From 578019682359a013eb4f894cf4532b1d8220df0a Mon Sep 17 00:00:00 2001 From: arthur Date: Thu, 13 Nov 2025 20:44:37 -0300 Subject: [PATCH] feat: added tutorial file --- .../maximum-subarray-sum-tutorial.pdf | Bin 0 -> 114447 bytes .../maximum-subarray-sum-tutorial.tex | 60 ++++++++++++++++++ maximum-subarray-sum/maximum-subarray-sum.pdf | Bin 152043 -> 78340 bytes maximum-subarray-sum/statement/tutorial.tex | 47 ++++++++++++++ 4 files changed, 107 insertions(+) create mode 100644 maximum-subarray-sum/maximum-subarray-sum-tutorial.pdf create mode 100644 maximum-subarray-sum/maximum-subarray-sum-tutorial.tex diff --git a/maximum-subarray-sum/maximum-subarray-sum-tutorial.pdf b/maximum-subarray-sum/maximum-subarray-sum-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..74c8748d73604a64a3152bcc84661caeb56e524c GIT binary patch literal 114447 zcma&LLv$`ow6z=Cwr$(CZQHhO+uCt|)E5jmRJ!0_?GFv^)bSh`sev9NM55&b_4hEc-W&duC~h*84M 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\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). \ No newline at end of file