From 189f75019b92374b029b159eeaaa98ac80ed5896 Mon Sep 17 00:00:00 2001 From: Daniel Saad Date: Sun, 10 May 2026 13:44:44 -0300 Subject: [PATCH] Updating lazy-frog files --- ...thur-dolival-analise-de-dados-tutorial.pdf | Bin 0 -> 115886 bytes ...thur-dolival-analise-de-dados-tutorial.tex | 46 +++++++ .../arthur-dolival-analise-de-dados.pdf | Bin 0 -> 196587 bytes .../arthur-dolival-analise-de-dados.tex | 54 ++++++++ .../arthur-dolival-decode-ways-tutorial.pdf | Bin 0 -> 102384 bytes .../arthur-dolival-decode-ways-tutorial.tex | 74 +++++++++++ decode-ways/arthur-dolival-decode-ways.pdf | Bin 0 -> 139688 bytes decode-ways/arthur-dolival-decode-ways.tex | 39 ++++++ ...rthur-dolival-delete-and-earn-tutorial.pdf | Bin 0 -> 90965 bytes ...rthur-dolival-delete-and-earn-tutorial.tex | 66 ++++++++++ .../arthur-dolival-delete-and-earn.pdf | Bin 0 -> 168083 bytes .../arthur-dolival-delete-and-earn.tex | 42 +++++++ .../arthur-dolival-edit-distance-tutorial.pdf | Bin 0 -> 103882 bytes 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b/analise-de-dados/arthur-dolival-analise-de-dados-tutorial.tex new file mode 100644 index 0000000..a443bff --- /dev/null +++ b/analise-de-dados/arthur-dolival-analise-de-dados-tutorial.tex @@ -0,0 +1,46 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} +\usepackage{graphicx} +\input{statement/preamble.tex} + +\title{ Tutorial: Análise de Dados} +\author{Leetcode 1223} +\date{} +\begin{document} +\maketitle +O problema pode ser resolvido por meio de uma abordagem de \textbf{programação dinâmica}, onde calculamos o número de sequências válidas de lançamentos de dados até cada posição, levando em consideração qual foi a última face sorteada e quantas vezes ela apareceu consecutivamente. + +A principal restrição é que cada face \( i \) (\(1 \le i \le 6\)) só pode aparecer no máximo \( d_i \) vezes seguidas. Assim, ao adicionar um novo lançamento à sequência, precisamos garantir que essa condição nunca seja violada. + +Definimos a seguinte estrutura de DP: + +\[ +dp[i][j] = \text{número de sequências válidas de comprimento } i \text{ cuja última face lançada é } j +\] + +Para lidar com o número de repetições consecutivas, expandimos a ideia considerando blocos consecutivos de uma mesma face. +Para cada face \( k \), podemos adicioná-la de \(1\) até \(d_k\) vezes, desde que a sequência anterior termine com uma face diferente de \(k\). + +Assim, a função de transição é dada por: +\[ +dp[i][k] = \sum_{j \ne k} \sum_{x = 1}^{\min(d_k, i)} dp[i - x][j] +\] +onde o termo \(dp[i - x][j]\) representa o número de sequências de comprimento \(i - x\) que terminam com uma face diferente de \(k\), às quais adicionamos \(x\) ocorrências consecutivas da face \(k\). + +As condições iniciais são: +\[ +dp[0][j] = 1, \quad \forall j \in \{1, 2, 3, 4, 5, 6\} +\] +representando a sequência vazia inicial, que serve como base para as transições. + +A resposta final será a soma de todas as formas possíveis de terminar a sequência de comprimento \(n\): +\[ +\sum_{j=1}^{6} dp[n][j] \bmod (10^9 + 7) +\] + +A complexidade temporal dessa solução é \(O(n \times 6 \times \text{max}(d_i))\), pois para cada posição e cada face consideramos até \(d_i\) possíveis comprimentos de repetições.\end{document} diff --git a/analise-de-dados/arthur-dolival-analise-de-dados.pdf b/analise-de-dados/arthur-dolival-analise-de-dados.pdf new file mode 100644 index 0000000000000000000000000000000000000000..04071cf0862f392a64305459d75a044c6e4a47bf GIT binary patch literal 196587 zcma&MLy#^E(4|?{E!(zj+qP}nwr%Sz6pt*e+8-B_C%zhgv168ddcR z@(IwKAah%|NLKEB`Pb!A9K;*4csl#68N(qW7uOS!^i{z1@6(vOlz`OTMxS00$%vF z1BZZaPfy2(=SUFemSb_5w4O$ep3n28gKbm9Jpp@7D&x~0WAzRtolGt17r*VU-na7z zu}en&aPx+vnsk`$t{s@8E!)+oW@Q-BtO0>T*_Bl8~Z73>Ee5nC3Xs34H|6&8_s~wz%z!<0NZ+_wqQtWqoPw^7rivyim6MT zyIp6y@$sS-+$hk<;`^dPe08h7m$n6=( 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confiante. + +Você concorda, achando que entendeu, até perceber que a máquina está literalmente lançando um dado de seis faces a cada segundo. + +Sim, você vai analisar \textbf{dados de verdade}. + +Mas há um problema: os dados do laboratório sofrem de \textit{fadiga de repetição}. +Se uma mesma face aparecer muitas vezes seguidas, o mecanismo emperra e a máquina trava, o que, segundo o manual, “não é recomendável”. + +Cada face $i$ ($1 \le i \le 6$) possui um limite $d_i$: o número máximo de vezes que ela pode aparecer consecutivamente antes de causar uma falha. + +Sua tarefa é determinar quantas sequências de lançamentos de comprimento $n$ podem ser geradas sem que a máquina quebre. + +Como o número de sequências válidas pode ser gigantesco, o gerente exige que o resultado seja dado módulo $10^9 + 7$. + +\Entrada + +A entrada contém um inteiro \(n\) que representa o número de lançamentos do dado, onde \(1 \leq n \leq 5000\). +Em seguida, são dados seis inteiros \(d_1, d_2, d_3, d_4, d_5, d_6\), cada um representando o número máximo de vezes consecutivas que a face correspondente pode aparecer, com \(1 \leq d_i \leq 15\). + +\Saida + +A saída consiste em um único inteiro representando o número total de sequências válidas de lançamentos, considerando o resultado módulo \(10^9 + 7\). + +\ExemploEntrada + +\begin{Exemplo} +\texttt{2} & \texttt{34}\\ +\texttt{1~1~2~2~2~3} & \\ +\rowcolor{gray!20}\texttt{2} & \texttt{30}\\ +\rowcolor{gray!20}\texttt{1~1~1~1~1~1} & \\ +\end{Exemplo} + + + +\Notas + +No primeiro caso de teste, \(n = 2\) e os valores \(d\) são \((1, 1, 2, 2, 2, 3)\). +As faces \(1\) e \(2\) não podem aparecer repetidas consecutivamente, enquanto as demais possuem limites mais altos, resultando em \(34\) sequências válidas. +Como há \(2\) lançamentos de dado e \(6\) faces, sem restrições haveria \(6 \times 6 = 36\) combinações possíveis. +Entretanto, as sequências \((1, 1)\) e \((2, 2)\) são inválidas devido às restrições de \(d_1 = 1\) e \(d_2 = 1\). +Logo, o total de combinações válidas é \(36 - 2 = 34\). \end{ProblemaAutor} +\end{document} diff --git a/decode-ways/arthur-dolival-decode-ways-tutorial.pdf b/decode-ways/arthur-dolival-decode-ways-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4027e7972f76d7fb588fc9edca75a8fb8dbfc6c8 GIT binary patch literal 102384 zcma&NV~j39*QI^8ZJ)Mn+tz8@-KTBawr$(CZQHip{XH}D&HQ;Md8d*}*8W>rm1|dJ zFVepvVzi8Otk9%$OT%l>tON`MwuTnaJUr0!GA1^rPG$s*94s6J|8qgpi4TCd{QBw+|f5$18;qi42P)((i18d zTd7s03)-~Jbx{88E&hA1;GnJSSd;ypTe~iN5K+|OalGu|+3#3mF-;n8zGl0lqoB{} zv}wBCE9K$;h8vnhGa8~8?sgLnXEbykuvElGFaEvRLJd+G9`T1ov{282jleWyS@pSr zsw_(hjLi3Kbgmw5C77P9DwKI+>mcNVwYh3dm7T-TbyaG#<}3L5 z;f1QVEl4r|*)%_7M?^;r+`d9PL?cGK3K5GINOldhzQ#gpQjS8qY@qw0O0I$4^t{Ak z(1h649ksP?Y)QXnB7>}TPWV7Xo$ET2QqA+Zb#=kcU2*B$$#%n{MwEv-hOI)Y!Xvk} z-i*}^KpYOn74KCc49t51^QmX#JksV2tU3Ec~>m!LnkX zLTwJ{#b3v3pZ;Jhz^UC5Yd5QYNr$snchf#DV|W z+@m8v)IKGbk<8!w{4uyJA+&~Z)o-;G7U{Duk+r^Fa@ynExj$Pt6nnK<9ZGvRMMto( 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+\title{ Tutorial: Decodificando Mensagens} +\author{Leetcode 91} +\date{} +\begin{document} +\maketitle +O problema pode ser resolvido utilizando uma abordagem de \textbf{programação dinâmica}, na qual calculamos, passo a passo, o número de formas possíveis de decodificar a sequência até cada posição. + +\subsection*{Modelagem} + +Cada dígito (ou par de dígitos consecutivos) da sequência pode representar uma letra do alfabeto latino, seguindo a regra: +\[ +1 \rightarrow A, \quad 2 \rightarrow B, \quad \ldots, \quad 26 \rightarrow Z +\] +Portanto, precisamos contar todas as maneiras válidas de interpretar a sequência numérica, considerando que apenas combinações entre $1$ e $26$ são válidas. + +\subsection*{Definição da DP} + +Definimos uma estrutura de DP com dois estados para cada posição $i$: +\begin{itemize} + \item $dp[i][0]$: número de mensagens possíveis terminando no caractere $i$ \textbf{quando o dígito atual não é concatenado} com o anterior; + \item $dp[i][1]$: número de mensagens possíveis terminando no caractere $i$ \textbf{quando o dígito atual é concatenado} com o anterior. +\end{itemize} + +\subsection*{Função de Transição} + +A função de transição é definida da seguinte forma: +\[ +dp[i][0] = +\begin{cases} +dp[i - 1][0] + dp[i - 1][1], & \text{se } 1 \leq s[i] \leq 9 \\[6pt] +0, & \text{caso contrário} +\end{cases} +\] + +\[ +dp[i][1] = +\begin{cases} +dp[i - 2][0] + dp[i - 2][1], & \text{se } 10 \leq 10 \cdot s[i-1] + s[i] \leq 26 \\[6pt] +0, & \text{caso contrário} +\end{cases} +\] + +\subsection*{Casos Base} + +Os casos base são: +\[ +dp[0][0] = 1, \quad dp[0][1] = 0, \quad dp[1][0] = 1, \quad dp[1][1] = 0 +\] +Isso ocorre porque, antes de processar qualquer caractere, há exatamente uma maneira “vazia” de formar uma mensagem válida, e o primeiro dígito pode, no máximo, representar uma única letra isolada. + +\subsection*{Resposta Final} + +O total de formas possíveis de decodificar a sequência é dado por: +\[ +dp[n][0] + dp[n][1] +\] +onde $n$ é o comprimento da sequência de entrada. + +\subsection*{Complexidade} + +\begin{itemize} + \item \textbf{Tempo:} $O(n)$, pois cada caractere é processado uma única vez; + \item \textbf{Espaço:} $O(n)$, podendo ser otimizado para $O(1)$ ao armazenar apenas os últimos dois estados. +\end{itemize}\end{document} diff --git a/decode-ways/arthur-dolival-decode-ways.pdf b/decode-ways/arthur-dolival-decode-ways.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3a5c07712bb60526a81870a9b1ef2cd28eb0f353 GIT binary patch literal 139688 zcma&NLy#p5&~1ClMwe}O*|u%lws8twHo9!vwr$(C)usEr7w?a_5%&#lj&@FRx&~_} zxuU2z9TPnpEcx8h@ER-|5hIa2XC9|@*L?530$g|na(U8QA443yI&V%1XANfc zvu62cITK=dc3nAI(f3QmA99a1T^7jP8I4I5RCwzl6Z4*xD-zZK2)j+r(7j!>Bv zdlFYqVr*d){~bD)+g|z;)?7THk-uyhaspW$->Ps57oYyvM@PGqWdWz*n15ri`*tFX zv%ajR%h2TIgMr@Q0}G=^Witw!Z6DV1$4z?kGv;+L5s~;#oteSwh=p5guMc)mkj=1M zYw6l!57c_w^}4j#fd+?eQ_kSEj5*f#Pc`JrpW6&?6*tyhWmG{Y;%ZA@C@d>fCO^B1 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correspondência segue a regra \( 1 \rightarrow A, 2 \rightarrow B, \ldots, 26 \rightarrow Z \). +Cada dígito ou combinação de dois dígitos consecutivos pode ser convertido em uma letra válida, desde que a sequência resultante obedeça às restrições de decodificação. +O objetivo é calcular o total de maneiras diferentes de decodificar a sequência numérica fornecida. + +\Entrada + +A entrada é composta por duas linhas. +A primeira linha contém um inteiro \( n \) (\( 1 \leq n \leq 100 \)), representando o tamanho da sequência numérica. +A segunda linha contém uma sequência de digitos de \( s \) de comprimento \( n \). + +\Saida + +A saída deve conter um único inteiro, representando o número total de maneiras possíveis de decodificar a sequência numérica \( s \) de acordo com o mapeamento \( 1 \rightarrow A, 2 \rightarrow B, \ldots, 26 \rightarrow Z \). + +\ExemploEntrada + +\begin{Exemplo} +\texttt{2} & \texttt{2}\\ +\texttt{12} & \\ +\rowcolor{gray!20}\texttt{3} & \texttt{3}\\ +\rowcolor{gray!20}\texttt{226} & \\ +\texttt{2} & \texttt{0}\\ +\texttt{06} & \\ +\end{Exemplo} + + + +\Notas + +Para a sequência \( s = "12" \), existem duas decodificações possíveis: "AB" (1, 2) e "L" (12). +Para a sequência \( s = "226" \), há três decodificações possíveis: "BZ" (2, 26), "VF" (22, 6) e "BBF" (2, 2, 6). +Para a sequência \( s = "06" \), não há nenhuma forma válida de decodificação.\end{ProblemaAutor} +\end{document} diff --git a/delete-and-earn/arthur-dolival-delete-and-earn-tutorial.pdf b/delete-and-earn/arthur-dolival-delete-and-earn-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..be8fc33bb8793771a29c0ea3a54cc572a47898df GIT binary patch literal 90965 zcma&NW0Npk&~7)jZQHhI?y+s#wr$(CZQHhOn|V%>_d_bDD%l@)b^m~_UR8Z9Qh5Tg6I(N9a{@+APF8~d^Fh&zT39=qI1fltB4Nz4NiVxU z8WE)wRtJoO%Uau*QO&w$arH5|I@`)Art;EQJu&~fk(JL)`F*t8`UQ-$wet=QCFdc(s1bxc-RZczX-MVd`g=+^?%Sc`Jn>hjNmDfHP<2H3HEZ)2hCK7e{ 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Deletar e Ganhar} +\author{Leetcode 740} +\date{} +\begin{document} +\maketitle +O problema pode ser resolvido utilizando uma abordagem clássica de \textbf{programação dinâmica}. +A ideia central é que, ao escolher um número \( x \), ganhamos uma certa quantidade de pontos, mas perdemos a possibilidade de escolher os números \( x - 1 \) e \( x + 1 \), já que eles devem ser removidos da sequência. + +\subsection*{Modelagem} + +Primeiro, agrupamos os números iguais da sequência e calculamos o total de pontos que seria obtido caso escolhêssemos todos os elementos de valor \( i \): +\[ +\text{points}[i] = (\text{quantidade de vezes que } i \text{ aparece}) \times i +\] +Assim, o problema passa a ser equivalente a escolher quais valores \( i \) maximizarão a soma total de pontos, respeitando a restrição de que, ao escolher \( i \), não podemos escolher \( i-1 \) nem \( i+1 \). + +\subsection*{Definição da DP} + +Definimos: +\[ +dp[i] = \text{pontuação máxima possível utilizando apenas os números de } 1 \text{ até } i +\] + +\subsection*{Função de Transição} + +A cada passo, temos duas escolhas: +\begin{itemize} + \item \textbf{Não escolher} o número \( i \): nesse caso, o resultado é o mesmo que \( dp[i - 1] \); + \item \textbf{Escolher} o número \( i \): ganhamos \( \text{points}[i] \), mas não podemos utilizar \( i - 1 \), então somamos com \( dp[i - 2] \). +\end{itemize} + +A função de transição é, portanto: +\[ +dp[i] = \max(dp[i - 1], \, dp[i - 2] + \text{points}[i]) +\] + +\subsection*{Casos Base} + +\[ +dp[0] = 0, \quad dp[1] = \text{points}[1] +\] + +\subsection*{Resposta Final} + +A pontuação máxima possível é: +\[ +dp[m] +\] +onde \( m \) é o maior valor presente na sequência original. + +\subsection*{Complexidade} + +\begin{itemize} + \item \textbf{Tempo:} \( O(m) \), onde \( m \) é o valor máximo na sequência; + \item \textbf{Espaço:} \( O(m) \), podendo ser otimizado para \( O(1) \) se armazenarmos apenas os dois últimos estados da DP. 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de Edição} +\author{} +\date{} +\begin{document} +\maketitle +\section{Solução do Problema} + +O problema de encontrar o menor número de operações para transformar uma string em outra, classicamente conhecido como Distância de Edição (\textit{Edit Distance}), pode ser resolvido de forma eficiente por meio de \textbf{programação dinâmica}. + +\subsection{Definição do Subproblema} + +Sejam $w_1$ e $w_2$ as duas strings de tamanhos $N$ e $M$, respectivamente. +Definimos o nosso estado da programação dinâmica como: + +$$dp[i][j] = \text{número mínimo de operações para transformar o sufixo } w_1[i \dots N-1] \text{ no sufixo } w_2[j \dots M-1].$$ + +Assim, $dp[0][0]$ representará a resposta final para as duas strings inteiras. + +\subsection{Função de Transição} + +Para determinar $dp[i][j]$, olhamos para os caracteres atuais $w_1[i]$ e $w_2[j]$ e tomamos uma decisão. + +Se os caracteres forem iguais ($w_1[i] == w_2[j]$), não precisamos realizar nenhuma operação. Apenas avançamos na análise de ambas as strings: + +$$dp[i][j] = dp[i+1][j+1]$$ + +Se os caracteres forem diferentes ($w_1[i] \neq w_2[j]$), temos três opções de edição. Avaliamos o resultado de aplicar cada uma delas e escolhemos a que produz o menor custo final. Cada operação tem um custo de $1$: + +$$dp[i][j] = 1 + \min \begin{cases} + dp[i+1][j] & \text{(Remoção do caractere } w_1[i]\text{)} \\ + dp[i+1][j+1] & \text{(Substituição de } w_1[i] \text{ por } w_2[j]\text{)} \\ + dp[i][j+1] & \text{(Inserção do caractere } w_2[j] \text{ em } w_1\text{)} + \end{cases}$$ + +\subsection{Casos Base} + +Os casos base ocorrem quando esgotamos os caracteres de uma das strings (ou seja, quando $i$ ou $j$ ultrapassam os limites): + +Se consumirmos toda a string $w_1$ ($i \ge N$), mas ainda restarem caracteres em $w_2$, a única forma de igualá-las é \textbf{inserir} todos os $M - j$ caracteres restantes: +$$dp[N][j] = M - j$$ + +Da mesma forma, se consumirmos toda a string $w_2$ ($j \ge M$), mas ainda restarem caracteres em $w_1$, a única forma de igualá-las é \textbf{remover} todos os $N - i$ caracteres que sobraram: +$$dp[i][M] = N - i$$\end{document} diff --git a/edit-distance/arthur-dolival-edit-distance.pdf 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Edição}{1}{256}{} + +O problema consiste em determinar o \textbf{menor número de operações} necessárias para transformar uma string em outra. + +As operações permitidas são as seguintes: + +\begin{itemize} + \item \textbf{Inserção}: inserir um caractere em qualquer posição; + \item \textbf{Remoção}: remover um caractere de qualquer posição; + \item \textbf{Substituição}: substituir um caractere por outro. +\end{itemize} + +O objetivo é calcular o menor custo possível para converter completamente uma dada string inicial em uma string final, utilizando apenas essas operações. + +\Entrada + +A entrada consiste de duas linhas: + +\begin{itemize} + \item A primeira linha contém dois inteiros \( n \) e \( m \) (\( 1 \leq n, m \leq 500 \)), representando os tamanhos das strings. + \item A segunda linha contém duas strings \( s_1 \) e \( s_2 \), ambas compostas apenas por letras minúsculas e com tamanhos exatos \( n \) e \( m \), respectivamente. +\end{itemize} + +\Saida + +Imprima um único inteiro representando o \textbf{menor número de operações} necessárias para transformar \( s_1 \) em \( s_2 \). + +\ExemploEntrada + +\begin{Exemplo} +\texttt{7~7} & \texttt{3}\\ +\texttt{estouro~calouro} & \\ +\rowcolor{gray!20}\texttt{11~8} & \texttt{7}\\ +\rowcolor{gray!20}\texttt{aniquilacao~intencao} & \\ +\end{Exemplo} + + + +\end{ProblemaAutor} +\end{document} diff --git a/estouro/arthur-dolival-estouro-tutorial.pdf b/estouro/arthur-dolival-estouro-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..42cffa347b869aa075f447b8f2a0a5c6ea5bd681 GIT binary patch literal 134011 zcma&OQ>-vRv!=Uj+qP}nwr$(|mTlX%ZQHhOuXX-8IeU^bdoFfgRaMe`S$Xp2Ns}sw zh|x0Au|ko~FORH4u@W#4*c(|v@$f*=%b41kyI2q~aWF6t{J#f^Ud+-*L84`OD8r8U_4ahJaNVRqL$&&|vRqw#?)SB56`cuPO3T^t z)0vH zJxz%!tNWtu%e_KQolQ3BrBdQoL(@75D%sU8DaQYo>3MCdvnIWl3B81ZIR)_p1zJLp zVs34&J8O$!pa(JR*`{hvc57Q5xUbXSHi+MJv4(^ljK!{eA^+eEnftjqCwlPt%IYiZ 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Nesses casos, como não há balões para estourar, a energia liberada é naturalmante nula: + +$$dp[i][j] = 0 \quad \text{para todo } i > j$$ + +Como a implementação iterativa inicializa a matriz completa com zeros, essa condição já é suprida por padrão. + +\subsection{Recuperação da Decomposição (Ordem das explosões)} + +Para além da energia máxima $dp[1][N]$, frequentemente queremos também a sequência de explosões que atinge esse valor. Para isso, mantemos uma matriz auxiliar \texttt{cut[i][j]} durante a computação, que armazena o índice $k$ responsável pelo maior ganho em $dp[i][j]$. + +Após preencher $dp$ e \texttt{cut}, criamos uma função recursiva para reconstruir a resposta. Ao chamarmos a função para o intervalo $[1, N]$, descobrimos o pivô $k = \texttt{cut}[1][N]$. Sabemos que $k$ é o \textit{último} balão estourado. Em seguida, chamamos recursivamente para os subintervalos $[1, k-1]$ e $[k+1, N]$. Ao empilharmos esses balões, obtemos a sequência cronológica invertida; basta inverter a lista no final para obtermos a sequência exata das explosões.\end{document} diff --git a/estouro/arthur-dolival-estouro.pdf b/estouro/arthur-dolival-estouro.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ff10561d169b6b4a2e09ea6828c388301bc9bc97 GIT binary patch literal 177288 zcmb5UV{;}9(6$-dcCOgAC!W~0C$??dp4ggLS8Ut1ZN2wy?Xy)|`+nH%PuZsY#LP_WP0ip01>u}soXm`D;XKxxwPc((Mp1zm8oMyVq^t@EyEO#M_&4e;oH|lk zTocWZiS;oUzeYhS%I_~01h&Az78Ow^ro{nJYyw}O?<9$vRHSc5=UY=t`Ton4iKKMm zG|AaBkt&f3;_8RA?Q4|83q5|CHO=a03SZKtovBV|K%$p6%7tvTf(5^3b@BDqgY$){ zI*l9qTA;oZfDuiqN!EOVbj?MBSSZpI{-TGHbfNyyzNbniW+{^c&#!eXyvko6*8F-i zIb#~#lGQe0Y9`W&l`d8)2PIRD$MM^5^Ci4Mvc=Fp5ko8U1gfLsAC^lM`Y52kEh<_uYzCfPMn%4|CS=KNay#On{OV4cnPi~D3^ zwqvmiWv&+M4ad*by+Vv7wee}jE5GN+NP4h!;84m4y>^k!i5 z=aJBk71>Rgxk?~&P*d=XyBa(OgQBvf!2J3R^lV0IeLJzq{un{-2-N(_XqXH#p<|Dw 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100644 index 0000000..34b2d4c --- /dev/null +++ b/flowers/arthur-dolival-flowers-tutorial.tex @@ -0,0 +1,49 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} +\usepackage{graphicx} +\input{statement/preamble.tex} + +\title{ Tutorial: O problema das Flores} +\author{} +\date{} +\begin{document} +\maketitle +O problema pode ser resolvido utilizando uma abordagem de \textbf{programação dinâmica}, onde calculamos o número de maneiras válidas de preencher o canteiro até cada posição, levando em conta a cor da última flor e o tamanho do bloco atual de flores consecutivas. + +A restrição principal é que não podem existir mais do que \(m\) flores consecutivas da mesma cor. Isso significa que, ao adicionar uma nova flor, precisamos garantir que a sequência resultante ainda seja válida, ou seja, o número de flores consecutivas da mesma cor não ultrapasse o limite imposto. + +Definimos a DP da seguinte forma: + +\[ +dp[i][c] = \text{número de maneiras válidas de formar um arranjo de comprimento } i \text{ cuja última flor tem cor } c +\] +onde \(c = 0\) representa a cor \texttt{V} (vermelha) e \(c = 1\) representa a cor \texttt{B} (branca). + +Para calcular \(dp[i][c]\), consideramos todos os blocos possíveis de flores consecutivas da mesma cor que terminam na posição \(i\). +Ou seja, podemos ter adicionado 1, 2, ..., até \(m\) flores consecutivas da cor \(c\), desde que o arranjo anterior termine com a cor oposta. + +A função de transição é dada por: + +\[ +dp[i][c] = \sum_{k=1}^{m} dp[i - k][1 - c] +\] +para todo \(i \ge k\), pois estamos adicionando um bloco de \(k\) flores da cor \(c\) sobre um arranjo válido de comprimento \(i - k\) que termina com a cor oposta. + +As condições base são: +\[ +dp[0][0] = dp[0][1] = 1 +\] +representando o caso vazio, que conta como uma configuração válida inicial para permitir as transições. + +O resultado final é a soma de todos os arranjos válidos que terminam em qualquer cor: +\[ +\text{Resposta} = dp[n][0] + dp[n][1] +\] + +Essa solução possui complexidade de tempo \(O(n \times m)\), já que para cada posição \(i\) consideramos até \(m\) comprimentos de blocos consecutivos possíveis. +O espaço necessário é \(O(n)\), podendo ser otimizado para \(O(m)\) se apenas os estados recentes forem mantidos.\end{document} diff --git a/flowers/arthur-dolival-flowers.pdf b/flowers/arthur-dolival-flowers.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e84a30c1fb4c194ec3eefb082aa2458d9cee92b3 GIT binary patch literal 174567 zcma%?Ly#^2(_Y{4-0>aTwr$(CZQHhO+qP|c=8kPMUsCx~sU(Z^s(abHQ+=K$krx)F zVW4G!BAH(vS%+f5r^mN5w1nd3hN6=;u{Cox$7f_`V8Q<%1w|)nVeM?>Ddn^!ir9Opb_cmCx$bfBuLhf1Orpk$EsLhgJHp9d 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Este ano, ele decidiu criar um canteiro linear de comprimento \(n\). Para cada uma das \(n\) posições no canteiro, Bino plantará exatamente uma flor, que pode ser ou vermelha (V) ou branca (B). + +Bino é um artista, e a estética é sua principal preocupação. Ele acredita que a beleza surge da variação, mas também da harmonia. Após muita deliberação, ele estabeleceu uma regra de ouro para seu jardim: \textbf{nunca devem existir mais do que \(m\) flores consecutivas da mesma cor}. + +Por exemplo, se \(n=5\) e \(m=2\), uma sequência como \texttt{V V B V V} é perfeitamente válida. No entanto, uma sequência como \texttt{V B B B V} é \emph{inválida}, pois contém um bloco de 3 flores brancas (\texttt{B B B}), o que excede o limite \(m=2\). Da mesma forma, \texttt{V V V V B} também é inválida por conter 4 flores vermelhas consecutivas. + +Bino está planejando o jardim e quer saber quantas opções de design ele realmente tem. Seu desafio é ajudar Bino a determinar o número total de arranjos distintos que ele pode criar para seu canteiro de comprimento \(n\), respeitando estritamente sua regra de monotonia (o limite de \(m\)). + +\Entrada + +A entrada contém dois inteiros separados por espaço, \(n\) e \(m\), onde \(n\) (\(1 \leq n \leq 10\,000\)) é o comprimento da sequência de flores e \(m\) (\(1 \leq m \leq 1\,000\)) é o número máximo permitido de flores iguais consecutivas. + +\Saida + +Imprima um único número inteiro: o número total de sequências de flores válidas de comprimento \(n\) que satisfazem a restrição de Bino. + +Como este número pode ser extremamente grande, sua resposta deve ser calculada e impressa \textbf{módulo \(10^9 + 7\)}. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{1~1} & \texttt{2}\\ +\rowcolor{gray!20}\texttt{2~2} & \texttt{4}\\ +\texttt{2~1} & \texttt{2}\\ +\end{Exemplo} + + + +\Notas + +Caso de teste 1: \(n=1, m=1\). +Para uma sequência de comprimento 1 existem duas opções: {vermelha} ou {branca}. Portanto, o número de sequências válidas é 2. + +Caso de teste 2: \(n=2, m=2\). +Como \(m \ge 2\), não há restrição efetiva para \(n=2\) além de que cada posição pode ser vermelha ou branca. Assim todas as \(2^2 = 4\) sequências são válidas.\end{ProblemaAutor} +\end{document} diff --git a/knapsack-zero-one/arthur-dolival-knapsack-zero-one-tutorial.pdf b/knapsack-zero-one/arthur-dolival-knapsack-zero-one-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..331c241963fe33f78dba5ee9a99903299f344c3a GIT binary patch literal 112647 zcma&NLyRy0({9ZQHhOoA;Z!^Cgq~liaCQ)uL9<=2UWuR6#_H zmXVGXigbQ?WF3l?fPui?$O?*w2Z~`b*k9?f0FTV)D7rcB3$dvg4JRE^0a0 z#57$?TS#hyie9hOo#RonFJ1oTlTXXuBo%Xh_Y*go>YS_VHy=`pIX~RbE!U=rz3uPk z;|`#rB#25*BwfZroRw5V5$&VOMxPN3^^Iu19)DTT8PrnE)UE4y9m>>H-g#E4>LKQN zdR}08FtiOW&>xS4(ODQ$l4W0><5!@E`5TkyYD%T13svTMR@J@@G0t{Em79?hW-1(*~E;~Ix2D#ryKC=(~*v0!m# z_TdZySFs{+-@WcK8Mhlyy=Nn%BsUVX$cE6fdmzz6^~?<_IZ4833+@*-!3aeGc;xhM zDMUiCt@Z-h1OyQ%JoIkKL0q6!s09x^; z9oAfD_{(cn2D&!W6`)I&O7GO&V9$Zbc`wTz*9+TwjE|56%`d;Z9uX5t?}i`>hRG?E z#O7;NKr&F0+jVyM*B>5wlh(VqX>GaFo%b`?IvlA4|4sDStnYe{Cnt!*3Y_I-&wlJk 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problema de encontrar a maior subsequência crescente (\textit{Longest Increasing Subsequence} - LIS) possui uma solução clássica em $O(N^2)$, mas quando $N$ é grande, podemos otimizá-la para \textbf{$O(N \log N)$} combinando \textbf{programação dinâmica com busca binária}. A grande sacada dessa otimização é mudar a perspectiva: em vez de guardar o tamanho da maior subsequência que termina em um índice, guardamos o \textbf{menor valor final} possível para uma subsequência de um determinado comprimento. + +\subsection{Definição do Subproblema} + +Seja $a$ o nosso vetor de números inteiros de tamanho $N$. +Definimos o nosso estado da programação dinâmica como: + +$$d[l] = \text{o menor valor que encerra uma subsequência crescente de comprimento exato } l.$$ + +O vetor $d$ terá uma propriedade muito importante: ele estará sempre rigorosamente ordenado de forma crescente. A resposta final será o maior $l$ tal que $d[l] \neq \infty$. + +\subsection{Função de Transição} + +Para calcular $d[l]$, iteramos sobre cada elemento $a[i]$ da nossa sequência original. Como queremos que os elementos finais de cada comprimento sejam os menores possíveis, tentamos usar $a[i]$ para melhorar o nosso vetor $d$. + +Como $d$ está ordenado, podemos usar \textbf{busca binária} (\texttt{upper\_bound} ou \texttt{lower\_bound}) para encontrar rapidamente a posição $l$ onde $a[i]$ deve ser inserido. Se $a[i]$ for maior que o final de uma subsequência de tamanho $l-1$ ($d[l-1] < a[i]$) e, ao mesmo tempo, puder substituir um final pior de tamanho $l$ ($a[i] < d[l]$), nós o atualizamos: + +$$d[l] = a[i]$$ + +Isso significa que encontramos uma subsequência crescente de comprimento $l$ que termina em $a[i]$, e esse final é melhor (menor) do que o que tínhamos registrado anteriormente. + +\subsection{Casos Base} + +Para que a lógica de busca e as comparações iniciais funcionem sem estourar os limites, inicializamos o vetor $d$ de tamanho $N+1$ da seguinte forma: + +$$d[0] = -\infty$$ +$$d[l] = \infty \quad \text{para todo } 1 \le l \le N$$ + +Isso representa que uma subsequência de tamanho $0$ termina em um valor infinitamente pequeno, enquanto os outros comprimentos ainda não foram alcançados. + +\subsection{Recuperação da Decomposição (Elementos da subsequência)} + +Para recuperar os elementos exatos da LIS nesta abordagem otimizada, o vetor $d$ sozinho não basta (pois ele pode conter uma mistura de elementos de diferentes subsequências válidas ao longo do tempo). Precisamos de duas estruturas de rastreamento: +\begin{itemize} + \item $pos[l]$: armazena o \textbf{índice original} do elemento que atualmente ocupa $d[l]$. + \item $p[i]$: armazena o predecessor do elemento de índice $i$, assim como na versão $O(N^2)$. +\end{itemize} + +Durante a transição, sempre que atualizamos $d[l] = a[i]$, nós também registramos que esse elemento se encontra no índice $i$ original fazendo $pos[l] = i$. +Ao mesmo tempo, sabemos que o elemento imediatamente anterior a ele na subsequência é o elemento que termina o comprimento $l-1$. Logo, conectamos o predecessor: $p[i] = pos[l-1]$. + +Ao final de todas as iterações, sabemos que \texttt{ans} é o comprimento máximo alcançado. O índice do último elemento dessa subsequência estará em $cur = pos[\texttt{ans}]$. A partir desse $cur$, reconstruímos a sequência retrocedendo $cur = p[cur]$ até atingir $-1$, e por fim invertemos a lista para exibi-la na ordem correta.\end{document} diff --git a/longest-increasing-subsequence-II/arthur-dolival-longest-increasing-subsequence-II.pdf b/longest-increasing-subsequence-II/arthur-dolival-longest-increasing-subsequence-II.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4221a62f33f584d10bd52c290d937d7a576c6975 GIT binary patch literal 155760 zcma&MQ?D=#ux7b!+qP}nwr$(CZQHhO+s3hwqF zNaffy5v4==wtaH`Y;Uh-U90+O+n#jcSE}Px5e__P;?+;(vQ@iQCDFax@Ok?C)xG}a zj(U1n5c|K+u@ z@`Y;WR)P5p4};|I$eN=`BC2-QSxD@h&8bOdebJ<3&G!&qX8S$2RHDtVrC^0Gsb%@* z)Cy)_`5g3!U7Ir5HBrK}{?_jLRp7kKuDVfg^Lxx5NXTwHk&@}?Ho34qF1+3kGZAgB zH=TqBhOWTYOvG?Mzq*By%=fyxjZ323jtb1S>t_LG6x(;1CI*WJ@4C;(+-rslfCDhu>>l9mt8F_hwYsQL;=AiM@{tga`?N8o1l-;(Vd-aZdjX=)A+s3n26DX+21u;PJ504)1g|XlX*BzH` zARsY15#|&al-U4pq*|94j(ZQZn)!y%Bp5?_Y24mgYxLdAzFr!}Y*tz~g>`0of@V`BS*c|K{7YEx=AaozEnF`&qRy@W_j8ve!^;nGl 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literal 0 HcmV?d00001 diff --git a/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence-tutorial.tex b/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence-tutorial.tex new file mode 100644 index 0000000..dcedfa9 --- /dev/null +++ b/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence-tutorial.tex @@ -0,0 +1,50 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} +\usepackage{graphicx} +\input{statement/preamble.tex} + +\title{ Tutorial: Maior subsequência Crescente} +\author{} +\date{} +\begin{document} +\maketitle +\section{Solução do Problema} + +O problema de encontrar a maior subsequência crescente (\textit{Longest Increasing Subsequence} - LIS) é um clássico que pode ser resolvido de forma eficiente por meio de \textbf{programação dinâmica}. A ideia central é focar no final da subsequência: para cada elemento da sequência original, calculamos qual é a maior subsequência crescente que termina exatamente nele. + +\subsection{Definição do Subproblema} + +Seja $a$ o nosso vetor de números inteiros com $N$ elementos, indexado de $0$ a $N-1$. +Definimos o nosso estado da programação dinâmica como: + +$$d[i] = \text{o comprimento da maior subsequência crescente que termina obrigatoriamente no índice } i.$$ + +A resposta final para o tamanho da maior subsequência não será necessariamente $d[N-1]$, mas sim o maior valor alcançado em todo o vetor $d$, ou seja, $\max(d[i])$ para $0 \le i < N$. + +\subsection{Função de Transição} + +Para determinar o valor de $d[i]$, precisamos olhar para todos os elementos anteriores à posição $i$, ou seja, um índice $j$ tal que $0 \le j < i$. + +Se encontrarmos um elemento anterior que seja estritamente menor que o elemento atual ($a[j] < a[i]$), isso significa que podemos estender a subsequência que terminava em $j$ anexando o elemento $a[i]$ ao final dela. O novo comprimento seria $d[j] + 1$. +Avaliamos todos os possíveis índices $j$ válidos e mantemos o que gera o maior comprimento: + +$$d[i] = \max_{\substack{0 \le j < i \\ a[j] < a[i]}} \big( d[j] + 1 \big)$$ + +\subsection{Casos Base} + +O caso base é bastante intuitivo: na pior das hipóteses, qualquer elemento isolado forma, por si só, uma subsequência válida de comprimento $1$. Portanto, a nossa inicialização padrão para todos os estados é: + +$$d[i] = 1 \quad \text{para todo } 0 \le i < N$$ + +\subsection{Recuperação da Decomposição (Elementos da subsequência)} + +Como o problema exige a impressão dos elementos que formam a subsequência, precisamos reconstruir o caminho ótimo. Para isso, utilizamos um vetor auxiliar $p[i]$ (de \textit{parent} ou predecessor), inicializado com $-1$. + +Durante a transição, sempre que atualizamos $d[i]$ com um valor maior proveniente de um $d[j] + 1$, registramos o índice de origem definindo $p[i] = j$. + +Após calcularmos todo o vetor $d$, localizamos o índice \texttt{pos} que contém o valor máximo absoluto de $d$. A partir de \texttt{pos}, recuperamos os elementos da subsequência percorrendo o caminho reverso: adicionamos $a[\texttt{pos}]$ à nossa lista de resposta e saltamos para o predecessor atualizando $\texttt{pos} = p[\texttt{pos}]$. Repetimos o processo até que $\texttt{pos}$ seja $-1$. Como resgatamos os elementos do final para o começo, basta inverter a lista resultante para apresentar a subsequência na ordem crescente original.\end{document} diff --git a/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence.pdf b/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c205b73444933816121e5132552e7f3ecbecae29 GIT binary patch literal 155446 zcma&MQ?D=#ux7b!+qP}nwr$(CZQHhO+s3992UKcRWsxd~h|x0A zu|kn9tc-3zu@W#4*c(|v@$f*=%b41kyI2q~u`sa`{J#Z7FJ@`uV(LUdFJ@!tVk%;4 zY;R%;#m5Kb?BZlHxH(!5{Sk=_wfmG7V$)RaqrQ~^?W-gzJV-eYQ zfi%iU>8bp6^~8!*A_18+?pXCogb*uMy!fwO^38pJZ~x~ee~U!Pq?E$c>5t8k z%CT!AN{94q`(*tYzh2$CR`pZ2J?X-)RL83#9C*;gtDnkct9GqQqIe^Dy4 zCR&Ed+1FpY9iD65S2^@&>WtqP8^eh?OePc4JRPQ&2e0ebQ$VlhbZfelCb_jKLEhjX zeRT^XneTNs8+W(67NgP?ld>3&%!pMO4rS9(9%uNq3i-v*SxM|E z$w^4_^G57JybIpmt<8^^{(Ne_qx#@$;6Ap!F_Va&i->)@_Sv$y2ftA8TM{;#tRc6s 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b/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence.tex new file mode 100644 index 0000000..2e49bac --- /dev/null +++ b/longest-increasing-subsequence/arthur-dolival-longest-increasing-subsequence.tex @@ -0,0 +1,42 @@ +\documentclass{maratona} + +\begin{document} +\begin{ProblemaAutor}{}{Maior subsequência Crescente}{1}{256}{} + +O problema consiste em determinar a maior subsequência crescente de uma sequência de números inteiros. +Uma subsequência é formada ao remover zero ou mais elementos da sequência original, sem alterar a ordem relativa dos elementos restantes. +A subsequência procurada deve conter pelo menos um elemento, e seus valores devem estar em ordem estritamente crescente, isto é, para todos os índices válidos \( i \) e \( j \) pertencentes à subsequência, se \( i < j \) então \( a_i < a_j \). +O objetivo é identificar essa subsequência de tamanho máximo e apresentar tanto o seu comprimento quanto os próprios elementos. + +\Entrada + +A entrada é composta por duas linhas. +A primeira linha contém um inteiro \( N \) (\( 1 \leq N \leq 1000 \)), representando o número de elementos da sequência. +A segunda linha contém \( N \) inteiros \( a_1, a_2, \ldots, a_N \) (\( -10^4 \leq a_i \leq 10^4 \)), separados por espaços, correspondentes aos elementos da sequência. + +\Saida + +A saída deve conter duas linhas. +A primeira linha deve conter um único inteiro representando o tamanho \( L \) da maior subsequência crescente. +A segunda linha deve conter \( L \) inteiros \( b_1, b_2, \ldots, b_L \), correspondentes aos elementos dessa subsequência, na mesma ordem em que aparecem na sequência original, separados por um espaço. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{5} & \texttt{5}\\ +\texttt{1~2~3~4~5} & \texttt{1~2~3~4~5}\\ +\rowcolor{gray!20}\texttt{4} & \texttt{3}\\ +\rowcolor{gray!20}\texttt{2~3~-1~4} & \texttt{2~3~4}\\ +\texttt{1} & \texttt{1}\\ +\texttt{0} & \texttt{0}\\ +\end{Exemplo} + + + +\Notas + +Para a sequência \( (1, 2, 3, 4, 5) \), toda a sequência já é estritamente crescente, portanto o tamanho da subsequência é \( L = 5 \) e ela contém os mesmos elementos. +Para a sequência \( (2, 3, -1, 4) \), uma das maiores subsequências crescentes possíveis é \( (2, 3, 4) \), com tamanho \( L = 3 \). +Para a sequência \( (0) \), há apenas um elemento, então a maior subsequência crescente é o próprio número \( 0 \), com tamanho \( L = 1 \). +\end{ProblemaAutor} +\end{document} diff --git a/longest-palindromic-subsequence/arthur-dolival-longest-palindromic-subsequence-tutorial.pdf b/longest-palindromic-subsequence/arthur-dolival-longest-palindromic-subsequence-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..83d2655e84b6227b99273ce24a9addf8f1faafba GIT binary patch literal 101646 zcma&NW2`Vt5N3I8+qP}nwr$(CZQHhOz1OyFW5300vYAPC`d3%#@003NNgq-L5iwdu zI#wvsh2_z8C{_Xn0(&DXC>|atdKptYa~BH&CRQd^g8%D4(TiEyxR^Q-(2Lm^x|oWX 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b/longest-palindromic-subsequence/arthur-dolival-longest-palindromic-subsequence.tex @@ -0,0 +1,39 @@ +\documentclass{maratona} + +\begin{document} +\begin{ProblemaAutor}{}{Maior Subsequência Palindrômica}{1}{256}{} + +O problema consiste em determinar a maior subsequência palindrômica de uma sequência de caracteres. +Uma subsequência é formada ao remover zero ou mais elementos da sequência original, sem alterar a ordem relativa dos elementos restantes. +A subsequência procurada deve ser um palíndromo, ou seja, deve ser idêntica quando lida de frente para trás ou de trás para frente. +O objetivo é identificar essa subsequência de tamanho máximo e apresentar tanto o seu comprimento quanto os próprios elementos. + +\Entrada + +A entrada é composta por duas linhas. +A primeira linha contém um inteiro \( n \) (\( 1 \leq n \leq 1000 \)), representando o número de caracteres da sequência. +A segunda linha contém uma string \( s \) de comprimento \( n \), composta por letras minúsculas do alfabeto. + +\Saida + +A saída é composta por uma linha, um único inteiro, que representa representando o tamanho \( L \) da maior subsequência palindrômica. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{5} & \texttt{4}\\ +\texttt{bbbab} & \\ +\rowcolor{gray!20}\texttt{4} & \texttt{2}\\ +\rowcolor{gray!20}\texttt{cbbd} & \\ +\texttt{1} & \texttt{1}\\ +\texttt{a} & \\ +\end{Exemplo} + + + +\Notas + +Para a sequência "bbbab", o tamanho da maior subsequência palindrômica é \( L = 4 \) e uma possível sequência é "bbbb". +Para a sequência "cbbd", a maior subsequência palindrômica é "bb", com tamanho \( L = 2 \). +Para a sequência "a", há apenas um elemento, então a maior subsequência palindrômica é o próprio caractere "a", com tamanho \( L = 1 \).\end{ProblemaAutor} +\end{document} diff --git a/maximum-subarray-sum/arthur-dolival-maximum-subarray-sum-tutorial.pdf b/maximum-subarray-sum/arthur-dolival-maximum-subarray-sum-tutorial.pdf new file mode 100644 index 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comumente conhecida como \textbf{Algoritmo de Kadane}. A ideia central é calcular, para cada posição $i$ da sequência, a maior soma de uma subsequência contínua que termina exatamente naquela posição. + +\subsection*{Modelagem} + +Dada uma sequência de entrada $A$ de tamanho $n$ (com índices de $0$ a $n-1$), queremos encontrar a maior soma possível $\sum_{k=i}^{j} A[k]$ para quaisquer índices $0 \leq i \leq j < n$. + +O algoritmo de Kadane resolve isso de forma linear, mantendo o controle da melhor subsequência encontrada até o momento. + +\subsection*{Definição da DP} + +Definimos $dp[i]$ como o valor da \textbf{maior soma de uma subsequência contínua que termina no índice $i$}. + +\subsection*{Função de Transição} + +Para calcular o valor de $dp[i]$, temos duas escolhas: + +A subsequência máxima que termina em $i$ é formada \textbf{apenas pelo elemento $A[i]$}. Isso ocorre se a soma da subsequência anterior ($dp[i-1]$) for negativa, pois estendê-la apenas diminuiria o valor. + +A subsequência máxima que termina em $i$ é uma \textbf{extensão da subsequência máxima que terminava em $i-1$}, adicionando $A[i]$. A soma seria $dp[i-1] + A[i]$. + +Selecionamos a maior dessas duas opções. Portanto, a função de transição é: + +$$dp[i] = \max(A[i], \quad dp[i-1] + A[i])$$ + +\subsection*{Casos Base} + +A primeira subsequência possível (que termina no índice $0$) deve obrigatoriamente conter $A[0]$. +O caso base é: + +$$dp[0] = A[0]$$ + +\subsection*{Resposta Final} + +É importante notar que $dp[n-1]$ é apenas a maior soma terminando na última posição, o que não é necessariamente a resposta final do problema. + +A subsequência de soma máxima pode terminar em qualquer índice $i$. Portanto, a resposta final é o \textbf{valor máximo encontrado em todo o array $dp$}. + +$$ \max_{0 \leq i < n} (dp[i])$$ + +\subsection*{Complexidade} + +\begin{itemize} + \item \textbf{Tempo:} $O(n)$, pois percorremos a sequência de entrada uma única vez para preencher o array $dp$. Encontrar o máximo do array $dp$ também leva $O(n)$, o que pode ser feito simultaneamente. + \item \textbf{Espaço:} $O(n)$ para armazenar o array $dp$. +\end{itemize} + +\textbf{Otimização de Espaço:} Note que o cálculo de $dp[i]$ depende apenas de $dp[i-1]$. Podemos otimizar o espaço para $O(1)$ mantendo apenas duas variáveis: uma para a "soma máxima terminando aqui" (equivalente a $dp[i-1]$) e outra para a "soma máxima global" (a resposta final).\end{document} diff --git a/maximum-subarray-sum/arthur-dolival-maximum-subarray-sum.pdf b/maximum-subarray-sum/arthur-dolival-maximum-subarray-sum.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5eef3356d2bacc306dc44afe13267b4fe441f7e8 GIT binary patch literal 152019 zcma&MQy&NVwr$&b%C=pnPT97td1uyq7rnYKI#6I-*cyh94~9|B%-+J)l8Bj$iH+$03m8TTD_d7HXCg)kTO(I9 zF*6efQ!^L=0T>roXEP%^7|)GsEm@aM4y3+I%{!@CF{*s9#N1?#S-J9aldJp#Jj*Ee zOl0wh)FqDozHTic60@oF96TT7qT_?%J`dsiZF4>!Rv$O+&wYzB*1@b(ebWrith6lk z_)Zb?jG8OIpXE{CqSToH+f$O=nX=h-^?vc}3C2@}<7;1Hy`B2c_lW6sz-k6yo)vhH z4>hzb>z+Y}1Mo`d*#`2l8OsV@+7F_cVpWp42=njq%J^kGR&rl~U;2y$HJdulc*j%) zN&~3$I2Zd*;?eC?|D}1oXP=GGR`CR*`>NS$n$!oSGEOSap6*E|BTSs-6sv`L8&bMg z8PruqS5-Uu)dP<~SOu^sn}=m&kDq|SKvO3a2NZ4`()tiH@$>n&{EBXsh?^VstE_5| 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subsequência escolhida deve conter pelo menos um elemento, e o objetivo é encontrar aquela cuja soma dos elementos seja máxima. + +\Entrada + +A entrada é composta por duas linhas. +A primeira linha contém um inteiro \( n \) (\( 1 \leq n \leq 10^5 \)), representando o número de elementos da sequência. +A segunda linha contém \( n \) inteiros \( a_1, a_2, \ldots, a_n \) (\( -10^4 \leq a_i \leq 10^4 \)), separados por espaços, correspondentes aos elementos da sequência. + +\Saida + +A saída deve conter um único inteiro, representando a maior soma possível obtida por uma subsequência contínua não vazia da sequência fornecida. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{5} & \texttt{15}\\ +\texttt{1~2~3~4~5} & \\ +\rowcolor{gray!20}\texttt{4} & \texttt{8}\\ +\rowcolor{gray!20}\texttt{2~3~-1~4} & \\ +\texttt{1} & \texttt{0}\\ +\texttt{0} & \\ +\end{Exemplo} + + + +\Notas + +Para a sequência \( (1, 2, 3, 4, 5) \), toda a sequência é positiva, portanto a maior soma é \( 15 \). +Para a sequência \( (2, 3, -1, 4) \), a subsequência com maior soma é \( (2, 3, -1, 4) \), totalizando \( 8 \). +Para a sequência \( (0) \), há apenas um elemento, logo a maior soma é \( 0 \). +\end{ProblemaAutor} +\end{document} diff --git a/min-sum-path-2d/arthur-dolival-min-sum-path-2d-tutorial.pdf b/min-sum-path-2d/arthur-dolival-min-sum-path-2d-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bbb2630842cfc46c3116397682a4a4abf057d790 GIT binary patch literal 94686 zcma&NQ>-{l@b`CY`}~e=+qP}nwr$(CZQHhO+kM`)|BL2r(r&VwnTx&Io#Z<^vmX+9 zVNn_eS{5jh`Q?#yC>DHrd^Bo?pzu7m-QAaj$#l5tspI?OmWJYmG3|K|b$9zdUb7gX3lm6!^tJo?2{4NV zPC`?#5Q277((Rh8bu4b!ZZ)@P=$t~wnz)**zML8*(P5NoXnb7ht(02)+-#ZJJZ@d_ z<@KQH=MjyPw7jAR%d-s#A{O^}ByP0<^=jPM(hqg?vumAlekDQ^FdCUD5K;}j$ZT!b zRGQ~LMggHBQ3twsH{|Q5BH6v$MTHo{KLEcqTeIry*iii`*9^G;-#Jw_*;r}*RYJax zAtNZ@PaEKx`QfZinY3=ZY>@_tJ6>(R50q-E>bM0_I9V~!XDQFdWW^KDD@ss4BWmsYjC7~S3?1c#AsK;NZ=r$zaiw^adSvxLGD2o_)eB=c+I%YM1k 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Sapo, seu programa deve imprimir 'impossible', sem as aspas simples. \ExemploEntrada + \begin{Exemplo} \texttt{4~4} & \texttt{14}\\ \texttt{1~1~4~2} & \texttt{impossible}\\ diff --git a/ra-preguicosa/src/tle.cpp b/ra-preguicosa/src/tle.cpp new file mode 100644 index 0000000..eb6a33a --- /dev/null +++ b/ra-preguicosa/src/tle.cpp @@ -0,0 +1,104 @@ +#include + +#define endl '\n' +#define INF 0x3f3f3f3f + +using namespace std; + +typedef pair pii; + +const int MAXN = 1e3 + 10; + +int C, L, W; +int Cr, Lr, Cs, Ls; +const int COSTS[5][5] = { + {7, 6, 5, 6, 7}, + {6, 3, 2, 3, 6}, + {5, 2, 0, 2, 5}, + {6, 3, 2, 3, 6}, + {7, 6, 5, 6, 7}, +}; +bitset alagado[MAXN]; +int g[MAXN][MAXN]; + +bool valid(int c, int l) { + return c > 0 && c <= C && l > 0 && l <= L && alagado[c][l] == 0; +} + +int h(int c, int l) { + return 0; +} + +void solve() +{ + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + g[i][j] = INF; + } + } + + priority_queue< + pair, + vector>, + greater<> + > pq; + + g[Cr][Lr] = 0; + pq.push({h(Cr, Lr), {Cr, Lr}}); + + while (!pq.empty()){ + auto [curCost, curCoords] = pq.top(); + pq.pop(); + + auto [c, l] = curCoords; + + if (curCost != g[c][l] + h(c,l)) continue; + + if (c == Cs && l == Ls){ + cout << g[c][l] << endl; + return; + } + + for (int i = -2; i <= 2; i++){ + for (int j = -2; j <= 2; j++){ + int nc = c + i, nl = l + j; + if (!valid(nc, nl)) continue; + + int cost = COSTS[i+2][j+2]; + int newG = g[c][l] + cost; + + if (newG < g[nc][nl]){ + g[nc][nl] = newG; + pq.push({newG + h(nc, nl), {nc, nl}}); + } + } + } + } + + cout << "impossible" << endl; +} + +int main() +{ + ios::sync_with_stdio(false); + cin.tie(nullptr); + + while (cin >> C >> L, C && L){ + cin >> Cr >> Lr >> Cs >> Ls; + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + alagado[i][j]=0; + } + } + cin >> W; + for (int i = 0; i < W; i++){ + int C1, C2, L1, L2; + cin >> C1 >> L1 >> C2 >> L2; + for (int k = C1; k <= C2; k++) + for (int l = L1; l <= L2; l++) + alagado[k][l] = 1; + } + solve(); + } + return 0; +} \ No newline at end of file diff --git a/ra-preguicosa/src/tle2.cpp b/ra-preguicosa/src/tle2.cpp new file mode 100644 index 0000000..9df82af --- /dev/null +++ b/ra-preguicosa/src/tle2.cpp @@ -0,0 +1,118 @@ +#include + +#define endl '\n' +#define INF 0x3f3f3f3f + +using namespace std; + +typedef pair pii; + +inline int get_int(){ + int ch, i; + while (((ch = getchar()) == ' ') || (ch == '\n')); + for (i = 0; ch >= '0' && ch <= '9'; ch = getchar() ) + i = 10 * i + (ch - '0'); + return i; +} + +const int MAXN = 1e3 + 10; + +int C, L, W; +int Cr, Lr, Cs, Ls; +const int COSTS[5][5] = { + {7, 6, 5, 6, 7}, + {6, 3, 2, 3, 6}, + {5, 2, 0, 2, 5}, + {6, 3, 2, 3, 6}, + {7, 6, 5, 6, 7}, +}; +bitset alagado[MAXN]; +int g[MAXN][MAXN]; + +bool valid(int c, int l) { + return c > 0 && c <= C && l > 0 && l <= L && alagado[c][l] == 0; +} + +int h(int c, int l) { + return 0; +} + +void solve() +{ + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + g[i][j] = INF; + } + } + + priority_queue< + pair, + vector>, + greater<> + > pq; + + g[Cr][Lr] = 0; + pq.push({h(Cr, Lr), {Cr, Lr}}); + + while (!pq.empty()){ + auto [curCost, curCoords] = pq.top(); + pq.pop(); + + auto [c, l] = curCoords; + + if (curCost != g[c][l] + h(c,l)) continue; + + if (c == Cs && l == Ls){ + cout << g[c][l] << endl; + return; + } + + for (int i = -2; i <= 2; i++){ + for (int j = -2; j <= 2; j++){ + int nc = c + i, nl = l + j; + if (!valid(nc, nl)) continue; + + int cost = COSTS[i+2][j+2]; + int newG = g[c][l] + cost; + + if (newG < g[nc][nl]){ + g[nc][nl] = newG; + pq.push({newG + h(nc, nl), {nc, nl}}); + } + } + } + } + + cout << "impossible" << endl; +} + +int main() +{ + ios::sync_with_stdio(false); + cin.tie(nullptr); + + while (C=get_int(), L=get_int(), C && L){ + Cr = get_int(); + Lr = get_int(); + Cs = get_int(); + Ls = get_int(); + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + alagado[i][j]=0; + } + } + W = get_int(); + for (int i = 0; i < W; i++){ + int C1, C2, L1, L2; + C1 = get_int(); + L1 = get_int(); + C2 = get_int(); + L2 = get_int(); + for (int k = C1; k <= C2; k++) + for (int l = L1; l <= L2; l++) + alagado[k][l] = 1; + } + solve(); + } + return 0; +} \ No newline at end of file diff --git a/ra-preguicosa/src/tle3.cpp b/ra-preguicosa/src/tle3.cpp new file mode 100644 index 0000000..3be7dce --- /dev/null +++ b/ra-preguicosa/src/tle3.cpp @@ -0,0 +1,118 @@ +#include + +#define endl '\n' +#define INF 0x3f3f3f3f + +using namespace std; + +typedef pair pii; + +inline int get_int(){ + int ch, i; + while (((ch = getchar()) == ' ') || (ch == '\n')); + for (i = 0; ch >= '0' && ch <= '9'; ch = getchar() ) + i = 10 * i + (ch - '0'); + return i; +} + +const int MAXN = 1e3 + 10; + +int C, L, W; +int Cr, Lr, Cs, Ls; +const int COSTS[5][5] = { + {7, 6, 5, 6, 7}, + {6, 3, 2, 3, 6}, + {5, 2, 0, 2, 5}, + {6, 3, 2, 3, 6}, + {7, 6, 5, 6, 7}, +}; +bitset alagado[MAXN]; +int g[MAXN][MAXN]; + +bool valid(int c, int l) { + return c > 0 && c <= C && l > 0 && l <= L && alagado[c][l] == 0; +} + +int h(int c, int l) { + return 0; +} + +void solve() +{ + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + g[i][j] = INF; + } + } + + priority_queue< + pair, + vector>, + greater<> + > pq; + + g[Cr][Lr] = 0; + pq.push({h(Cr, Lr), {Cr, Lr}}); + + while (!pq.empty()){ + auto [curCost, curCoords] = pq.top(); + pq.pop(); + + auto [c, l] = curCoords; + + if (curCost != g[c][l] ) continue; + + if (c == Cs && l == Ls){ + cout << g[c][l] << endl; + return; + } + + for (int i = -2; i <= 2; i++){ + for (int j = -2; j <= 2; j++){ + int nc = c + i, nl = l + j; + if (!valid(nc, nl)) continue; + + int cost = COSTS[i+2][j+2]; + int newG = g[c][l] + cost; + + if (newG < g[nc][nl]){ + g[nc][nl] = newG; + pq.push({newG, {nc, nl}}); + } + } + } + } + + cout << "impossible" << endl; +} + +int main() +{ + ios::sync_with_stdio(false); + cin.tie(nullptr); + + while (C=get_int(), L=get_int(), C && L){ + Cr = get_int(); + Lr = get_int(); + Cs = get_int(); + Ls = get_int(); + for (int i = 1; i <= C; i++){ + for (int j = 1; j <= L; j++){ + alagado[i][j]=0; + } + } + W = get_int(); + for (int i = 0; i < W; i++){ + int C1, C2, L1, L2; + C1 = get_int(); + L1 = get_int(); + C2 = get_int(); + L2 = get_int(); + for (int k = C1; k <= C2; k++) + for (int l = L1; l <= L2; l++) + alagado[k][l] = 1; + } + solve(); + } + return 0; +} \ No newline at end of file diff --git a/rod-cutting/arthur-dolival-rod-cutting-tutorial.pdf b/rod-cutting/arthur-dolival-rod-cutting-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a0f303ddc0c759780ee4b984c167008296871140 GIT binary patch literal 128068 zcmb5UQ;;r9)Mi<>ZQHhO+qP|b{uSS5HP>X2y=SGZ(p% zm;^mD13L`);>y?t3_B4Mk)w$X3?Cm1qnx>erJEHI3ky3J(f=+OMhR;>H**&vMhQD( zH*+y_Q%5s%7y$toS2q`PV|y5{tyOL5gza{e{x8j05q-sbxjj;nXqjnrrfCTCA@>x9|$?UP&@GX_wzDHg3G?$KI=q#tG#^-0mw|Or~aI0~1(T_yYdkQB9{= 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z#PegMCyQ&Wl{|KL{On|Tm!k&q=b0WWn~k?0etBHGk@;m5MV{+8=s`1P*tg3LVB;Q> z(|*f_6Qk=yFO|*;1D{nJ`<3KT)%^80b#^gya)H34COVLfo0FRxotj!qK^*-*00Zfm AQ~&?~ literal 0 HcmV?d00001 diff --git a/rod-cutting/arthur-dolival-rod-cutting-tutorial.tex b/rod-cutting/arthur-dolival-rod-cutting-tutorial.tex new file mode 100644 index 0000000..92d7d8c --- /dev/null +++ b/rod-cutting/arthur-dolival-rod-cutting-tutorial.tex @@ -0,0 +1,55 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} +\usepackage{graphicx} +\input{statement/preamble.tex} + +\title{ Tutorial: Barras e Barras} +\author{} +\date{} +\begin{document} +\maketitle +\section{Solução do Problema} + +O problema do corte de barras (\textit{Rod Cutting}) pode ser resolvido de forma eficiente por meio de \textbf{programação dinâmica}. +A ideia é decompor o problema em subproblemas menores: para cada comprimento \(L\) queremos saber qual é a melhor receita máxima que podemos obter cortando (ou não cortando) uma barra de comprimento \(L\). + +\subsection{Definição do Subproblema} + +Seja \(p[1 \dots n]\) o vetor de preços onde \(p[i]\) é o preço de vender um pedaço de comprimento \(i\). +Definimos: + +\[ +dp[L] = \text{o maior receita obtida cortando uma barra de comprimento } L. +\] + +Assim, \(dp[L]\) representa a resposta ótima para uma barra de tamanho \(L\). + +\subsection{Função de Transição} + +Para determinar \(dp[L]\) consideramos a primeira peça que retiramos da barra: se escolhemos cortar uma peça de tamanho \(x\) (com \(1 \le x \le L\)), obtemos imediatamente o preço \(p[x]\) e ficamos com o subproblema da barra restante de tamanho \(L-x\), cujo melhor valor é \(dp[L-x]\). Logo: + +\[ +dp[L] = \max_{1 \le x \le L} \big( p[x] + dp[L-x] \big). +\] + +Em particular, se não fizermos cortes (i.e., vendermos a barra inteira), isso corresponde ao caso \(x=L\) e é coberto pela fórmula acima. + +\subsection{Casos Base} + +O caso base é natural: + +\[ +dp[0] = 0, +\] + +isto é, uma barra de comprimento zero tem receita zero. + +\subsection{Recuperação da Decomposição (pedaços)} + +Para além do valor máximo \(dp[n]\), frequentemente queremos também uma decomposição concreta (lista de comprimentos dos pedaços) que atinja esse valor. Para isso mantemos, durante a computação de \(dp\), um vetor auxiliar \texttt{cut[L]} que armazena o tamanho \(x\) do primeiro pedaço que produz o ótimo para \(dp[L]\). Após preencher \(dp\) e \texttt{cut}, reconstruímos os pedaços repetindo a operação: tomar \(x=\texttt{cut}[L]\), imprimir \(x\), e reduzir \(L \leftarrow L - x\), até \(L=0\). +\end{document} diff --git a/rod-cutting/arthur-dolival-rod-cutting.pdf b/rod-cutting/arthur-dolival-rod-cutting.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9920bf0253542f962328bff2f6e475d745663257 GIT binary patch literal 172033 zcma&ML$EMVvt@a0+qP|e*S2ljwr$(CZQHhOqyDPu7ttNBBD&5f&wLM7=Gvt4!lJZ{ zbgWRM^UEXaP^<(D1a^j&P&_` zgh;$Kv=Dgu#1)p!Y|Ws-Nt$fa&^0TPZE1^}@fTuB;UkvZbm6m+A_26B>ks{RuT&dh%8wH^qXsQ4FN zS`th0;qSiH>>^Q|Se9ac&R&1U|Dc}F!s5$uvl&X^vy=v2$j0u|Vkdko!o`rU<@g8< z5&l?6rR9v6QuVB`0Rkdaz|{8~5!a6}JPvkkYx7C^ ztda~;0vuNY9fbT^0B@sX;2Zgd42CaDF@NA~DsipeW{;Q!trktQX0ozRw;NdMRuD<` z%IJPLa>unk>$}tM#!)@WN*dx)`PorGp_g>8&!md6fz9^{c!OcasuKdo{J1}$A+R=|2HqLiK5`+o z6w@_>>tCUxBvgyZg;4WxeP44u#U{b>45oju$Z4G;jg$e-JhjPu}K~zFft%9ON;Y&%)G}7(OVSC?>CUhSO zfJsK0fXefQOM$NdSywx1S&3%n8 zqtfEgP$V&QYaH*67cRAo+=i+Z1^ivq59YQaMBzQF51^NgW18RfMAB2=E$`S(o*1WT zog@_8yu>zH9qQh0_qgphjmf=w7Wbyn*t9KXs?kb@xqYiW``Y*jt0UvAkdvb`V^YmW zM8M1@)!~smlMvqPPrKtu1ZquMz&OmWZ>o1}Zi;g|xO2flV?@35+RUigq1#fN1O(!a zS1MspYR)_LgNr*zx1l2BtSd8=PC3F!bd(LT@zIe@p%w(DKPJNT>jBI#TxSb8#eg{X zrd+r2M9UV7#m~Zz>_A&n1;myZ^`y#bb@YyzFYwL*6j|bDIiKI~_v+(;N~Dx9TheU!v*O6*JNONCg(lNa&Gh>QtLNKy;%YlHy^WJ2_v zX%LbN-m#|`w}{|xEh~hPy>F=vmo!GgB75Kx;7{q~PY<81u!6zV6*``tMC2lXz%n*%?OvsXPNKn^+<>5iD!lxvtOK-HmfNE33-JsLNmGtZ6(mAt$*kyaPKte2}F8AsK= z7({)K@K#w~sJAk5U`-40Q&AOkXyj^Lx zbV{bznm!{#S_)sw^6`3mbAIl0`^%InppIz>HGraUV3Z>uzIt`=-rkx?uxANSJs=a2 z4kbAcid04x8%bLe*E#bdrZ^#J8Qs(O1eRmz+#^1pCNO|&hgPnQ9vLeEaoQDrDi#&> z?t93B`d4POsa|;`{;`?zNH*aRyvwfjfk6^*CM)!mD);r<)(ZFX z(%0&Em-CT-)5R{U*{;{&#+nwWrrH`|s|Els@%corFft7sL$U zWE>nw08`^Q1j}j-ETFPc85PKP5-HJPX!Ll%y9<20n0{Q%cFI^gDA$h8$UKGg4?X}{ zU`g$0KkVj*IjGtgX=&$~1D21+Nc9t(pF2!6M@$aYpX}SPOWj6CDf%L}{(tAV;oTi` z3uC6C>Xc+YC`&sd^FhHGcc$^^;}ZX{@gtCowvNlRPXk<@!(ddL?e`ox3W`0Nn9Dq+ zId311lgyz)5V~R5Y~`U4X!tnyXP(ISrNJ`L{vM1u;_UYnB{sQy*Vij5V-Dg&{kfmP zZ$+M=g5nBlTAKRyXwbHf@|2bfOk#gKf`3u$j7<5z#NLNiM*aXhn|<$~Ol*z+AIkgB z^gjvlKT!DJP?3d={eP1p!T&!h`Y7=_Z4)5eyiijtte4#r$|_zA%bISRRL`CP<#FKS z0h$TOj`s28i;S>V=b+B(nfxQeerI8Q!Ti~?j(;peVFsS0A${d3z>|0_pKA)Rco}mYp z2J;gw-h=#TRKZw++6hlB`H0G`+(mxt;Sk2?dm*6fBs+Qbk*aScJ#Kz%p+w^K;k!!_ zJ~fnk7rdWK=M?u^QY*D`-}i_|z*WgVP82Z10eKEVqjnhx9H+*}i{9<~YV1^5m^cIW zOAJT^Tplt%XmZf%O$s>b#$rcI82(*8J6jq&_7-&z1G_KVNxhC^jxpiGw8`{XefZxA z&Y!Xw5iOl1D?Dv@l=IpD18j-N;}+;iZ2Ch?kvl*r9ycU0Ltq)r0j@m&?v=d7Dt(k+>|Pv2^=)qO~?2lNjV z7i6h0(v<;Dvi~<^J-{%h^rjnAp39`K^g%KC-PwFb z#4bbr2eE}_cQ&N{16)GBrSab*!p#2PBEm?($il|>U%Lep0UJ9r=YReF6R7_h3E0`0 znf`B=ju<1c72O2m}Zcl7wBawB0O4;Sx6lN<~nR z3MyPmMdf&1klyo$ZLhhXzm{G*lNxp})7%W+rn8Kk?riHf*bmg}cs2%=V(L_U8h}I) zii>k||9=1o;^F`xNF-;+!Tk*o>wBJ#vPKwKz+#<}zuzlXJE~DlHk8llY+jDOUA_52%)8d4_e2PKXT8Po0kwSuay1PRN z=r90@HwV$o@%n?~TtP7S8|a(6rQ`0aO#tYWLto2T03BxogcIXDr41lh#J%_z8USPi z0k(y77~0dtY2WCZ1K7Ix7lg72z=JydKv(~u@8j*y9RTzL?fZp%9sN)vY`?TIFR!9q z1O|3`4(Q+lyxN1n_`o9eC0t@Ke_yQI{z#yH$yMzrQ%46f~yIsorR|fI{L{{C) z`Cw?V;Dn4o=$#p4B$3JR^=cqj)0yP{>%3;QaZXC|;K);uD%+`v-tL62P zdbL;O{M0dB;}!Ue*4np;F~FYiV`N8u?{R@+pb-Eep(7)q1NLtNFvw|%?bpE)=u)iP z5x=KlsRiZ1Lq!0sA5Hpy4BQyB(>vnBEaQUp7vc|md;c!o%SGV8fUOVMGyq_y18=8% z?=3i3rvK4_us(^CKe`TrX{1Yq1AyvuXA6*};He23SARr4a*SF6|iIE1^r%ij5m_y3v; 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znBp5$T~S>ONcO-#>z{pCiUEh+G^n RC@9E*VGJT7S_o~1{{p15BUJzZ literal 0 HcmV?d00001 diff --git a/subset-sum/arthur-dolival-subset-sum-tutorial.tex b/subset-sum/arthur-dolival-subset-sum-tutorial.tex new file mode 100644 index 0000000..59147ac --- /dev/null +++ b/subset-sum/arthur-dolival-subset-sum-tutorial.tex @@ -0,0 +1,45 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} +\usepackage{graphicx} +\input{statement/preamble.tex} + +\title{ Tutorial: Prateleira Dourada} +\author{} +\date{} +\begin{document} +\maketitle +\section{Solução do Problema} + +O problema de organizar os livros para preencher exatamente a prateleira é uma aplicação clássica do problema da \textbf{Soma de Subconjuntos} (\textit{Subset Sum}), que é uma variação do problema da Mochila (0/1 \textit{Knapsack}). Podemos resolvê-lo de forma eficiente utilizando \textbf{programação dinâmica}. A ideia é rastrear todos os comprimentos parciais possíveis que conseguimos formar com as combinações de livros avaliados até o momento. + +\subsection{Definição do Subproblema} + +Seja $T$ o comprimento total da prateleira. +Definimos o nosso estado da programação dinâmica como um vetor booleano: + +$$dp[j] = \text{1 se for possível formar o comprimento exato } j \text{ com algum subconjunto dos livros.}$$ + +O nosso objetivo final é descobrir se, após testar todas as combinações de livros, o estado $dp[T]$ se torna verdadeiro. + +\subsection{Função de Transição} + +Para atualizar os nossos estados, iteramos e avaliamos um livro de cada vez. Seja $x$ a espessura do livro atual. +A lógica é simples: se já descobrimos que é possível formar uma pilha de livros com espessura exata $j$ (ou seja, $dp[j] == 1$), então, ao colocar esse novo livro na pilha, também seremos capazes de alcançar a espessura $j + x$. + +A transição, pode ser descrita como: + +$$dp[j + x] = dp[j + x] \lor dp[j]$$ + +\textbf{Detalhe crucial de implementação:} Para garantir que o bibliotecário use cada livro \textbf{no máximo uma vez}, precisamos percorrer as capacidades de trás para frente (começando em $T - x$ e diminuindo até $0$). Se iterássemos da esquerda para a direita, um livro poderia ser considerado mais de uma vez, o que não é o correto nesse problema. +\subsection{Casos Base} + +O caso base representa a situação em que não selecionamos livro algum. A soma das espessuras de uma prateleira vazia é zero. Portanto, é sempre possível "formar" o comprimento $0$: + +$$dp[0] = 1$$ + +Todos os outros comprimentos no vetor de $1$ a $T$ nascem como $0$ (falsos/impossíveis), pois ainda precisamos descobrir se eles podem ser alcançados utilizando os livros da pilha.\end{document} diff --git a/subset-sum/arthur-dolival-subset-sum.pdf b/subset-sum/arthur-dolival-subset-sum.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e867ac7cc64345a5cde63aa459dc7eceb038600e GIT binary patch literal 174506 zcma&NQ*2xl-K^_b&#XnJ zARl48548jf<%hA-$N5 zp^K@Asj&Ipo9|0qUhGUVnjl1q zF2E#o7DhR0@$$+m>pSt+#nsO+W;W0LlnRh{Nlb8QILw)%pO;2{J4^49;oi0XHFue2 zp0;LAFpFkG$1Litxi6GqyV=D5le~d(qHm^zmHt7k)RN&U#$mnZo2HxdeJaD`jK~{l znfI4EMwHE-$u=2kz$`R*&{96T@4TvEcSTkoxUF~G{{q3cr6xgEL*J|fYFAsQHpoFV zQMX(O%-G_&PIfDXwHl5MFj@vWvE`l2bDNv1-_O|*B{UW)S8-K{Cbbp5*5>-T>aoO* zUbi;;`x1jqWd^rFXtSRR_x;J>_N`2=mwqJ#uHrOK*NiI11lmUs+vI7T4S^w*2XKM! zd3r8|a89LE@hYItikpVJsA@HMLMVexx8o)*|3HLHx94LU(pXsjY%87rEWNaBKRq{} 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+\begin{ProblemaAutor}{}{Férias}{1}{256}{Codeforces 363 (Div. 1)} + +Miguel tem \(n\) dias de férias e pretende aproveitá-los para estudar programação e também praticar esportes. Para cada um desses dias, ele sabe antecipadamente duas informações: se a academia estará aberta e se haverá uma competição online disponível. Assim, cada dia se enquadra em uma das quatro situações possíveis: sem academia e sem competição, sem academia mas com competição, com academia mas sem competição, ou com ambos disponíveis. + +Em cada dia, Miguel deve escolher exatamente uma ação: descansar, participar da competição (caso ele exista naquele dia) ou ir à academia (caso esteja aberta). Seu objetivo é descansar o mínimo possível. No entanto, ele impõe a si mesmo uma regra importante: \textbf{não repetir a mesma atividade em dias consecutivos}. Em outras palavras, Miguel não aceita fazer esporte dois dias seguidos, nem participar de competições em dias consecutivos, embora descansar possa ocorrer repetidamente sem qualquer restrição. + +Dado o calendário das férias, determine o menor número de dias de descanso que Miguel será obrigado a ter. + +\Entrada + +A primeira linha contém um inteiro positivo \(n\) (\(1 \le n \le 100\)), o número de dias de férias. + +A segunda linha contém \(n\) inteiros \(a_1, a_2, \dots, a_n\) (\(0 \le a_i \le 3\)), separados por espaços, onde cada \(a_i\) descreve as opções do \(i\)-ésimo dia: +\begin{itemize} + \item \(a_i = 0\): academia fechada e sem competição; + \item \(a_i = 1\): academia fechada e há competição; + \item \(a_i = 2\): academia aberta e sem competição; + \item \(a_i = 3\): academia aberta e há competição. +\end{itemize} + +\Saida + +A saída consiste de um único número inteiro, o número mínimo possível de dias em que Miguel terá que descansar, obedecendo à restrição de não repetir a mesma atividade em dias consecutivos. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{4} & \texttt{2}\\ +\texttt{1~3~2~0} & \\ +\rowcolor{gray!20}\texttt{7} & \texttt{0}\\ +\rowcolor{gray!20}\texttt{1~3~3~2~1~2~3} & \\ +\texttt{2} & \texttt{1}\\ +\texttt{2~2} & \\ +\end{Exemplo} + + + +\Notas + +No primeiro caso de teste, Miguel pode participar da competição no dia 1 e ir à academia no dia 3. Descansando nos dias 2 e 4. + +No segundo caso de teste, Miguel pode participar da competição nos dias 1, 3, 5 e 7 e ir à academia nos demais. + +No terceiro caso de teste, como Miguel não pode ir à academia em dois dias seguidos ele será obrigado a descansar em um dos dias.\end{ProblemaAutor} +\end{document}

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e o comprimento de cada pedaço resultante deve ser um número inteiro de polegadas. + +É fornecida uma tabela de preços \( p_i \), onde \( p_i \) representa o preço de venda de uma barra de comprimento \( i \). + +O objetivo é determinar qual é a \textbf{maior receita possível} ao cortar (ou não cortar) a barra original, bem como uma decomposição válida cujos comprimentos somados sejam exatamente \( n \) e cujo valor total seja igual à receita máxima. + +\Entrada + +A entrada é composta por duas linhas. +A primeira linha contém um inteiro \( n \) (\( 1 \le n \le 1000 \)), representando o comprimento da barra de aço. +A segunda linha contém \( n \) inteiros positivos \( p_1, p_2, \ldots, p_n \), onde \( 1 \le p_i \le 10000 \), e \( p_i \) indica o preço de uma barra de comprimento \( i \). + +\Saida + +A saída é composta por duas linhas. +Na primeira linha deve ser impresso um inteiro representando a \textbf{receita máxima} que pode ser obtida cortando a barra. +Na segunda linha deve ser impressa uma sequência de inteiros positivos representando os comprimentos dos pedaços utilizados, cuja soma deve ser exatamente \( n \). + +É permitido imprimir os pedaços em qualquer ordem. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{8} & \texttt{22}\\ +\texttt{1~5~8~9~10~17~17~20} & \texttt{2~6}\\ +\rowcolor{gray!20}\texttt{4} & \texttt{5}\\ +\rowcolor{gray!20}\texttt{1~2~3~5} & \texttt{4}\\ +\texttt{8} & \texttt{24}\\ +\texttt{3~5~8~9~10~17~17~20} & \texttt{1~1~1~1~1~1~1~1}\\ +\end{Exemplo} + + + +\Notas + +No primeiro caso, a barra possui tamanho 8 e os preços fornecidos indicam o valor de venda de cada tamanho possível. A saída informa que a decomposição escolhida foi em pedaços de tamanhos 2 e 6, que somam exatamente o tamanho total da barra. O preço do pedaço de tamanho 2 é 5 e o preço do pedaço de tamanho 6 é 17, totalizando 22. Não existe nenhuma outra forma de cortar a barra que produza um valor maior do que esse, portanto a solução apresentada é ótima. + +No segundo caso, a barra possui tamanho 4. O preço de venda para a barra inteira de tamanho 4 é 5. Qualquer tentativa de cortar a barra em pedaços menores resulta em um valor total inferior, já que combinações como 1+3, 2+2 ou quatro pedaços de 1 produzem lucro máximo igual a 4. Assim, o melhor é não realizar nenhum corte e vender a barra inteira. Dessa forma, o valor ótimo é 5 e a decomposição correspondente consiste apenas no pedaço de tamanho 4. +\end{ProblemaAutor} +\end{document} diff --git a/subset-sum/arthur-dolival-subset-sum-tutorial.pdf b/subset-sum/arthur-dolival-subset-sum-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..63f69b1a062b27d402ff861bcdcc42c0e0e689e7 GIT binary patch literal 88458 zcma&NL$EMR&}MmU+qP}nwr$(C-fP>oZQHhOW4@l4KVl}P7oCfW%uQ`-bMmP;qzWQp zw2XADP^9zABkNGC1PlcBMpjTfJW%v9rgr8o76eSp%p3&&cR|sMS=zXmIuX!|*%-Q* zikKSPo0vlJ@j*GeIGGySLV0ZVXiERvZb$6>QipjJA+*b&?}2%*#~up@8Nj;VUfm5? zn!&U#O!azUh~d1pmq)ECPxWGX1=*O1v`Hk=9I349t^B?C9I0|2)u|v_p=wfJYg5-c zt)Pvr;-Obo9hg73M0deDvN>jzJ*FjZ7z~>le&=cInTT(VquynM)@j6=_A)ngT`ZUn0 zIP7SACsX2?LffiF|4=3M%{D+QOS(3x0!oc+@|&375TMz}*ObDiNV(0Yq`H2V#{ezg zG4}SY!_E^uIH7JO+?xX?J*(Q`7v|r5%hLYS$1c*uMQhuFLM!Wrnf;3QuXqXmH)7P> ziPnqXWd=R_Gi7*(^&FrB>Mr!gcC84ZeceBK^7g8U_3)cz6&Axo8qC=cM(a#kHdyY~ z%bjG$R7&)33R+E{4$d`e-*j*tamI0q1*Y8#(cKi#zgT(|T>r>7R+E=Vuf=_6-TPj& 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peso[i] <= w: + dp[i][w] = max(dp[i-1][w], dp[i-1][w - peso[i]] + valor[i]) + else: + dp[i][w] = dp[i-1][w] +\end{verbatim} + +O resultado final é dado por $dp[n][W]$, que representa o maior valor possível que pode ser obtido sem ultrapassar a capacidade da mochila. +\end{document} diff --git a/knapsack-zero-one/arthur-dolival-knapsack-zero-one.pdf b/knapsack-zero-one/arthur-dolival-knapsack-zero-one.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ae35a87316c25a5463a5ac9ed41a7716b6ea2454 GIT binary patch literal 158485 zcma&MQ;;wKuVC4>ZQHhO+qP}nwr$()uWj45J^#+`o!Y89RWo@?DtXMqN$QX)h=|cL z(y>C3&M%LyL$MMt5ZD`8LGkcF(aV_HnY&mJFflT*6a3GBq8GEYaWQowpck_-bTJh% zHMTb~h2rCba&~btHME8D*sRr*PTpce>bp|Emsrt)*o`l7S|^dC+R!DZlvjnz#IcAx zDkP3VR&>(&c-ewwp>zT(_WU4JxqW|l7w|w|CdRP-0 zIP5{Nq@9tuAMMip{spIJ6Zos#mjS%E`?rF&NnzF!^tW|*bGQadesb$gR%%PgqKviD zLsg(@%1$Ij=@kkS*Mpl+`Iv_=QP?EaVBSCVV|`PldR;Q)lm` zA`7M;#82d^jDP97bzS9KJ}(7d4OaD7S~7ZGn@Y?JB}X7g;NI(k)WCWNeTD0;FHvM{b*Kwe(V- 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b/knapsack-zero-one/arthur-dolival-knapsack-zero-one.tex @@ -0,0 +1,51 @@ +\documentclass{maratona} + +\begin{document} +\begin{ProblemaAutor}{}{Knapsack Problem}{1}{256}{} + +O problema consiste em determinar o maior valor total que pode ser obtido ao selecionar um subconjunto de itens para colocar em uma mochila com capacidade limitada. +Cada item possui um peso e um valor associados, e a mochila só pode suportar um peso total máximo. +O objetivo é escolher um conjunto de itens de forma que a soma de seus pesos não ultrapasse a capacidade máxima da mochila e que a soma de seus valores seja a maior possível. +Cada item pode ser escolhido no máximo uma vez. + +\Entrada + +A entrada é composta por \( N + 1 \) linhas. +Na primeira linha, há dois inteiros \( N \) e \( W \) (\( 1 \leq N \leq 100 \), \( 1 \leq W \leq 10^5 \)), representando respectivamente o número de itens e a capacidade máxima da mochila. +Cada uma das próximas \( N \) linhas contém dois inteiros \( w_i \) e \( v_i \) (\( 1 \leq w_i \leq W \), \( 1 \leq v_i \leq 10^9 \)), representando respectivamente o peso e o valor do \( i \)-ésimo item. + +\Saida + +Imprima um único inteiro representando o valor total máximo que pode ser obtido sem exceder a capacidade da mochila. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{3~8} & \texttt{90}\\ +\texttt{3~30} & \\ +\texttt{4~50} & \\ +\texttt{5~60} & \\ +\rowcolor{gray!20}\texttt{5~5} & \texttt{5}\\ +\rowcolor{gray!20}\texttt{1~1} & \\ +\rowcolor{gray!20}\texttt{1~1} & \\ +\rowcolor{gray!20}\texttt{1~1} & \\ +\rowcolor{gray!20}\texttt{1~1} & \\ +\rowcolor{gray!20}\texttt{1~1} & \\ +\texttt{6~15} & \texttt{17}\\ +\texttt{6~5} & \\ +\texttt{5~6} & \\ +\texttt{6~4} & \\ +\texttt{6~6} & \\ +\texttt{3~5} & \\ +\texttt{7~2} & \\ +\end{Exemplo} + + + +\Notas + +Para o conjunto de itens \( (w_i, v_i) = \{(3, 30), (4, 50), (5, 60)\} \) e capacidade \( W = 8 \), a melhor escolha é pegar os itens de peso 3 e 5, totalizando valor 90. +Para o conjunto de itens \( (1, 1), (1, 1), (1, 1), (1, 1), (1, 1) \) e capacidade \( W = 5 \), todos os itens podem ser colocados, resultando em valor total \( 5 \). +Para o conjunto \( (6, 5), (5, 6), (6, 4), (6, 6), (3, 5), (7, 2) \) e capacidade \( W = 15 \), a melhor combinação alcança valor máximo 17. +\end{ProblemaAutor} +\end{document} diff --git a/longest-common-subsequence/arthur-dolival-longest-common-subsequence-tutorial.pdf b/longest-common-subsequence/arthur-dolival-longest-common-subsequence-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c3311063817fd990ed45e3fb0de124f2951b8f0c GIT binary patch literal 122094 zcma&NL$ELalWciy+qP}n_^xf+wr$(CZQHhO>-}$HdSW8F7hScfTGt{oBTkVjh=|cL z(y>C3&M%LwL$MMt5ZD=7Lh#G%<>c&WVqgR1zImf59k<<%*!!jStQaHX^ij$&iq|a%L$a<9JiTpApKq~X zgh^|mddU`u^SGrnA}*b2FgKL@G8m<;i6YuqIqvDLa-Bz~_kAxGj>;U&OsbZKpHCT1 zJ0o?K$N;VLa%tnx!DiyH{wAz+GrNW@>GCjFc@g#^rl6~hQ!DniWiaFNaHR>Q*lxR} ze6V%ZJMt^syE4AvP%xS;HRfprZZUHN;JIuBoVLy=L;T 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zy9g{3&3%s2(@u9a5?U_%LDf7R;5C#xNw+xLA2J|kkUgZc&@z_OAin6NbYA(M+r;6j rRN==ar6c`yesq|?*#CZ={x}CeTrkNEq=mwuQV=AqkdT^zI_ 0 && j > 0) { + + // Caso 1: os caracteres são iguais → fazem parte da LCS + if (s1[i - 1] == s2[j - 1]) { + lcs = s1[i - 1] + lcs; // adiciona no início da string + i--; + j--; + } + // Caso 2: se o valor de cima é maior, movemos para cima + else if (dp[i - 1][j] > dp[i][j - 1]) { + i--; + } + // Caso 3: caso contrário, movemos para a esquerda + else { + j--; + } +} + +\end{verbatim} + +Ao final desse processo, a string \texttt{lcs} conterá uma maior subsequência comum entre as duas strings.\end{document} diff --git a/longest-common-subsequence/arthur-dolival-longest-common-subsequence.pdf b/longest-common-subsequence/arthur-dolival-longest-common-subsequence.pdf new file mode 100644 index 0000000000000000000000000000000000000000..45507c33aaae18e7feb7b6a566e6df24bb6ec8a7 GIT binary patch literal 186279 zcma&sV{}%+kii)QNyz%*N2g 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comprimento da \textbf{maior subsequência comum} (Longest Common Subsequence - LCS) entre duas strings fornecidas. +Uma \textit{subsequência} é uma sequência que pode ser obtida a partir de uma string original removendo-se zero ou mais caracteres, sem alterar a ordem relativa dos caracteres restantes. +Dadas duas strings \( s_1 \) e \( s_2 \), deseja-se determinar \textbf{o comprimento da maior subsequência comum} e também \textbf{uma subsequência comum de comprimento máximo} presente em ambas. + +\Entrada + +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. + +\Saida + +A saída consiste em duas linhas. +Na primeira linha, deve ser impresso o comprimento da maior subsequência comum. +Na segunda linha, deve ser impressa uma subsequência comum de comprimento máximo. + +\ExemploEntrada + +\begin{Exemplo} +\texttt{5~3} & \texttt{3}\\ +\texttt{abcde~ace} & \texttt{ace}\\ +\rowcolor{gray!20}\texttt{3~3} & \texttt{3}\\ +\rowcolor{gray!20}\texttt{abc~abc} & \texttt{abc}\\ +\texttt{3~3} & \texttt{0}\\ +\texttt{abc~hhh} & \texttt{}\\ +\end{Exemplo} + + + +\Notas + +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, logo a maior subsequência comum é "abc", que possui tamanho 3. + +Para \( s_1 = abc \) e \( s_2 = hhh \), não há caracteres em comum, e portanto a maior subsequência comum é a palavra vazia, que tem tamanho 0. +\end{ProblemaAutor} +\end{document} diff --git a/longest-increasing-subsequence-II/arthur-dolival-longest-increasing-subsequence-II-tutorial.pdf 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