From 6542a90f82f4d92a58870e7711dab223111fff99 Mon Sep 17 00:00:00 2001 From: arthur Date: Wed, 3 Dec 2025 20:44:14 -0300 Subject: [PATCH] feat: new statement files. Generator generated. New TLE solution --- longest-common-subsequence/input/1 | 1 + longest-common-subsequence/input/10 | 1 + longest-common-subsequence/input/100 | 1 + longest-common-subsequence/input/101 | 1 + longest-common-subsequence/input/102 | 1 + longest-common-subsequence/input/103 | 1 + longest-common-subsequence/input/104 | 1 + longest-common-subsequence/input/11 | 1 + longest-common-subsequence/input/12 | 1 + longest-common-subsequence/input/13 | 1 + longest-common-subsequence/input/14 | 1 + longest-common-subsequence/input/15 | 1 + longest-common-subsequence/input/16 | 1 + longest-common-subsequence/input/17 | 1 + longest-common-subsequence/input/18 | 1 + longest-common-subsequence/input/19 | 1 + longest-common-subsequence/input/2 | 1 + longest-common-subsequence/input/20 | 1 + longest-common-subsequence/input/21 | 1 + longest-common-subsequence/input/22 | 1 + longest-common-subsequence/input/23 | 1 + longest-common-subsequence/input/24 | 1 + longest-common-subsequence/input/25 | 1 + longest-common-subsequence/input/26 | 1 + longest-common-subsequence/input/27 | 1 + longest-common-subsequence/input/28 | 1 + longest-common-subsequence/input/29 | 1 + longest-common-subsequence/input/3 | 1 + longest-common-subsequence/input/30 | 1 + longest-common-subsequence/input/31 | 1 + longest-common-subsequence/input/32 | 1 + longest-common-subsequence/input/33 | 1 + longest-common-subsequence/input/34 | 1 + longest-common-subsequence/input/35 | 1 + longest-common-subsequence/input/36 | 1 + longest-common-subsequence/input/37 | 1 + longest-common-subsequence/input/38 | 1 + longest-common-subsequence/input/39 | 1 + longest-common-subsequence/input/4 | 1 + longest-common-subsequence/input/40 | 1 + longest-common-subsequence/input/41 | 1 + longest-common-subsequence/input/42 | 1 + longest-common-subsequence/input/43 | 1 + longest-common-subsequence/input/44 | 1 + longest-common-subsequence/input/45 | 1 + longest-common-subsequence/input/46 | 1 + longest-common-subsequence/input/47 | 1 + longest-common-subsequence/input/48 | 1 + longest-common-subsequence/input/49 | 1 + longest-common-subsequence/input/5 | 1 + longest-common-subsequence/input/50 | 1 + longest-common-subsequence/input/51 | 1 + longest-common-subsequence/input/52 | 1 + longest-common-subsequence/input/53 | 1 + longest-common-subsequence/input/54 | 1 + longest-common-subsequence/input/55 | 1 + longest-common-subsequence/input/56 | 1 + longest-common-subsequence/input/57 | 1 + longest-common-subsequence/input/58 | 1 + longest-common-subsequence/input/59 | 1 + longest-common-subsequence/input/6 | 1 + longest-common-subsequence/input/60 | 1 + longest-common-subsequence/input/61 | 1 + longest-common-subsequence/input/62 | 1 + longest-common-subsequence/input/63 | 1 + longest-common-subsequence/input/64 | 1 + longest-common-subsequence/input/65 | 1 + longest-common-subsequence/input/66 | 1 + longest-common-subsequence/input/67 | 1 + longest-common-subsequence/input/68 | 1 + longest-common-subsequence/input/69 | 1 + longest-common-subsequence/input/7 | 1 + longest-common-subsequence/input/70 | 1 + longest-common-subsequence/input/71 | 1 + longest-common-subsequence/input/72 | 1 + longest-common-subsequence/input/73 | 1 + longest-common-subsequence/input/74 | 1 + longest-common-subsequence/input/75 | 1 + longest-common-subsequence/input/76 | 1 + longest-common-subsequence/input/77 | 1 + longest-common-subsequence/input/78 | 1 + longest-common-subsequence/input/79 | 1 + longest-common-subsequence/input/8 | 1 + longest-common-subsequence/input/80 | 1 + longest-common-subsequence/input/81 | 1 + longest-common-subsequence/input/82 | 1 + longest-common-subsequence/input/83 | 1 + longest-common-subsequence/input/84 | 1 + longest-common-subsequence/input/85 | 1 + longest-common-subsequence/input/86 | 1 + longest-common-subsequence/input/87 | 1 + longest-common-subsequence/input/88 | 1 + longest-common-subsequence/input/89 | 1 + longest-common-subsequence/input/9 | 1 + longest-common-subsequence/input/90 | 1 + longest-common-subsequence/input/91 | 1 + longest-common-subsequence/input/92 | 1 + longest-common-subsequence/input/93 | 1 + longest-common-subsequence/input/94 | 1 + longest-common-subsequence/input/95 | 1 + longest-common-subsequence/input/96 | 1 + longest-common-subsequence/input/97 | 1 + longest-common-subsequence/input/98 | 1 + longest-common-subsequence/input/99 | 1 + .../longest-common-subsequence-tutorial.pdf | Bin 0 -> 93319 bytes .../longest-common-subsequence-tutorial.tex | 67 ++++++++++++++++++ .../longest-common-subsequence.pdf | Bin 23830 -> 72026 bytes .../longest-common-subsequence.tex | 27 ++++++- longest-common-subsequence/problem.json | 2 +- longest-common-subsequence/src/TLE.cpp | 27 +++++++ longest-common-subsequence/src/ac.cpp | 1 + longest-common-subsequence/src/generator.cpp | 1 + longest-common-subsequence/src/validator.cpp | 8 ++- .../statement/description.tex | 3 + .../statement/input.tex | 2 + .../statement/notes.tex | 5 ++ .../statement/output.tex | 1 + .../statement/tutorial.tex | 54 ++++++++++++++ 118 files changed, 296 insertions(+), 6 deletions(-) create mode 100644 longest-common-subsequence/longest-common-subsequence-tutorial.pdf create mode 100644 longest-common-subsequence/longest-common-subsequence-tutorial.tex diff --git a/longest-common-subsequence/input/1 b/longest-common-subsequence/input/1 index 7b1cf19..849fadc 100644 --- a/longest-common-subsequence/input/1 +++ b/longest-common-subsequence/input/1 @@ -1 +1,2 @@ +5 3 abcde ace diff --git a/longest-common-subsequence/input/10 b/longest-common-subsequence/input/10 index e25f864..dff109c 100644 --- a/longest-common-subsequence/input/10 +++ b/longest-common-subsequence/input/10 @@ -1 +1,2 @@ +1 5 s tzfpn diff --git a/longest-common-subsequence/input/100 b/longest-common-subsequence/input/100 index 9c85d48..3b9875e 100644 --- a/longest-common-subsequence/input/100 +++ b/longest-common-subsequence/input/100 @@ -1 +1,2 @@ +259 820 vavjwmfgrtevacmqkxzbqlisadomincrcfltxuxivbzfuzcytqxfptithfyfwjzcirltwowbwiwseotxexyktdxscgnvrhsjyieqspbvwatvydpkkoicaoiphihhgcvsiofqrjampsxbyawvomgkipniryirvlhmdzgvwbhyvznyecyvlwnszgudgfqmrkpdhdaqxxnjlndwzodecefrkrakgcxoocywyccsvzmfjnisnakfjzxajbfdpjjaqjkxyve 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b/longest-common-subsequence/input/101 @@ -1 +1,2 @@ +517 119 bjyyzrcoqejddbtyrwbngzfgeliwbeijudqbobcvaucuqxmxnnurqmvmjrzjefwumewavzhjneolbgmcmfvlhfofkwryrgjamuduqxtifwttpnuoidsagvypmjdrxmrkuvcxjzvcfnnegiabbgienuxjcszotokihudxtvkeqbnpsswzsaxegdypvolzlknmxxpkvsfggcwxserkbbnfyzuxhabqrxfciskcoeuyvoyjrmxyttxyiagzbcfipcljtpaesropmikietlutuvwlhndriivjhsrmtzkxcbgrgejfipyotialnqyxwqattrpekaibzixwednqxqmnroweeiocwwcqlnkyehesqdxgjvagpgolspfxtrputvbmtpdqgznunlqcxjpengadhbctxrpqdxeqqwpesydalbquebjnuxqydaisbnoddvjdtiwojangfcumftwxcupfzqkkfkuazyyfrkfdpikxkvelyljywodspdgdpfqupydpqeewowtm ddtpjnewotptdzmuwohedbzbpejlrulozlrbrhfwubpgmiqmuopaxffhsntyrtsimuxuwlibhqnsfukgybkdrqxizoatulfefkjzezgyfysofrngunbajpu diff --git a/longest-common-subsequence/input/102 b/longest-common-subsequence/input/102 index e4256f7..44c48d8 100644 --- a/longest-common-subsequence/input/102 +++ b/longest-common-subsequence/input/102 @@ -1 +1,2 @@ +799 496 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nkkmnifgxpigarinognscdgelmuyoriihaygkhbcsamsqjrpoxsbnagwacouiluoplmawlumiikyhghurxhlxxyozofpfihwixpcfyllnwbgrbshyqogvqhduwbccccjlusuyxoflojluqlttujkbckqwjbtnhomjbykupmttcoknyrdxuabnfvodyojjgtimgpyutamfuzlxmtewrkzolvrhlgctsarztvgjvwthvazmqshtxnbevriuebzcsexujwpjxpezgjgquawdwuuoidxkudmfebjtfydlnonoziyetxyfdnizhadejsqtmiaxnwppzbewuweuqsjnoiwhevcmvrrpvo diff --git a/longest-common-subsequence/input/104 b/longest-common-subsequence/input/104 index 366fc73..9bb94ed 100644 --- a/longest-common-subsequence/input/104 +++ b/longest-common-subsequence/input/104 @@ -1 +1,2 @@ +1000 1000 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diff --git a/longest-common-subsequence/input/11 b/longest-common-subsequence/input/11 index 4a05c96..287a072 100644 --- a/longest-common-subsequence/input/11 +++ b/longest-common-subsequence/input/11 @@ -1 +1,2 @@ +2 2 ue cz diff --git a/longest-common-subsequence/input/12 b/longest-common-subsequence/input/12 index ff7b33c..4041e67 100644 --- a/longest-common-subsequence/input/12 +++ b/longest-common-subsequence/input/12 @@ -1 +1,2 @@ +5 5 civdq pnxnn diff --git a/longest-common-subsequence/input/13 b/longest-common-subsequence/input/13 index 71cde93..e15933e 100644 --- a/longest-common-subsequence/input/13 +++ b/longest-common-subsequence/input/13 @@ -1 +1,2 @@ +4 5 duhz quudh diff --git a/longest-common-subsequence/input/14 b/longest-common-subsequence/input/14 index a1c424d..5507957 100644 --- a/longest-common-subsequence/input/14 +++ b/longest-common-subsequence/input/14 @@ -1 +1,2 @@ +3 4 ncw jpcm diff --git a/longest-common-subsequence/input/15 b/longest-common-subsequence/input/15 index ce04f70..4c89dcc 100644 --- a/longest-common-subsequence/input/15 +++ b/longest-common-subsequence/input/15 @@ -1 +1,2 @@ +3 4 jmc bnjf diff --git a/longest-common-subsequence/input/16 b/longest-common-subsequence/input/16 index 65ddede..366030e 100644 --- a/longest-common-subsequence/input/16 +++ b/longest-common-subsequence/input/16 @@ -1 +1,2 @@ +4 3 xkgj trx diff --git a/longest-common-subsequence/input/17 b/longest-common-subsequence/input/17 index f546226..71f4256 100644 --- a/longest-common-subsequence/input/17 +++ b/longest-common-subsequence/input/17 @@ -1 +1,2 @@ +4 2 iqqu qh diff --git a/longest-common-subsequence/input/18 b/longest-common-subsequence/input/18 index fcd6c73..d0f6b9f 100644 --- a/longest-common-subsequence/input/18 +++ b/longest-common-subsequence/input/18 @@ -1 +1,2 @@ +4 3 xxrx fhk diff --git a/longest-common-subsequence/input/19 b/longest-common-subsequence/input/19 index f617b0f..3a696c5 100644 --- a/longest-common-subsequence/input/19 +++ b/longest-common-subsequence/input/19 @@ -1 +1,2 @@ +3 5 unf pcoil diff --git a/longest-common-subsequence/input/2 b/longest-common-subsequence/input/2 index a3be0a1..6068019 100644 --- a/longest-common-subsequence/input/2 +++ b/longest-common-subsequence/input/2 @@ -1 +1,2 @@ +3 3 abc abc diff --git a/longest-common-subsequence/input/20 b/longest-common-subsequence/input/20 index d44391c..e32ff8a 100644 --- a/longest-common-subsequence/input/20 +++ b/longest-common-subsequence/input/20 @@ -1 +1,2 @@ +1 4 k arve diff --git a/longest-common-subsequence/input/21 b/longest-common-subsequence/input/21 index e91e9c6..f553f14 100644 --- a/longest-common-subsequence/input/21 +++ b/longest-common-subsequence/input/21 @@ -1 +1,2 @@ +1 4 u stul diff --git a/longest-common-subsequence/input/22 b/longest-common-subsequence/input/22 index 0e60f7e..c0ada5a 100644 --- a/longest-common-subsequence/input/22 +++ b/longest-common-subsequence/input/22 @@ -1 +1,2 @@ +4 4 lwmf nyxr diff --git a/longest-common-subsequence/input/23 b/longest-common-subsequence/input/23 index 4f51a6e..f6b6332 100644 --- a/longest-common-subsequence/input/23 +++ b/longest-common-subsequence/input/23 @@ -1 +1,2 @@ +3 4 wzh ewcz diff --git a/longest-common-subsequence/input/24 b/longest-common-subsequence/input/24 index fda2bbd..97d69e9 100644 --- a/longest-common-subsequence/input/24 +++ b/longest-common-subsequence/input/24 @@ -1 +1,2 @@ +2 1 oz t diff --git a/longest-common-subsequence/input/25 b/longest-common-subsequence/input/25 index 81fbfdb..df953a3 100644 --- a/longest-common-subsequence/input/25 +++ b/longest-common-subsequence/input/25 @@ -1 +1,2 @@ +3 1 djw f diff --git a/longest-common-subsequence/input/26 b/longest-common-subsequence/input/26 index 060ef55..0caa810 100644 --- a/longest-common-subsequence/input/26 +++ b/longest-common-subsequence/input/26 @@ -1 +1,2 @@ +1 3 k jxk diff --git a/longest-common-subsequence/input/27 b/longest-common-subsequence/input/27 index fabbeaa..b33efd6 100644 --- a/longest-common-subsequence/input/27 +++ b/longest-common-subsequence/input/27 @@ -1 +1,2 @@ +1 5 m swkdk diff --git a/longest-common-subsequence/input/28 b/longest-common-subsequence/input/28 index a8fead3..9810dbc 100644 --- a/longest-common-subsequence/input/28 +++ b/longest-common-subsequence/input/28 @@ -1 +1,2 @@ +3 5 agw sufee diff --git a/longest-common-subsequence/input/29 b/longest-common-subsequence/input/29 index c666712..cf9dd6e 100644 --- a/longest-common-subsequence/input/29 +++ b/longest-common-subsequence/input/29 @@ -1 +1,2 @@ +2 4 nc cepi diff --git a/longest-common-subsequence/input/3 b/longest-common-subsequence/input/3 index f4dc35c..03386dc 100644 --- a/longest-common-subsequence/input/3 +++ b/longest-common-subsequence/input/3 @@ -1 +1,2 @@ +3 3 abc hhh diff --git a/longest-common-subsequence/input/30 b/longest-common-subsequence/input/30 index 8dfc578..9d3d86f 100644 --- a/longest-common-subsequence/input/30 +++ b/longest-common-subsequence/input/30 @@ -1 +1,2 @@ +1 1 z o diff --git a/longest-common-subsequence/input/31 b/longest-common-subsequence/input/31 index ffc57ee..785503a 100644 --- a/longest-common-subsequence/input/31 +++ b/longest-common-subsequence/input/31 @@ -1 +1,2 @@ +3 4 rzf wbqo diff --git a/longest-common-subsequence/input/32 b/longest-common-subsequence/input/32 index e7f5e98..dac667a 100644 --- a/longest-common-subsequence/input/32 +++ b/longest-common-subsequence/input/32 @@ -1 +1,2 @@ +5 2 drmkl za diff --git a/longest-common-subsequence/input/33 b/longest-common-subsequence/input/33 index 9fc85d2..41c15e4 100644 --- a/longest-common-subsequence/input/33 +++ b/longest-common-subsequence/input/33 @@ -1 +1,2 @@ +2 5 rq lalxz diff --git a/longest-common-subsequence/input/34 b/longest-common-subsequence/input/34 index 41eeb78..e6c97ba 100644 --- a/longest-common-subsequence/input/34 +++ b/longest-common-subsequence/input/34 @@ -1 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a/longest-common-subsequence/input/39 b/longest-common-subsequence/input/39 index 1c40de4..5f65717 100644 --- a/longest-common-subsequence/input/39 +++ b/longest-common-subsequence/input/39 @@ -1 +1,2 @@ +13 10 ndfqjamzkptts dlrkjoyrpx diff --git a/longest-common-subsequence/input/4 b/longest-common-subsequence/input/4 index fe078b7..4bedc3e 100644 --- a/longest-common-subsequence/input/4 +++ b/longest-common-subsequence/input/4 @@ -1 +1,2 @@ +2 5 bm ydiad diff --git a/longest-common-subsequence/input/40 b/longest-common-subsequence/input/40 index e9e95f9..9c32cb4 100644 --- a/longest-common-subsequence/input/40 +++ b/longest-common-subsequence/input/40 @@ -1 +1,2 @@ +6 17 ceahgi vsjoadmkfnkswrawk diff --git a/longest-common-subsequence/input/41 b/longest-common-subsequence/input/41 index f4da613..3445cdd 100644 --- a/longest-common-subsequence/input/41 +++ b/longest-common-subsequence/input/41 @@ -1 +1,2 @@ +11 3 cmcciabzbrs zjq diff --git a/longest-common-subsequence/input/42 b/longest-common-subsequence/input/42 index c5f2209..48d4369 100644 --- a/longest-common-subsequence/input/42 +++ b/longest-common-subsequence/input/42 @@ -1 +1,2 @@ +20 11 iqydptpkcsdgcqjshznd nnrywfrnrgh diff --git a/longest-common-subsequence/input/43 b/longest-common-subsequence/input/43 index 2efbaea..2700aed 100644 --- a/longest-common-subsequence/input/43 +++ b/longest-common-subsequence/input/43 @@ -1 +1,2 @@ +19 5 qrxpxnlkcwdpdkswvtx wwyoo diff --git a/longest-common-subsequence/input/44 b/longest-common-subsequence/input/44 index 2a14a2c..9ae4bb3 100644 --- a/longest-common-subsequence/input/44 +++ b/longest-common-subsequence/input/44 @@ -1 +1,2 @@ +4 20 nexe xouyfjhnwpqfxexzxfol diff --git a/longest-common-subsequence/input/45 b/longest-common-subsequence/input/45 index 13b5165..3f5cbdb 100644 --- a/longest-common-subsequence/input/45 +++ b/longest-common-subsequence/input/45 @@ -1 +1,2 @@ +9 11 blpncetyh xkbosccskxb diff --git a/longest-common-subsequence/input/46 b/longest-common-subsequence/input/46 index 2927592..5a9f9bc 100644 --- a/longest-common-subsequence/input/46 +++ b/longest-common-subsequence/input/46 @@ -1 +1,2 @@ +20 20 rosavnpxzoeoyyghbbqk utpwmhqwkvirmlfcpidl diff --git a/longest-common-subsequence/input/47 b/longest-common-subsequence/input/47 index ec3c008..7709c7c 100644 --- a/longest-common-subsequence/input/47 +++ b/longest-common-subsequence/input/47 @@ -1 +1,2 @@ +10 18 kzmnttaifc nwezesomkxhaiafmvk diff --git a/longest-common-subsequence/input/48 b/longest-common-subsequence/input/48 index 175eea9..8ff5a06 100644 --- a/longest-common-subsequence/input/48 +++ b/longest-common-subsequence/input/48 @@ -1 +1,2 @@ +8 12 isyrbtul xbcjmornquma diff --git a/longest-common-subsequence/input/49 b/longest-common-subsequence/input/49 index 3807e4e..dbc8134 100644 --- a/longest-common-subsequence/input/49 +++ b/longest-common-subsequence/input/49 @@ -1 +1,2 @@ +14 8 rhieqggrivouwf ghdfallc diff --git a/longest-common-subsequence/input/5 b/longest-common-subsequence/input/5 index 427d55b..f8c8500 100644 --- a/longest-common-subsequence/input/5 +++ b/longest-common-subsequence/input/5 @@ -1 +1,2 @@ +3 3 oue byf diff --git a/longest-common-subsequence/input/50 b/longest-common-subsequence/input/50 index 5b3bfc5..cd43445 100644 --- a/longest-common-subsequence/input/50 +++ b/longest-common-subsequence/input/50 @@ -1 +1,2 @@ +3 14 ngi mvprzaeapqmilw diff --git a/longest-common-subsequence/input/51 b/longest-common-subsequence/input/51 index 3e25976..c6ce819 100644 --- a/longest-common-subsequence/input/51 +++ b/longest-common-subsequence/input/51 @@ -1 +1,2 @@ +3 1 gog i diff --git a/longest-common-subsequence/input/52 b/longest-common-subsequence/input/52 index bc8c496..c36ce75 100644 --- a/longest-common-subsequence/input/52 +++ b/longest-common-subsequence/input/52 @@ -1 +1,2 @@ +1 12 s qxzjztvcgvzl diff --git a/longest-common-subsequence/input/53 b/longest-common-subsequence/input/53 index 55796e4..9d73f59 100644 --- a/longest-common-subsequence/input/53 +++ b/longest-common-subsequence/input/53 @@ -1 +1,2 @@ +14 9 wnezbwxmpobmrv kwvvwquud diff --git a/longest-common-subsequence/input/54 b/longest-common-subsequence/input/54 index ab3374b..5c7ae8e 100644 --- a/longest-common-subsequence/input/54 +++ b/longest-common-subsequence/input/54 @@ -1 +1,2 @@ +83 149 hmmzheqxiwhfuxavlwftgyxxhpygspvwdnvmgnwnpkdrnznzvrkmjjmwisyrdkodoafwsjfuvmeewpdwpiy hoxebjibxphiefgtsawcdivtltrshjqnkkmdtjgscnozmojnhigippjemzzzbcvoyplxenffmfdzdiojuodgbulvivhtbhayeeebiactyaovqbivudydgxwsgmhlracaayipsojleqhpygshcvxww diff --git a/longest-common-subsequence/input/55 b/longest-common-subsequence/input/55 index 49d6486..8bf1042 100644 --- a/longest-common-subsequence/input/55 +++ b/longest-common-subsequence/input/55 @@ -1 +1,2 @@ +803 953 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a/longest-common-subsequence/input/56 b/longest-common-subsequence/input/56 index c36a04b..178bee6 100644 --- a/longest-common-subsequence/input/56 +++ b/longest-common-subsequence/input/56 @@ -1 +1,2 @@ +623 892 mqgxhcfcmrakgrojgxizwxascasinzxcdxewculofitylhxhzxmjdlisupdylshahcipzoptullyisxsczjqawagnngcggiwkhnarjisvcdpzggtlludvcsjxeubfgyjdimtpftgytkfntylnflbfutsyovxmligoaiqzlxwfrsljucdxssamgcifdljashcjjwhlkqnipxzjglptedzxmthvzgruhhckufdecakrrzoqfqcduzhjsdaskqglqlnnbkuuytqthfcjraawtbccurpzpumcsmqhebhqrfigamblzxycvevluzcuqxvrjgekyaofvrvcohldavkjiykhccmnjpdhkkimpsmvcyqxlgwlbhldkjlzwthkfnizebvlgozrnwwvxdckgnckqtkqidhoociwrpkxesldtgxtaurglbgmpyqlpzbnsbzhanbiypuziqurwrbrbxmhbrdpdrerbobajuldlwrpkrtfblspnkxyitiohlpgjlcktlaqsalnoiphbtuyrrghseznkmzukaiihatzttiylactadtninwcsufbynjkxaajnagfevscfuhoajuqmisgbbsxqbylmraszacpliuglqyuzbnizx 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a/longest-common-subsequence/input/63 b/longest-common-subsequence/input/63 index 91a05d3..fb3a4e8 100644 --- a/longest-common-subsequence/input/63 +++ b/longest-common-subsequence/input/63 @@ -1 +1,2 @@ +264 286 idgdledutxclalkbnjnwmgsbotnpgrvzqrgjkwpkadhcsccxycukreswgwzyjorwcbokbmkpftucrejbrlpnqhmccqjnyvegkvpeatdzwcwabgkzekkdyfgubjdfdjnilummwxgvhhcfgrzinrodtidtvluobawqmxeamumkscvlhmivfubjcmblwkoxskirgqgnilltztieaipfrrshtqlhxxibaqmqhmbkaimrkzvgewnjpevfqgywzppbieigzkkmppls ofxblcyjvesuhovqvuygolnpqfpyrwyiubiwqjdjhvjnttzfnjimryvpyhoflqtfrrtldujwpinxmjmcmwzmdluubvwspacmhcwandvpqyiqtozipddjlatrprdnygmbaynlfzmbmtrytsqsxsuejlgooqeyucqvjfsmhxmhxbcbdqskznxnnphsoqjfmaqklcmckfgieszpirgotzgvlhinlogmtfnrqnxobylzddfyfpytcdezitduqhozqxqxxtxgpzkkbxmkuhdomjnrftojtrfyup diff --git a/longest-common-subsequence/input/64 b/longest-common-subsequence/input/64 index a79c227..74e19ed 100644 --- a/longest-common-subsequence/input/64 +++ b/longest-common-subsequence/input/64 @@ -1 +1,2 @@ +443 600 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b/longest-common-subsequence/input/72 index 3f38142..33985a0 100644 --- a/longest-common-subsequence/input/72 +++ b/longest-common-subsequence/input/72 @@ -1 +1,2 @@ +431 337 mvccqcyjgoohkqxxtvxzmvfjjxyjwlbqmeqwwlienzkdbhswgfbkhfygltsucdijozwaiewpixapyazfztksjeoqjugjfhdbqzuezbuajfvvffkwprroyivfoocvslejffgxuiofisenroxoeixmdbzonmreikpflciwsbafrdqfvdfojgoziiibqhwwsvhnzmptgirqqulkgmyzrfekzqqujmdumxkudsgexisupedisgmdgebvlvrpyfrbrqjknrxyzfpwmsxjxismgdyjzarvzldolzkvbuwxxlamrbwsciocflkfktpylozedwtpvklbxsywciseurivfkqrfxrsvolbrgfltdykhcchurrjjxhmcrgyqgrqhpyyiyhhpaliuoipytxipngxkzkddbuukffgwybfthhzgwwskbcctjo ksjzxsvrolxlfevugxnqmjpagyamsixzamlgstqdjoeqswmrdkaybkzxyjdtqdrhwqkwvdeyjwqhqhcedewwubpbufomialzbuoiretrdhuujwmoqswwjynejckhwreyjvojcdflrezwydwtxpbvkqmravdtwuorzkibosotwskqoewstbkhmdupptgipsisohbdkrwxdziynoxiutgypvbmxrdshlagmbbfljtocyzlzpznsnukflfovwzxujusylfyvwjbfgnzxxesukrtolqopcztfmpjlzkwoegjtvjxtukgzxhmalimjqkulwzqbcfiruqrhuepxtgol diff --git a/longest-common-subsequence/input/73 b/longest-common-subsequence/input/73 index ccdc575..cf474bf 100644 --- a/longest-common-subsequence/input/73 +++ b/longest-common-subsequence/input/73 @@ -1 +1,2 @@ +583 340 zczbczthkytvcpblwurnxinegezvyenihuvoxzvuuhpxljzhsyjwwkqvdfbwfuezchmtignisvtlqhgcpcazpcsgwpiifhgkqoblhhakugnymxtnrjehyxordmmxrowhpvsqfafvwqktopvmnrznroflnpbknjxfxgikjlpbjhifwkdjcamfnzjuumcuzkkmjzlmoapwgiemrjcsclxujryyhrsmkemofrozcvvwlhahdpuzlolxdjpvawzpvfoexqpvhmiteycflhucwljdqnrnhvqjqjbjeyrzuvedqeohmtefdhzhhgtiflpxtcqpjclewgygplzwiadfgumieroepsneydwlmivfblcqgohwhshmmcdelgauzhwlbfmpvhgviwzetvzwauhyqgeqfmdkfitdsfneqzdtidmmskwzixaqyofvpaoeetamubwyokgdygukwbsmhbxbmmuxjluzcrbduomseetuhggxosheourrluzeyzageiizmxjdqvjxukvfcipfywlntlerokolmkxjiuulwmxdaamguelyfzofchjsjdhfgdkuconchsvjanc 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--git a/longest-common-subsequence/input/75 b/longest-common-subsequence/input/75 index 184cbd5..4f60063 100644 --- a/longest-common-subsequence/input/75 +++ b/longest-common-subsequence/input/75 @@ -1 +1,2 @@ +395 100 mjkaxwzybtbcvarxfismnsogdgjnsriffenzybnmhfbrgtytvthknhrmfymjgekymfgacytrcqrlqwxrzrzrphesafjuhcmaimiviafcueyudnqbetezjidpqbvwlllufgkqmmceoagaystqbpmhewwlsvqrvajxqhxobzpstexllewonfxhrjelszndpncaeprygrxjxhefggxdsdzgkrembblqwybjqsdmlpqnccfvmoohhjmohpopmtrncmzngwxwewuxtkwohsonitgfstroqnebyakduapwxnnjdwlrewmamhutnumrmsbalcrblivsatxmxqsytnolmoiionfetxjmlexxsgadnycdvjpysgsqraixdduwyqlpuvsowgxrwldckbb qslrfrjjmjueclkjkoaacserycqwyegseezihuzgqtczenndmjvckbybqgbwmujmeekyenhuulzewaconrsqticsgzalrfwdtvdb diff --git a/longest-common-subsequence/input/76 b/longest-common-subsequence/input/76 index dbab2f1..270ec3a 100644 --- a/longest-common-subsequence/input/76 +++ b/longest-common-subsequence/input/76 @@ -1 +1,2 @@ +700 251 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a/longest-common-subsequence/input/77 b/longest-common-subsequence/input/77 index 44b304a..f564266 100644 --- a/longest-common-subsequence/input/77 +++ b/longest-common-subsequence/input/77 @@ -1 +1,2 @@ +289 711 dktacthnqxfkuqhghwbytbozlwktvmrmnpyjwwxkwgtowlcvnkylsdhziacqvsaifkichyfsgbkragwpkmddtnbtnxghfieqwapaddxujvawiuzgppobwrbzwvecbspsgbeddicqlwkigyzcthysqxkdqqivzffwdznnbwimsjledfdaoetvkrzxwrxhuchbyuldkkaqpmlzlxlqibtjavvbwwlalaezyvarzfcagfncjaukiantbhhyulzrlpoxhdkqejvpnjtqwknuqlnnynoovtadkzhop 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a/longest-common-subsequence/input/82 b/longest-common-subsequence/input/82 index 3d256b3..1bf2fa4 100644 --- a/longest-common-subsequence/input/82 +++ b/longest-common-subsequence/input/82 @@ -1 +1,2 @@ +430 919 cwjxnmsnisxrjrwxrsqhrafednqiziimxkqanchosdchnespbtrgbvyobflzacfaqugltbbaeertmtxrzsucowvwhcxxdolcenliuoczikvhxgazfwduxqgedrhkttmjtlomfhmoxmrxeanffeosdxhclhuzhfjgdvugnwyagutzmzctbqlvqdpipagteqjnyieuhnuabjxkdramnagctlbgrqhhxqgiuzwlgkkyreudsypfndrllmdhksjpobzbdbkapanixjpnjpsrdvvbdivawegyncyqlkahddssjlysxwgmjgxjncluscmbpdxsxfqpknzfkuzjhajuvqnlnuslqlyiyqblycerjgohootboanzfqdymnzdbtvmqjhcgbrtkaweaansjhotsawfrnoszpvosqfvuuswuyhturvnve 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b/longest-common-subsequence/input/83 index f0f1bee..684e821 100644 --- a/longest-common-subsequence/input/83 +++ b/longest-common-subsequence/input/83 @@ -1 +1,2 @@ +160 529 bncvpbunimbeyupwgpthtvfdlcftiytkqcfalemiauywussylgjjwrsdrxxwasbfdhekbfzakezeonjhtfnlhiuazqspezyoahzqlgtiuthidxohbeszoptnttdhmzzehulzfuhvtemafaasseidfiqzfxstiwss fqjyziwuhwlmuhqxhtgkgmozoqkmwessciiiyoewijgipxtndifkuxvkndxsctfpjneurbgffqyxwkzwhpubyybkxgbhjfujyrvetxilzdgpllccpopqtynpbkeworxablzmokxtejewpfpcbqeumksnvxkjrinfzzmsarxpygfixwfqhktjfjfkkbrztkmmrnoyyktzmezrncbstnwacinynunvfanplqddrrdnewuamyscfvxystpetjkynrcbpszdibsgducvfaikkyspikslfjgfbwgheecdvhiuckjjccrnyxrggcnbelauqvghsooaaaelbbctjoavreldouhzfngmlxteyeidgbbfathqvosyalihspogldlysvjgtgrfiskexmbqwgskwjjiuotsjvgryxsxuuxairslzlpgrquxvqciazcyblbkkavaszzpcliguqdurwnbhmlabyqtuivzoptpoifvictburxsogakgxrimsthwprvujkdfdodscewvadydskfd diff --git a/longest-common-subsequence/input/84 b/longest-common-subsequence/input/84 index 1258005..0c4499e 100644 --- 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a/longest-common-subsequence/input/88 b/longest-common-subsequence/input/88 index 2d81adc..14c17f1 100644 --- a/longest-common-subsequence/input/88 +++ b/longest-common-subsequence/input/88 @@ -1 +1,2 @@ +767 778 ghyqgkajjxbfzxpbapklybdqztpwioubezkyzkjqlvxfbwefyctouieagynuzwfimmjcivfqnmfczzofhahmagjkxwwzixulvisumwfqzgrmrczotrgzkhkzcwzcmchfuahuczrrymjwyfselxmfeywdygtxfmgdppxtmxdefewuivwvafcrskiypyscizklcrovowlfnokconlpxmnkwomjdjeuvoclizcuktcsyrrnzlyxzalqeofahsbdmfmyjwqjygifsrurzriqdyceoxrxjhmvbgcqukdytlllikjoqkrhiyrbcvvogvimqxahjxxxgdzexqyxiglgrtqqmegackvtxogvtkybnsnydlzuhlnqxhzhvhzscnreoecawkpknjhcsnofnsegrjfohvlruqfvhdrsftuwrcrhzwdxirqqumlpxignhyrpazmrvsoqveobbrualpqjpyhrfdiloyrooyznbzpouchxzuvcvezmrbrucxtwikkibxaiidlkicperxyjqzpdyylumlcyoarwsowihkbyfjhfcopkfdotfacajqhipicqvigbdlfbkqmlcveviedfkzpqehfdagrmrihqrvqphyneilacalxezkaabyfnknntodzrucperedfcgvqpaacryxkpcfrytxpnfgghamvssaiherhudytmzttoenkzgtqlftcfkluahhocrllxaaxqufawzilcfxksdhuutotrtwssioxcpblilturghghcmehla 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b/longest-common-subsequence/input/9 index 342d2f6..a17816b 100644 --- a/longest-common-subsequence/input/9 +++ b/longest-common-subsequence/input/9 @@ -1 +1,2 @@ +3 5 tsz flegs diff --git a/longest-common-subsequence/input/90 b/longest-common-subsequence/input/90 index f735460..0a82d56 100644 --- a/longest-common-subsequence/input/90 +++ b/longest-common-subsequence/input/90 @@ -1 +1,2 @@ +524 129 rdrmabxteyjbmjnxpbmvjphyblphxvileqhnadtdhmpamqqwvfflstmljtwsruqdnpsdqrtkmbzbmampqahegtavgqnproocuprelwrsyvahcgwbndtjmazuviacogostuzcekjzpojdzncsnirkbhvqmgkthyvhtcpcnohvnhhfggkjtrzjbvkryikrkvsyuvknmatsxoeveqajowuhhvgijbxzmzkogvksonlgtsxwbegnhrvdycmbbzacipebzvkfkjelllzfetdwbpwscvgezyloorytzmpulvyqvrpawqbhpmjmmtcayhrfxqtvfkvyusftmrwgpbwbltmrtcafjazftbpvufrkdpvqrdfaolgnxwbhcpclluwcltalkzbuupwylqsumybogjwtjidhxkjchbbyznwukbdcitctztlvuzyigmdakwfmpaytnbiyaoweeqqzqsyovqfrqtomeksntpyoqzxcvnnwrnjyukvpijehiwshlendifinnocqexcmweka 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b/longest-common-subsequence/input/93 index 40eb549..8cbd663 100644 --- a/longest-common-subsequence/input/93 +++ b/longest-common-subsequence/input/93 @@ -1 +1,2 @@ +981 960 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a/longest-common-subsequence/input/94 b/longest-common-subsequence/input/94 index d2b12f8..026e4a0 100644 --- a/longest-common-subsequence/input/94 +++ b/longest-common-subsequence/input/94 @@ -1 +1,2 @@ +729 991 ljmmjrfohptfkgxgxjgjblnwynprhctbfevszahlgwvgbxxopqouyrfvwofbmhaawvwtrbxassftxmnkagmhzwzforymjmhbqyapkkxkvxhjflblopdssipykdwstvrgjnutvgepbueewrzpjyuwrzffexmamynrdydwhofisslvzzjtruwntvsstxjqgxvtiirmsdxebpjosppfgzdqfjpfjtpfdlalmpkifxbezjgryuvcmhdpdvlwnhewrfjlhhqubuvpwdveqwfcgiofztnseqwddemnuhpvbkctmflwfblgmxjajhtnxllsbsgwblctcyvwrsmqcpvswhaxuykxjownmhmwqhzhybghkwcofyshwchhtadxejhqbuebbzureuvsdnkktgagqydkowwmvxthwwxusbybvcvzrifthstmtkgonaczuiawowgpivpxmpalembimpiajpqntcvtluwwicmrhbgpifqqsncrpfanifqniywtgqyarmzsbcjhjihnqhmscstaoqmwedeakxogengpczsnmztdzrdlauqkxyckinlvfxovowyibxxnyzqlfrhpilsukixboezrcyymidremlujwrihruuybzemfjnvpqduwrojoucgkwnbhjbjtevtibjtkpunhebpsonljaylemwmmrzewdyqdixnfcgcjpzmzukkguzunemitvztmkkajawlefmsskqsh 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--git a/longest-common-subsequence/input/95 b/longest-common-subsequence/input/95 index 45580fc..3b15dc1 100644 --- a/longest-common-subsequence/input/95 +++ b/longest-common-subsequence/input/95 @@ -1 +1,2 @@ +937 356 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+\author{} +\date{} +\begin{document} +\maketitle +\section{Solução do Problema} + +O problema da maior subsequência comum (\textit{Longest Common Subsequence}, LCS) pode ser resolvido por meio de \textit{programação dinâmica}. +A ideia central é decompor o problema em subproblemas menores, de forma que a solução ótima final seja construída a partir das soluções ótimas desses subproblemas. + +\subsection{Definição do Subproblema} + +Sejam duas strings \( s_1 \) e \( s_2 \), de tamanhos \( n \) e \( m \), respectivamente. +Definimos: + +\[ +dp[i][j] = \text{o comprimento da maior subsequência comum entre } s_1[1..i] \text{ e } s_2[1..j]. +\] + +Isto é, \( dp[i][j] \) representa a resposta do problema considerando apenas os primeiros \( i \) caracteres de \( s_1 \) e os primeiros \( j \) caracteres de \( s_2 \). + +\subsection{Função de Transição} + +Ao analisar um par de posições \( (i, j) \), temos duas possibilidades principais: + +\begin{itemize} + \item Se os caracteres atuais são iguais, isto é, \( s_1[i] = s_2[j] \), então podemos estender uma subsequência comum encontrada anteriormente: + \[ + dp[i][j] = dp[i-1][j-1] + 1. + \] + + \item Caso contrário, não é possível usar ambos os caracteres ao mesmo tempo, então devemos escolher o melhor resultado entre ignorar um dos dois: + \[ + dp[i][j] = \max(dp[i-1][j], \, dp[i][j-1]). + \] +\end{itemize} + +Assim, a função de transição pode ser resumida como: + +\[ +dp[i][j] = +\begin{cases} +dp[i-1][j-1] + 1, & \text{se } s_1[i] = s_2[j] \\ +\max(dp[i-1][j], \, dp[i][j-1]), & \text{caso contrário} +\end{cases} +\] + +\subsection{Casos Base} + +Os casos base seguem a lógica natural do problema: + +\[ +dp[0][j] = 0 \quad \forall j \ge 0 +\] +\[ +dp[i][0] = 0 \quad \forall i \ge 0 +\] + +Isto representa que, se uma das strings tiver tamanho zero, nenhuma subsequência comum pode ser formada.\end{document} diff --git a/longest-common-subsequence/longest-common-subsequence.pdf 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zXhLbn6d%~VXn4XA{1n?*m%IJtc7k$jn{|sQA{mK2zaG(9sF& zwz2f&$N2{)SA!aL@+-1E@@x_o>P9K7&2@H@@3N_Q%ko0y)AZ~E_OD+^e+?`jUOM_q zeD`>(wcz1bS`m8LG7_oB`47Clsg6D%eti{|rs#(Y? z!{YuYW(PK~Bh9`)*LkJ)BiJ2>ITP2&P9$r_hgY;!eOne28F=dJDzTTU;&YU4jkhIiFS{1?AQE9 zb;TpNzoBp7FV#>scizC0jw_zqR}yKQr;?J(U~&*50uQ7bziyq|mf%7BBjKKu?Q*){ z1T3{;Oyy0^x|3}1@rhAgab)IT+5_T!FPoqhhqRp`&%J7J9VUT`<~Y)ayQ=!@MShmE z6SEum&C%-25z+-K^eu*V=kn7CF80YU#*=azkL#u5XV$(kdm6N{Rl3=VA9A7@jf8Vj z*GA%ko<=J6PnFu6_TQkVDD)>-2N(vlz0)|W%hxWnn56d@t$<5N>OfLh+DA}cGa(AP zD$M$T!1Ky0x8v9D4y*<@^kv%;wnucx5!Ft-yf!ppv!2?>|FvBx%vnojwA691!ZxEc zxKMp+e7)amC~mrO`2Iuty3Y9SyN24u^YmchV+{zw?DKSYs8xm9bRnV1G5n-T9T(S9 z)^z7=Y~+vS_E)W~R%4mBn^mXhdfpFsnZL4}x3DA9IRE`(4cY9z$wo8=#ZZrGsIE_;>Z%^V|Nkz#wjTQL^iipFFeaeD#=dMinB@jP8(Uv7a{pwP zXZkyi99xU6jp@f#*(Q5Sd0QtjoiUB6c?UE4ReE@%{5b9S`D+f#<+7&6dy5(H*YTVw ziJYKU>*D3Dt?JS1KTI0A;TXHKgB+!Lu|LG^9~dgRo4;{ATUxhA*TG;UzJ!>!rek1X z$n$`|Hi{(8_NL+Tzh5mH-Mwqn@N=d~7_MEvkG1#vfaA>p&=LPyBKZ;S#%A6`{mEhO z5K(0})Aw}4_3|lu>^r_ZL@MBW&_(R3ujMbz@vqs5d zOLh6yr51?T3nluueM}ctjB^_e@0Yu!G5yYh{Hgis?{k9r3iFzTXX$P67s-#LW&K+3 z;ZE1mWu=cV9C@e|TU>v9Enog-%mR@dDCy;Xy!?63(}#1`f8wH~tIi2bvh3ezlyxrH zx_*s5ck$kZpkR6*#?#LukQLzR=KB} + +using namespace std; + +int solve(string &s1, string &s2, int i = 0, int j = 0) +{ + if (i == s1.size() || j == s2.size()) + return 0; + + if (s1[i] == s2[j]) + { + return (solve(s1, s2, i + 1, j + 1) + 1); + } + return max(solve(s1, s2, i, j + 1), solve(s1, s2, i + 1, j)); +} + +int main() +{ + int n, m; + cin >> n >> m; + string s1, s2; + cin >> s1 >> s2; + + cout << solve(s1, s2) << endl; + + return 0; +} \ No newline at end of file diff --git a/longest-common-subsequence/src/ac.cpp b/longest-common-subsequence/src/ac.cpp index 54327a7..3696ec2 100644 --- a/longest-common-subsequence/src/ac.cpp +++ b/longest-common-subsequence/src/ac.cpp @@ -16,6 +16,7 @@ int solve(string &s1, string &s2, int i = 0, int j = 0) } int main(){ + int n, m; cin >> n >> m; string s1, s2; cin >> s1 >> s2; memset(memo, -1, sizeof(memo)); diff --git a/longest-common-subsequence/src/generator.cpp b/longest-common-subsequence/src/generator.cpp index 0af0d47..ee639a8 100644 --- a/longest-common-subsequence/src/generator.cpp +++ b/longest-common-subsequence/src/generator.cpp @@ -14,6 +14,7 @@ template void append(vector &dest, const vector &orig) { string output_tc(const string &x, const string &y) { ostringstream oss; + oss << x.size() << " " << y.size() << endl; oss << x << " " << y << endl; return oss.str(); } diff --git a/longest-common-subsequence/src/validator.cpp b/longest-common-subsequence/src/validator.cpp index 237c9cf..3e434ca 100644 --- a/longest-common-subsequence/src/validator.cpp +++ b/longest-common-subsequence/src/validator.cpp @@ -6,9 +6,13 @@ using namespace std; int main(int argc, char* argv[]) { registerValidation(argc, argv); - inf.readToken("[a-z]{1, 1000}", "s1"); + int n = inf.readInt(1, 1000, "n"); inf.readSpace(); - inf.readToken("[a-z]{1, 1000}", "s2"); + int m = inf.readInt(1, 1000, "m"); + inf.readEoln(); + inf.readToken("[a-z]{" + to_string(n) + "}", "s1"); + inf.readSpace(); + inf.readToken("[a-z]{" + to_string(m) + "}", "s2"); inf.readEoln(); inf.readEof(); return 0; diff --git a/longest-common-subsequence/statement/description.tex b/longest-common-subsequence/statement/description.tex index e69de29..659f97b 100644 --- a/longest-common-subsequence/statement/description.tex +++ b/longest-common-subsequence/statement/description.tex @@ -0,0 +1,3 @@ +O problema consiste em determinar o comprimento da maior subsequência comum entre duas strings. +Uma subsequência é uma sequência que pode ser obtida a partir da string original removendo-se zero ou mais caracteres, sem alterar a ordem relativa dos restantes. +Dadas duas strings \( s_1 \) e \( s_2 \), o objetivo é encontrar o tamanho da maior subsequência que aparece em ambas. diff --git a/longest-common-subsequence/statement/input.tex b/longest-common-subsequence/statement/input.tex index e69de29..feb3a0f 100644 --- a/longest-common-subsequence/statement/input.tex +++ b/longest-common-subsequence/statement/input.tex @@ -0,0 +1,2 @@ +A entrada é composta por duas linhas. Na primeira linha, há dois inteiros \( n \) e \( m \) (\( 1 \leq n, m \leq 1000 \)), representando respectivamente os tamanhos das strings \( s_1 \) e \( s_2 \). +Na segunda linha, há duas strings \( s_1 \) e \( s_2 \), cada uma composta apenas por letras minúsculas do alfabeto, com tamanhos \( n \) e \( m \), respectivamente. \ No newline at end of file diff --git a/longest-common-subsequence/statement/notes.tex b/longest-common-subsequence/statement/notes.tex index e69de29..e38825d 100644 --- a/longest-common-subsequence/statement/notes.tex +++ b/longest-common-subsequence/statement/notes.tex @@ -0,0 +1,5 @@ +Para as strings \( s_1 = abcde \) e \( s_2 = ace \), a maior subsequência comum é "ace", que possui tamanho 3. + +Para \( s_1 = abc \) e \( s_2 = abc \), ambas as strings são idênticas, então a maior subsequência comum tem tamanho 3. + +Para \( s_1 = abc \) e \( s_2 = hhh \), não há caracteres em comum, e portanto a maior subsequência comum tem tamanho 0. diff --git a/longest-common-subsequence/statement/output.tex b/longest-common-subsequence/statement/output.tex index e69de29..7a38938 100644 --- a/longest-common-subsequence/statement/output.tex +++ b/longest-common-subsequence/statement/output.tex @@ -0,0 +1 @@ +Imprima um único inteiro representando o comprimento da maior subsequência comum entre \( s_1 \) e \( s_2 \). diff --git a/longest-common-subsequence/statement/tutorial.tex b/longest-common-subsequence/statement/tutorial.tex index e69de29..fb768e7 100644 --- a/longest-common-subsequence/statement/tutorial.tex +++ b/longest-common-subsequence/statement/tutorial.tex @@ -0,0 +1,54 @@ +\section{Solução do Problema} + +O problema da maior subsequência comum (\textit{Longest Common Subsequence}, LCS) pode ser resolvido por meio de \textit{programação dinâmica}. +A ideia central é decompor o problema em subproblemas menores, de forma que a solução ótima final seja construída a partir das soluções ótimas desses subproblemas. + +\subsection{Definição do Subproblema} + +Sejam duas strings \( s_1 \) e \( s_2 \), de tamanhos \( n \) e \( m \), respectivamente. +Definimos: + +\[ +dp[i][j] = \text{o comprimento da maior subsequência comum entre } s_1[1..i] \text{ e } s_2[1..j]. +\] + +Isto é, \( dp[i][j] \) representa a resposta do problema considerando apenas os primeiros \( i \) caracteres de \( s_1 \) e os primeiros \( j \) caracteres de \( s_2 \). + +\subsection{Função de Transição} + +Ao analisar um par de posições \( (i, j) \), temos duas possibilidades principais: + +\begin{itemize} + \item Se os caracteres atuais são iguais, isto é, \( s_1[i] = s_2[j] \), então podemos estender uma subsequência comum encontrada anteriormente: + \[ + dp[i][j] = dp[i-1][j-1] + 1. + \] + + \item Caso contrário, não é possível usar ambos os caracteres ao mesmo tempo, então devemos escolher o melhor resultado entre ignorar um dos dois: + \[ + dp[i][j] = \max(dp[i-1][j], \, dp[i][j-1]). + \] +\end{itemize} + +Assim, a função de transição pode ser resumida como: + +\[ +dp[i][j] = +\begin{cases} +dp[i-1][j-1] + 1, & \text{se } s_1[i] = s_2[j] \\ +\max(dp[i-1][j], \, dp[i][j-1]), & \text{caso contrário} +\end{cases} +\] + +\subsection{Casos Base} + +Os casos base seguem a lógica natural do problema: + +\[ +dp[0][j] = 0 \quad \forall j \ge 0 +\] +\[ +dp[i][0] = 0 \quad \forall i \ge 0 +\] + +Isto representa que, se uma das strings tiver tamanho zero, nenhuma subsequência comum pode ser formada. \ No newline at end of file