Λͱ͢Δɻ ࡼʹण͕࢘͋Δ߹ɺࡼͷण࢘Λݸ৯Δɻࡼʹण͕࢘ແ͍߹ɺԿߦΘͳ͍ɻ ͯ͢ͷण͕࢘ແ͘ͳΔ·Ͱͷૢ࡞ճͷظΛٻΊ͍ͯͩ͘͞ɻ ೖྗͯ͢Ͱ͋Δɻ 1,2,…, N i (1 ≤ i ≤ N) ai (1 ≤ ai ≤ 3) 1,2,…, N i i i i 1 ≤ N ≤ 300 1 ≤ ai ≤ 3 IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT EQ<J><K><L> ͷࡼ ͷࡼ ͷࡼʹ͓͚Δૢ࡞ճͷظ ai = 1 i ai = 2 j ai = 3 k w ํͱղྫ EPVCMFEGT MMJ MMK MML \ JG EQ<J><K><L> SFUVSOEQ<J><K><L> JG J K L SFUVSO EPVCMFJKL EPVCMF J K L EPVCMFSFT OJKL JG L SFT EGT J K L EPVCMF LJKL JG K SFT EGT J K L EPVCMF KJKL JG J SFT EGT J K L EPVCMF JJKL SFUVSOEQ<J><K><L> SFT ^ w શࡼ֬ͰબΕΔͷͰɺ ͷॱ൪ Өڹ͠ͳ͍ w ঢ়ଶͱͯ͠ࡼ͝ͱͷૢ࡞ճͷظ Λߟ͑Δ ai -
۠ؒ%1 /ඖͷεϥΠϜ͕ԣҰྻʹฒΜͰ͍·͢ɻ࠷ॳɺࠨ͔ΒJ൪ͷεϥΠϜͷେ͖͞B@JͰ͢ɻ ଠ܅ɺͯ͢ͷεϥΠϜΛ߹ମͤͯ͞ඖͷεϥΠϜʹ͠Α͏ͱ͍ͯ͠·͢ɻ εϥΠϜ͕ඖʹͳΔ·Ͱɺଠ܅࣍ͷૢ࡞Λ܁Γฦ͠ߦ͍·͢ɻ ࠨӈʹྡΓ߹͏ඖͷεϥΠϜΛબͼɺͦΕΒΛ߹ମͤͯ͞৽͍͠ඖͷεϥΠϜʹ͢Δɻ ߹ମલͷඖͷεϥΠϜͷେ͖͞Λ Y͓ΑͼZͱ͢Δͱɺ߹ମޙͷεϥΠϜͷେ͖͞Y ZͱͳΔɻ
͜ͷͱ͖ɺଠ܅Y ZͷίετΛࢧ͏ɻ ͳ͓ɺ߹ମͷલޙͰεϥΠϜͨͪͷҐஔؔมΘΒͳ͍ɻ ଠ܅͕ࢧ͏ίετͷ૯ͷ࠷খΛٻΊ͍ͯͩ͘͞ɻ ೖྗͯ͢Ͱ͋Δɻ 2 ≤ N ≤ 400 1 ≤ ai ≤ 109 w ࠨӈͷ۠ؒͷঢ়ଶΛཧ͢Δ%1 ྫEQ<-FGU><3JHIU> ۠ؒ<-FGU 3JHIU ͷղ &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ/4MJNFTIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@O w ํͱղྫ w ٯʹʮεϥΠϜ͕྾͢ΔࡍʹɺεϥΠϜ ͷେ͖͚ͩ͞ίετ͕͔͔Δʯͱߟ͑Δ w ྾ޙͷͭεϥΠϜͷେ͖͕͞࠷খͱͳ ΔҐஔͰׂ͢Δ ૯ ૯ ૯ -
۠ؒ%1ίʔυ ྦྷੵΛԿߟ͑ͣʹॻ͚ΔΑ͏ʹ͢ΔʂIUUQTRJJUBDPNESLFOJUFNTBCFEFGGD IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT EQ<M><S> ۠ؒ<M S ʹ͓͚Δίετ૯ͷ࠷খ C<J> ۠ؒ<J>ͷྦྷੵ
C<> B<> GPS JOUJ J O J C<J> C<J> B<J> ྦྷੵ͔Β۠ؒऔಘ MMHFU MMM MMS \ MMSFT C<S> JG M SFT C<M> SFUVSOSFT ^ MMEGT MMM MMS \ JG M S SFUVSO JG WJTJUFE<M><S> SFUVSOEQ<M><S> WJTJUFE<M><S> USVF MMSFT */' ׂΛͯ͢ࢼ͠ ࠷ίετ͕খ͍͞ҐஔͰׂ GPS JOUJ M J S J SFT NJO SFT EGT M J EGT J S HFU M S SFUVSOEQ<M><S> SFT ^ -
%1 πϦʔ%1 ݸͷౡ͕͋Γ·͢ɻౡʹ ͔Β ·Ͱͷ൪߸͕͍͍ͭͯ·͢ɻ·ͨɺ ݸͷڮ͕͋Γ·͢ɻ ൪ͷڮ
൪ͷౡͱ ൪ͷౡΛͭͳ͍Ͱ͍·͢ɻ Ͳͷౡ͔ΒͲͷౡڮΛ͍͔ͭ͘ܦ༝ͯ͠౸ୡͰ͖Δ͜ͱ͕Θ͔͍ͬͯ·͢ɻ ͢͵͚܅ɺͦΕͧΕͷౡΛന·ͨࠇʹృΔ͜ͱʹ͠·ͨ͠ɻ ͨͩ͠ɺ྆ͷౡ͕ࠇͰృΒΕ͍ͯΔΑ͏ͳڮ͕͍͚͋ͬͯ·ͤΜɻ ৭ͷృΓํ͕Կ௨Γ͋Δ͔ɺ Ͱׂͬͨ༨ΓΛٻΊ͍ͯͩ͘͞ɻ Ͳͷౡ͔ΒͲͷౡڮΛ͍͔ͭ͘ܦ༝ͯ͠౸ୡͰ͖Δ N 1 N N − 1 i ai bi 109 + 7 1 ≤ N ≤ 105 1 ≤ ai , bi ≤ N w ߏ্Ͱܭࢉ͢Δ%1 "U$PEFS#FHJOOFS$POUFTU%ృΓֆIUUQTBUDPEFSKQDPOUFTUTBCDUBTLTBCD@E TUBUJDJOUEQ<><> GVODUJPO WPJE JOU JOU EGT <> JOUV JOUQSF \ EQ<V><> EQ<V><> GPS JOUW(<V> \(<V><W> VͷࢠW JG W QSF DPOUJOVF ༿ͳΒDPOUJOVF EGT W V ࢠWͷEQ<W>< >Λߋ৽ EQ<V><> EQ<V><> EQ<W><> EQ<W><> .0% ന ࠇ EQ<V><> EQ<V><> EQ<W><> .0% നͷΈ ^ SFUVSO ^ EGT DPVU EQ<><> EQ<><> .0% FOEM IUUQTBUDPEFSKQDPOUFTUTBCDTVCNJTTJPOT EQ<V><> VΛͱ͢Δ෦ʹ ؚ·ΕΔશΛന·ͨࠇͰృΔ௨Γ EQ<V><> V͕നͷ߹ʹ͓͚Δ্ه w ํͱղྫ w ෦ͷJʹ͓͚Δ௨ΓΛ͑Δ ԽࣜΛཱͯͯʮUSFF%1ʯΛղ͘% ృΓֆ w Լهࢀর -
ήʔϜ%1 ݸͷਖ਼͔ΒͳΔू߹ ͕͋Γ·͢ɻ ଠ܅ͱ࣍܅͕࣍ͷήʔϜͰউෛ͠·͢ɻ ࠷ॳʹɺ ݸͷੴ͔ΒͳΔࢁΛ༻ҙ͠·͢ɻೋਓ࣍ͷૢ࡞Λަޓʹߦ͍·͢ɻ ઌखଠ܅Ͱ͢ɻ
ͷݩΛͻͱͭબͼɺࢁ͔Βͪΐ͏ͲݸͷੴΛऔΓڈΔɻ ઌʹૢ࡞Λߦ͑ͳ͘ͳͬͨਓ͕ෛ͚Ͱ͢ɻ ೋਓ͕࠷దʹߦಈ͢ΔͱԾఆͨ͠ͱ͖ɺͲͪΒ͕উ͔ͭΛఆ͍ͯͩ͘͠͞ɻ ೖྗͯ͢Ͱ͋Δɻ N A = a1 , a2 , …, aN K A x x 1 ≤ N ≤ 102 1 ≤ K ≤ 105 1 ≤ a1 < a2 < ⋯ < aN ≤ K &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ,4UPOFTIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@L IUUQTBUDPEFSKQDPOUFTUTEQTVCNJTTJPOT EQ<J> Γͷੴ͕Jݸʹ͓͚Δ ଠͷউͪঢ়ଶϑϥά w ํͱղྫ w , ͷ࣌ʜෛ͚ঢ়ଶ ଠͷෛ͚ w , Jͷ࣌ʜ, dJͰෛ͚ঢ়ଶ͕ ͋ΔͳΒ ෛ͚Λ૬खʹԡ͚͠Δ w ࠨӈͷ۠ؒͷঢ়ଶΛཧ͢Δ%1 ྫEQ<-FGU><3JHIU> ۠ؒ<-FGU 3JHIU ͷղ GPS J J L J \GPS K K O K \ B<K>ݸͷੴΛऔΕΔͳΒ औΔ EQ<J> USVF͕ͭͰ͋Ε USVF JG JB<K> \ EQ<J>c EQ<JB<K>> ^ ^^ -
෦ྻ%1 ͞ ͷจࣈྻ ͕༩͑ΒΕΔɻ ͷ෦จࣈྻ ۭจࣈؚΉ ͱͯ͠ߟ͑ΒΕΔͷͷݸΛ্͑͛Αɻ
͕ͨͩ͑͠ͱͯେ͖͘ͳΔ͜ͱ͕͋ΔͷͰɺݸΛ Ͱׂͬͨ͋·ΓΛٻΊΑɻ ͷ֤จࣈখจࣈΞϧϑΝϕοτͷΈ छྨ n S S 1 ≤ n ≤ 105 S 2JJUB෦จࣈྻͷ্͑͛෦ྻ%1จࣈྻͷ෦จࣈྻΛॏෳͳࠪ͘͢Δ%1ͷಛू ղͳ͠ EQ<J> Γͷੴ͕Jݸʹ͓͚Δ ଠͷউͪঢ়ଶϑϥά w ํͱղྫ w ෦จࣈྻʹؔͯ͠ܭࢉ͢Δ%1 ྫ͋Δจࣈྻʹ͓͍ͯ෦จࣈྻͷଘࡏΛ্͑͛ WFDUPS WFDUPS JOU OFYU DBMD/FYU 4 EQ<> ۭจࣈؚΉ GPS JOUJ J O J \ GPS JOUK K K \ JG OFYU<J><K> O DPOUJOVF EQ<OFYU<J><K> >ʹEQ<J>ΛՃࢉ BEE EQ<OFYU<J><K> > EQ<J> ^ ^ MPOHMPOHBOT GPS JOUJ J O J BEE BOT EQ<J> DPVU BOT FOEM OFYU<J><D> 4ͷJจࣈҎ߱Ͱ ࠷ॳʹD͕ొ͢ΔJOEFY WFDUPS WFDUPS JOU DBMD/FYU DPOTUTUSJOH4 \ JOUO JOU 4TJ[F WFDUPS WFDUPS JOU OFYU O WFDUPS JOU O GPS JOUJ O J J \ GPS JOUK K K OFYU<J><K> OFYU<J ><K> OFYU<J><4<J>B> J ^ SFUVSOOFYU ^ EQ<J> จࣈྻ4ͷ͏͔ͪΒJ൪ͷ෦ ΑΓಘΒΕΔ෦จࣈྻͷݸ ͨͩ͠ ಉ͡෦จࣈྻΛੜ͢ΔͳΒબJOEFY͕ࣙॻॱ࠷খʹͳΔΑ͏ʹ -
ૠೖ%1 ҎԼͷ݅Λຬͨ͢จࣈྻͷݸΛNPE ͰٻΊΑɻ BΛ ݸɺCΛ ݸɺ[Λ
ݸؚΉ ଞͷจࣈؚ·ͳ͍ ɻ ಉ͡จࣈ͕ྡΓ߹͏͜ͱͳ͍ɻ ͷ͏ͪ গͳ͘ͱҰͭΑΓେ͖͍ͱ͢Δɻ freq1 freq2 freq26 0 ≤ freqi ≤ 10 freqi w ॱྻจࣈྻʹૠೖ͢Δ͜ͱͰભҠ͢Δ%1 5ZQJDBM%1$POUFTU0จࣈྻIUUQTUEQDDPOUFTUBUDPEFSKQUBTLTUEQD@TUSJOH EQ<J><K> Jจࣈ·Ͱʹ͓͍ͯ ࿈ଓ͍ͯ͠ΔՕॴ OHͳՕॴ ͕ Kݸ͋ΔΑ͏ͳจࣈྻͷݸ w ํͱղྫ w จࣈͷؒΛݟ͕ͯ྆ಉจࣈ ͳΒOH ҧ͏จࣈͳΒPL w GSFR<J>ͷจࣈΛׂͯ͠ PLOHͳՕॴʹೖΕ͍ͯ͘ "MHPPHMF͍͢όΧࢽ EFpOFSFQJ J B C GPS JOUJ JOU B J JOU C J EFpOFSFQ J O SFQJ J O JOUTPMWF \ D DPNCJOBUJPO ߹ͷ ύεΧϧͷࡾ֯ܗͳͲͰٻΊΔ G<J> GSFR<J> TVN<J> J൪·Ͱͷ૯จࣈ SFQ J O TVN<J > TVN<J> G<J> EQ<><> SFQ J O SFQ K TVN<J> SFQJ L G<J> SFQ M NJO K L BEE EQ<J ><KM G<J>L> NVM EQ<J><K> NVM D<G<J>><L> NVM D<K><M> D<TVN<J> K><LM> SFUVSOEQ<O><> ^ ׂL ݱࡏͷPLͳՕॴͷTVN<J> K ࣍ͷOHͳՕॴͷKM GSFR<J>L dp[i + 1][j − l + freq[i] − k] + = -
ൈڈ%1 w ૠೖ%1ͱٯʹൈڈ͢Δ%1 "3$&4PSUFEBOE4PSUFEIUUQTCFUBBUDPEFSKQDPOUFTUTBSDUBTLTBSD@D ͦΕͧΕ ͔Β ͷ͕ ͭͣͭॻ͔Εͨന͍Ϙʔϧͱࠇ͍Ϙʔϧ͕߹Θͤͯ
ݸҰྻʹฒ ΜͰ͍·͢ɻ ࠨ͔Β ݸͷϘʔϧʹॻ͍ͯ͋Δ Ͱɺ৭ Ͱද͞Ε·͢ɻ ͷͱ͖Ϙʔϧ͕ന͍͜ͱΛɺ ͷͱ͖Ϙʔϧ͕ࠇ͍͜ͱΛද͠·͢ɻ ਓؒͷߴڮ܅࣍ͷΑ͏ͳඪΛୡ͍ͨ͠Ͱ͢ɻ Λຬͨ͢ҙͷͷ ʹରͯ͠ɺ͕ॻ͔Εͨന͍Ϙʔϧͷํ͕͕ॻ͔Εͨ ന͍ϘʔϧΑΓࠨʹ͋Δ Λຬͨ͢ҙͷͷ ʹରͯ͠ɺ͕ॻ͔Εͨࠇ͍Ϙʔϧͷํ͕͕ॻ͔Εͨ ࠇ͍ϘʔϧΑΓࠨʹ͋Δ ඪΛୡ͢ΔͨΊʹߴڮ܅࣍ͷΑ͏ͳૢ࡞͕Ͱ͖·͢ɻ ྡΓ߹͏ೋͭͷϘʔϧΛεϫοϓ͢ΔඪΛୡ͢ΔͨΊʹඞཁͳૢ࡞ճͷ࠷খΛٻΊͯͩ͘ ͍͞ɻ ·ͨ ͳΒ 1 N 1 2N i (1 ≤ i ≤ 2N) ai ci ci = W ci = B 1 ≤ i < j ≤ N (i, j) i j 1 ≤ i < j ≤ N (i, j) i j 1 ≤ N ≤ 2000 1 ≤ ai ≤ N ci = W ci = B i ≠ j (ai , ci ) ≠ (aj , cj ) -
ൈڈ%1ίʔυ "3$&4PSUFEBOE4PSUFEIUUQTCFUBBUDPEFSKQDPOUFTUTBSDUBTLTBSD@D EQ<C><X> ࠇͷC നͷX·Ͱઌ಄ʹ࣋ͬͯ དྷ͍ͯΔ߹ͷૢ࡞ճͷ࠷খ w ํͱղྫ w
നͱࠇʹ͍͔ͭͯΒॱ൪ʹ ઌ಄ʹऔͬͯ͘Δૢ࡞Λߦͬ ͍ͯ͘ w ॏཁߟʮനͷX·Ͱɺࠇͷ C·Ͱઌ಄ʹऔ͖ͬͯͨ߹ ʹΔϘʔϧͷྻҰҙʹఆ ·Δʯ ͭ·ΓɺനͱࠇͷͲͷ·Ͱ ઌ಄ʹऔ͖͔ͬͯͨͰঢ়ଶΛ వΊΔ͜ͱ͕Ͱ͖ͦ͏ w ͦΕΛLFZͱͨ͠%1ͩͱ 0 /? Ͱɺߋ৽0 Ͱߦ ͚ͦ͏ IUUQTBUDPEFSKQDPOUFTUTBSDTVCNJTTJPOT JOUG JOUC JOUX \ JG C /BOEX / SFUVSO JG EQ<C><X> SFUVSOEQ<C><X> JOUSFT JOG JG C / \ JOUDOU JG JEY#<C > DOU TN#<C ><JEY#<C >> JG JEY#<C > DOU TN8<X ><JEY#<C >> DINJO SFT G C X DOU ^ JG X / \ JOUDOU JG JEY8<X > DOU TN#<C ><JEY8<X >> JG JEY8<X > DOU TN8<X ><JEY8<X >> DINJO SFT G C X DOU ^ SFUVSOEQ<C><X> SFT ^ ·Μ·Μ·Μ w Լهࢀর -
͢%1 w EQ<J >͔ΒEQ<J>Λ෮ݩ͢Δ͜ͱͰΛղ͘%1 "3$%จͷଟ͍ߴڮళIUUQTBSDDPOUFTUBUDPEFSKQUBTLTBSD@ ߴڮళͰ छྨͷ͕ചΒΕ͍ͯ·͢ɻʮͲͷछྨͷ͕͍ͭ͋͘Δ͔ʯͷใ͕༩͑ ΒΕΔͷͰɺʮ߹ܭ
ݸͷΛબͿํ๏ʯͷΛٻΊͯԼ͍͞ɻͨͩ͠ɺಉ͡छྨͷ۠ ผ͠ͳ͍͜ͱͱ͠·͢ɻ ͍ɺ͜Εগ͠؆୯ա͗ΔͷͰɺͪΐͬͱͨ͠จՃ͠Α͏ɻ ͔ΒͳΔ ݸͷ จΛ༻ҙͨ͠ͷͰɺͦΕͧΕʹ͍ͭͯʮ छྨͷΛͪΐ͏Ͳ ݸબͳ͚ΕͳΒͳ͍ͱ ͖ɺ߹ܭ ݸͷΛબͿํ๏ʯͷΛٻΊͯԼ͍͞ɻ N M k, x Q ki xi M 1 ≦ N ≦ 2000 1 ≦ M ≦ 2000 1 ≦ Q ≦ 500,000 1 ≦ ai ≦ M 1 ≦ ki ≦ N 1 ≦ xi ≦ aki EQ<J><K> J൪·ͰͰKݸͷΛબͿ௨Γ EQ<J><K> /छྨͷબͼํ͔Βͭר͖͢༻ͷྻ w ํͱղྫ w ॏෳ߹ͤΛٻΊΔ w ͷݟ͍ͯ͘ॱ൪Θͳ͍ ʹணͯ͠ ࣜมܗ w มܗʹΑΓʮ/छྨͷΛબ Ϳબͼํ͔Βͭר͖͢ʯ͜ ͱ͕Ͱ͖Δ LNKQTCMPHπόαͷඋ DUZMTQSPCMFNTPMWJOHNBZPLP`TEJBSZ w Լهࢀর -
͢%1ίʔυ "3$%จͷଟ͍ߴڮళIUUQTBSDDPOUFTUBUDPEFSKQUBTLTBSD@ '03 J / EQ<J><> ͜͜௨ৗ௨ΓEQΛຒΊΔ
'03 J / '03 K . \ EQ<J ><K > EQ<J ><K> EQ<J><K > JG K"<J> EQ<J ><K > .0%EQ<J><K"<J>> EQ<J ><K > .0% ^ લ͔ΒஞҰܭࢉ͢ΔͷΛΊͯ ͢%1Ͱܭࢉ͢Δ '03 J / '03 K . \ EQ<J><K> EQ</><K> JG K EQ<J><K> .0%EQ</><K> JG K"<J> EQ<J><K> EQ<J><K"<J>> EQ<J><K> .0% ^ EQ<J><K> J൪·ͰͰKݸͷΛબͿ௨Γ EQ<J><K> /छྨͷબͼํ͔Βͭר͖͢༻ͷྻ -
࿈݁%1 w ࿈݁ؔʹجͮ͘%1 ྫEQ<J><K> Jྻ·ͰΛͬͨ࣌ʹ Jྻͷ Ϛεಉ࢜ͷ࿈͕݁ؔKͰ͋ΔΑ͏ͳృΓํ w
6OJPOpOEͰ࿈݁͢Δ 5ZQJDBM%1$POUFTU4ϚεIUUQTUEQDDPOUFTUBUDPEFSKQUBTLTUEQD@HSJE ͢͵͚܅ɺ ͷϚεΛനͱࠇʹృΓ͚Δ͜ͱʹͨ͠ɻ ࠨ্ͷϚεͱӈԼͷϚεࠇʹృΔɻ ࠨ্ͷϚε͔Βɺࠇ͍ϚεΛ্ԼࠨӈʹͨͲͬͯɺӈԼͷϚεʹߦ͘͜ͱ͕Ͱ͖Δɻ ݅Λຬͨ͢ృΓ͚ํͷݸΛNPE ͰٻΊΑɻ H × W 2 ≤ H ≤ 6 2 ≤ W ≤ 100 EQ<J><K> Jྻ·Ͱݟͨ࣌ɼJྻͷ֤ Ϛεͷ࿈͕݁ؔKͰ͋ΔΑ͏ͳృΓํ w ํͱղྫ w ͦΕͧΕͷϚεͷ࿈݁ؔΛͪͳ͕ Β ྻΛࠨ͔Βӈʹݟ͍ͯ͘%1Λߟ͑ Δ w ਐCJU ͰҎԼͷΑ͏ʹߟ͑Δ ന ࠇ͔ͭ ͱ࿈݁ d ࠇ͔ͭ ͱ࿈݁Ͱͳ͍ɽ ಉ͡൪߸͕ৼΒΕ͍ͯΕɼͦΕΒͷ Ϛε࿈݁Ͱ͋Δͱߟ͑Δɽ = 7H ͍͢όΧࢽLNKQTCMPHTJNF[J@UBOͷه GPS JOUJ J X J GPS JOUK K QX<I> K \࣍ঢ়ଶ JG DVS<K> DPOUJOVF GPS JOUL L I L \ VOJPO@pOEVG I WFDUPS JOU Q I VG࿈݁ؔΛٻΊΔ GPS JOUM M I M \Q࣍ঢ়ଶͷ֤ߦͷ࿈݁ؔΛอଘ JG L M \࣍ঢ়ଶͷMߦ͕ࠇ JG KQX<M> Q<M> ݱঢ়ଶ͕࿈݁ͳΒ࣍࿈݁ GPS JOUN M N I N \࿈݁ؔΛVGͰ·ͱΊΔ JG L N \ JG N M VGVOJUF M N MͱM ߦ͕ࠇͳΒ࿈݁ FMTF\ JOUV KQX<M> ݱঢ়ଶͷMߦ Nߦͷ࿈݁ؔ W KQX<N> JG V V W VGVOJUF M N ݱঢ়ଶͰಉ͡࿈݁ؔ ^ʜͳΒ࣍ঢ়ଶͰ࿈݁ -
5SJF%1 w 5SJFͷ্Ͱܭࢉ͢Δ%1 "U$PEFSఱԼҰϓϩάϥϚʔίϯςετຊઓ$ͨΜͨ͘͝͞Μ IUUQTUFOLBpOBMPQFODPOUFTUBUDPEFSKQUBTLTUFOLB@@pOBM@D จࣈྻ4ͱɺཁૉ.ͷ୯ޠͷू߹1 \1@ 1@ ʜ
1@.^͕༩͑ΒΕ·͢ɻ୯ޠ1@Jɺ ͷॏΈ8@JΛ͍࣋ͬͯ·͢ɻ จࣈྻ4͔Βɺ1ʹؚ·ΕΔ୯ޠΛॏͳΓ߹Θͳ͍Α͏ʹऔΓग़͢͜ͱΛߟ͑·͢ɻ୯ޠͷॏΈͷ ૯͕࠷େΛͱΔΑ͏ʹऔΓग़͢ͱ͖ɺͦͷ࠷େ͍ͭ͘Ͱ͠ΐ͏͔ʁ ͳ͓ɺಉ͡୯ޠΛෳճऔΓग़ͨ͠߹ɺͦΕΒͷ୯ޠผʑʹ͑Δ͜ͱͱ͠·͢ɻ ≤c4c≤ ≤.≤ ≤c1Jc≤ ≤8J≤ 4 1Jӳখจࣈ͔ΒͳΔจࣈྻ w ํͱղྫ w ༩͑ΒΕΔ୯ޠͯ͢Ͱ5SJFΛͭ͘ Δ w จࣈྻ4ͷJจࣈT@JΛࠨͱ͢Δ4ͷ ෦จࣈྻͰ ୯ޠͱશҰக͢Δ ͷΛߟ͑Δ w ୯ޠͷ͕͞ߴʑͳͷͰ ෦จ ࣈྻͷબͼํ௨Γ͔͠ͳ͍ IPHFDPEFS w Լهࢀর EQ<J> Jจࣈ·Ͱݟͨ࣌ͷ࠷େ 3&1 J / \ 5SJF DVS USJF SFQR L \ JG J L / DPOUJOVF DIBSUBSHFU T<J L> 5SJF OFYU JG DVS 5SJF \ OFYU USJFpOE UBSHFU DVS DVS OFYU DINBY EQ<J L> DVS 5SJF EQ<J> DVS TDPSFEQ<J> ^FMTFDINBY EQ<J L> EQ<J> ^^ -
શํҐ%1 w ҙͷΛࠜͱͨ͠ʹ͓͚Δ%1 "J[V0OMJOF+VEHF%JBNFUFSPGB5SFFIUUQKVEHFVBJ[VBDKQPOMJOFKVEHFEFTDSJQUJPOKTQ JE (3-@@"MBOH KQ ඇෛͷॏΈΛͭແͷTͷܘΛٻΊ͍ͯͩ͘͞ɻͷ࠷ԕؒͷڑΛͷܘͱ͍͍·͢ɻ
1 ≤ n ≤ 100,000 0 ≤ wi ≤ 1,000 FJͷه w Լهࢀর w ํͱղྫ w దͳΛࠜͱͯ͠ Λ༗ ͱΈͳ͢ w ҙͷΛࠜͱͨ͠෦ʹ ͍ͭͯ ෦ͷࠜʹlඞཁͳ ใ%@JΛٻΊΔ %@J*൪ͷ͔Βͦͷͷ෦ Ͱ࠷ԕ͍ ༿ ·Ͱͷڑ w ҙͷΛࠜͱͨ͠શମʹ ͍ͭͯ ͷղΛٻΊΔ WPJEEGT JOUJEY JOUQBS \GPS FEHFFH<JEY> \ JG FUP QBS DPOUJOVF EGT FUP JEY EJTU<JEY> NBY EJTU<JEY> EJTU<FUP> FDPTU ^^ JOUEGT JOUJEY JOUE@QBS JOUQBS \ WFDUPS QBJS JOU JOU E@DIJME E@DIJMEFNQMBDF@CBDL GPS FEHFFH<JEY> \ JG FUP QBS E@DIJMEFNQMBDF@CBDL E@QBS FDPTU FUP FMTFE@DIJMEFNQMBDF@CBDL FDPTU EJTU<FUP> FUP ^ TPSU E@DIJMESCFHJO E@DIJMESFOE JOUSFU E@DIJME<>pSTU E@DIJME<>pSTU GPS FEHFFH<JEY> \ JG FUP QBS DPOUJOVF SFU NBY SFU EGT FUP E@DIJME<E@DIJME<>TFDPOE FUP>pSTU JEY ^ SFUVSO SFU ^ EGT / DPVU EGT / FOEM -
ΠϯϥΠϯ%1 w ߋ৽Օॴ͕গͳ͍߹ʹ ଞͷྖҬΛ͍·Θ͢͜ͱͰߴԽ͢Δ%1 w 4FHUSFF#*5Λ༻͍ͯߦ͏ &EVDBUJPOBM%1$POUFTU%1·ͱΊίϯςετ2'MPXFSTIUUQTBUDPEFSKQDPOUFTUTEQUBTLTEQ@R
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