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+\section{Solução do Problema} + +O problema consiste em encontrar o \textbf{menor custo} necessário para percorrer uma grade (\textit{matriz}) de inteiros positivos. +A grade possui \( n \) linhas e \( m \) colunas, e cada célula contém um número positivo que representa o custo de passar por ela. + +O objetivo é sair da célula \textbf{superior esquerda} da grade e chegar à célula \textbf{inferior direita}, movendo-se apenas para a \textbf{direita} ou para \textbf{baixo}. +A soma dos valores das células visitadas deve ser a menor possível. + +Este problema pode ser resolvido de forma eficiente utilizando \textbf{programação dinâmica}. + +\subsection{Definição do Subproblema} + +Seja uma matriz de custos \( C \) de tamanho \( n \times m \). Definimos: + +\[ +dp[i][j] = \text{o menor custo para chegar à célula } (i,j) \text{ a partir de } (0,0). +\] + +Ou seja, \( dp[i][j] \) representa o custo mínimo para alcançar exatamente a posição \( (i, j) \). + +\subsection{Função de Transição} + +Como só é possível mover-se para a \textbf{direita} ou para \textbf{baixo}, a célula \( (i,j) \) só pode ser alcançada a partir de: + +\begin{itemize} + \item \( (i-1, j) \): movimento vindo de \textbf{cima}; + \item \( (i, j-1) \): movimento vindo da \textbf{esquerda}. +\end{itemize} + +Assim, o custo mínimo para chegar a \( (i,j) \) é dado por: + +\[ +dp[i][j] = C[i][j] + \min(dp[i-1][j],\ dp[i][j-1]). +\] + +\subsection{Casos Base} + +\begin{itemize} + \item Primeira célula: + \[ + dp[0][0] = C[0][0] + \] + \item Primeira linha (somente movimentos para a direita): + \[ + dp[0][j] = dp[0][j-1] + C[0][j] + \] + \item Primeira coluna (somente movimentos para baixo): + \[ + dp[i][0] = dp[i-1][0] + C[i][0] + \] +\end{itemize} + +\subsection{Construção da Tabela} + +Após inicializar os casos base, preenchermos o restante da matriz \( dp \) utilizando: + +\[ +dp[i][j] = C[i][j] + \min(dp[i-1][j],\ dp[i][j-1]). +\] + +Ao término do processo, o valor da última posição: + +\[ +dp[n-1][m-1] +\] + +representa o \textbf{menor custo possível} para ir da célula inicial até a célula final.\end{document} diff --git a/min-sum-path-2d/min-sum-path-2d.pdf b/min-sum-path-2d/min-sum-path-2d.pdf index 1993289c08a329b5ea2c286c8483713ca2167496..25c845668596760520e4d9ea8a0e8348ec103ed8 100644 GIT binary patch delta 38336 zcmZs?Wmp_tw=GKJu8jqEhsNFA-Gf7L*8rh$5=gMZf;++8LvRAYo#5`lAvlNkle_P| zXaA{ZR@YOqYRx&um}^v*0Pv%^2=UbLTo6tkKogA#YM=}(@uB~C(BEfNx6%vzWGy7x zaA37)zj$!G+9t}_l=#g`(RfJh@)8ZZkk$TRMRhtxagq`_lp;8ewa(Wt;i={B#^-Ve zt8UP*`|XCG&jTMfHTM*karE&`{8Iwdw|+&D=|cZQGd`dA?A9klmu9`)Cm#ltRf?Z~ za+<|mp+7oT9y>O2iFp#K`y1+i|5$lEw@n~3f847u$BleG*K#gyur^ZMr>M^uJvrLPP;`6(;qdgjCZS#swnLCi(Oe7h?7c3 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