From 97973cdae547f0318bb942c1864f8ceca9004df0 Mon Sep 17 00:00:00 2001 From: arthur Date: Thu, 9 Apr 2026 21:55:17 -0300 Subject: [PATCH] feat: new tutorials added. --- subset-sum/problem.json | 2 +- subset-sum/statement/tutorial.tex | 30 +++++++++++++++ subset-sum/subset-sum-tutorial.pdf | Bin 0 -> 88460 bytes subset-sum/subset-sum-tutorial.tex | 43 +++++++++++++++++++++ subset-sum/subset-sum.pdf | Bin 84239 -> 87619 bytes subset-sum/subset-sum.tex | 2 +- vacations/statement/tutorial.tex | 46 ++++++++++++++++++++++ vacations/vacations-tutorial.pdf | Bin 0 -> 110992 bytes vacations/vacations-tutorial.tex | 59 +++++++++++++++++++++++++++++ vacations/vacations.pdf | Bin 87215 -> 87215 bytes 10 files changed, 180 insertions(+), 2 deletions(-) create mode 100644 subset-sum/subset-sum-tutorial.pdf create mode 100644 subset-sum/subset-sum-tutorial.tex create mode 100644 vacations/vacations-tutorial.pdf create mode 100644 vacations/vacations-tutorial.tex diff --git a/subset-sum/problem.json b/subset-sum/problem.json index 007e634..923f86a 100644 --- a/subset-sum/problem.json +++ b/subset-sum/problem.json @@ -1,7 +1,7 @@ { "version": "1.0", "problem": { - "title": "", + "title": "Prateleira Dourada", "event": "", "time_limit": 1.0, "memory_limit_mb": 256, diff --git a/subset-sum/statement/tutorial.tex b/subset-sum/statement/tutorial.tex index e69de29..16d9303 100644 --- a/subset-sum/statement/tutorial.tex +++ b/subset-sum/statement/tutorial.tex @@ -0,0 +1,30 @@ +\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. \ No newline at end of file diff --git a/subset-sum/subset-sum-tutorial.pdf b/subset-sum/subset-sum-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dd6969abc30e2982da7a31a791366a6900c27dec GIT binary patch literal 88460 zcma&NQ_L_t)M)wGwr$(CZM=_d+qP}nwr$(Ct@+Mm{$wU|F3wKc?B4XQ-D|g5qzWQp zw2XADP^9zABkNGC1PlarhL%t~JW%v9CbnkI<^)X4%xnbzcRS_3}D@FukQLU z$;PoNO80nTjN-hul|`zmNbg{63A#EFWfe=PHC$HJRr+=MydK8X+Nva2rfg(w#j$Ih zR?tRe@z5))4$L2%qB~@vl%$W=*6BVmZD?jhwe?{899`{_auc_>@5uIWw;vDP?CEtk z)R)M{%&^*aD36Bnx`jwdZz_^uU9N=7n_whGl;_%l*{K z*10=G+im+2K|f5ovl#btDB}KPTPVTc^rUGwX`K&@%RSWHuBdpU8@H}O z{n`sUr(mGlVm}ySV1cQR2R&DCOjfOEo@7dNAuOg6Rq<_ImQyR~WVknY-`7P)OG{)? 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RATJM*kp*12qG^Bv`~&^0Y{&or literal 0 HcmV?d00001 diff --git a/subset-sum/subset-sum-tutorial.tex b/subset-sum/subset-sum-tutorial.tex new file mode 100644 index 0000000..4d4cf4c --- /dev/null +++ b/subset-sum/subset-sum-tutorial.tex @@ -0,0 +1,43 @@ +\documentclass[10pt]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amsthm,amssymb} +\usepackage{fullpage} +\usepackage{url} +\pagenumbering{gobble} +\usepackage{hyperref} + +\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/subset-sum.pdf b/subset-sum/subset-sum.pdf index 5c45a4c329633f986e72a0ef60e7e6aae0b31484..00cb3576058f309ea8d50c92b2a295d76177a9a3 100644 GIT binary patch delta 10194 zcmajEbyU<}@HfsbB}=orbP7nrE)CKR(p}OaNFyCfcV84xN{~hn7ez`!5Cl;|B^0C^ zqyl|X?W7Vd0aatyUM^po3; zwqYP=Q;;ifo+M#fAKe6JvM`Ty!^QcI8gB03LfXzTGY6Fd?)tzac~r+6q2T?){?sR_ zn9FCuY?6mRQWsYns+|mY1_~eck5rzPU+jkl)a3HpzW(#@!AjF^<`-YAU#*8(-b>!^ zo?YjMtH=FcNBRedA$;#rnzJx2@7o}-6>)|MmT3!UXB zEORwlyH8%1ugV2<-ScX}=!Dfz*8COroQ~fSSs$zYks#^}UeCpA2p?@VNWHe`qCT=) zNY>cru*DKGikIpOWKHTx)_H}#Zu>IY!R&KqZCIHQ`tD_S7}L)-QJ zEtbzEGT&ASxlu6rV`47wh_^>nXLCD>L^S;)f59s6BD9Lpg}(If%v%=vO$DNTqIKw< 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