From 33d87c5809dddcac177f84cb2846f6f720dcc53e Mon Sep 17 00:00:00 2001 From: Daniel Saad Date: Wed, 20 May 2026 14:45:08 -0300 Subject: [PATCH] fix: inclusao-de-subintervalos solutions --- ...val-inclusao-de-subintervalos-tutorial.pdf | Bin 0 -> 81794 bytes ...val-inclusao-de-subintervalos-tutorial.tex | 49 +++++++++++++ .../inclusao-de-subintervalos-tutorial.pdf | Bin 0 -> 126447 bytes .../inclusao-de-subintervalos-tutorial.tex | 49 +++++++++++++ inclusao-de-subintervalos/src/caio.cpp | 31 ++++++++ inclusao-de-subintervalos/src/wa.cpp | 66 ++++++++++++++++++ 6 files changed, 195 insertions(+) create mode 100644 inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.pdf create mode 100644 inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.tex create mode 100644 inclusao-de-subintervalos/inclusao-de-subintervalos-tutorial.pdf create mode 100644 inclusao-de-subintervalos/inclusao-de-subintervalos-tutorial.tex create mode 100644 inclusao-de-subintervalos/src/caio.cpp create mode 100644 inclusao-de-subintervalos/src/wa.cpp diff --git a/inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.pdf b/inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..584eca62047e20d26ed124aa8526c978c50db5d4 GIT binary patch literal 81794 zcma&sQ;;x9&?e}%ZQHhO+qP}nw)<<_wr$(C?L9NGf5dL=Tx?xs)pf-ine`N@f`}L` zBONOg>HPA@Iut7b1A)Df6%-E-6upe8owWY^m zFxEF~drjIp7B>tnx+ZV!9$CkjzSdE?g#T#I22I%XldEUeV3KXLJN2`C`D`l5%9yYx zZwb*^nz~x&`gVUmZ-Gy!gJX&pciPyn7<;v(15*qRI`^h}-K{X@UO6W^-I}Pay>Z=I zPlV-M)mHQ2*fdt%wi+9BIdIu`cDQzF2Y3Ab;Gl786qu{XAb$V~5~;RGSgeW}h~HbbaWZ3`T``J3z7!ECnFk%^)P77sl*FO%|)LTT!vf3;pSwzXEf$eXT|P@SvaYS zj|TWbltm=gZgHXcMGIt2TsMWYq^8O)gds;51mLgpE+4$XI1aud2MDtiV`QVd&4Laq z)(POevJFGd!nmQM`QtB6hJ_Sw$C#HXIya@Pzi1i=u`tTck`=qvEmiGV9GNFSZG=@{VTfHrW@ z7^YGPS{)aD=4ve1?E?rDJhduwgg(VBlhUn+Eu(|E&`PLL~5!;kunl!W-mC 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literal 0 HcmV?d00001 diff --git a/inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.tex b/inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-tutorial.tex new file mode 100644 index 0000000..85ca4a8 --- /dev/null +++ b/inclusao-de-subintervalos/arthur-dolival-inclusao-de-subintervalos-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: Inclusão de Subintervalos} +\author{} +\date{} +\begin{document} +\maketitle +Nesse problema, devemos ordenar os intervalos $(a_i, b_i)$: + +\begin{itemize} + \item Em ordem crescente de $a_i$. + \item Em caso de empate, em ordem decrescente de $b_i$. +\end{itemize} + +Em seguida, devemos iterar sobre os intervalos ordenados, mantendo o maior valor de $b_i$ +visto até o momento. Se encontrarmos um intervalo $(a_i, b_i)$ tal que $b_i$ seja menor ou +igual ao maior valor de $b_i$ visto até o momento, então esse intervalo é coberto por um +intervalo anterior e podemos descartá-lo. Caso contrário, atualizamos o maior valor de +$b_i$ visto até o momento. + +O algoritmo abaixo implementa essa lógica. + +\begin{algoritmo} +\caption{\Call{interval-cover}{$I$}} +\KwIn{Lista de intervalos $I[0,n-1]$} +\KwOut{Lista de intervalos $I'$ não cobertos por nenhum outro intervalo} +$I' \gets \emptyset$\; +\Call{sort}{$I$} \tcp{Ordena os intervalos com critérios especificados} +$\max_b \gets -\infty$\; +\ForEach{$(a_i, b_i) \in I$}{ + \If{$b_i > \max_b$}{ + $I'.\Call{append}{(a_i,b_i)}$\; + $\max_b \gets b_i$\; + } +} +\Return{$I'$}\; +\end{algoritmo} + + +\section*{Análise} + +O passo de ordenação tem complexidade $\Theta(n \lg n)$ e domina o custo do algoritmo.\end{document} diff --git a/inclusao-de-subintervalos/inclusao-de-subintervalos-tutorial.pdf b/inclusao-de-subintervalos/inclusao-de-subintervalos-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ba04b3371de9f951f546ce6a2c22b0fca8b41180 GIT binary patch literal 126447 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0000000..f344978 --- /dev/null +++ b/inclusao-de-subintervalos/src/caio.cpp @@ -0,0 +1,31 @@ +#include + +using namespace std; + +int main() { + int n; + cin >> n; + + priority_queue> pq; + int a, b; + for (int i = 0; i < n; i++) { + cin >> a >> b; + pq.push({-a, b}); + } + + int count = 0; + int end = -1; + while (!pq.empty()) { + auto [_, e] = pq.top(); + pq.pop(); + + if (e > end) { + count++; + end = e; + } + } + + cout << count << endl; + + return 0; +} \ No newline at end of file diff --git a/inclusao-de-subintervalos/src/wa.cpp b/inclusao-de-subintervalos/src/wa.cpp new file mode 100644 index 0000000..0ea10e6 --- /dev/null +++ b/inclusao-de-subintervalos/src/wa.cpp @@ -0,0 +1,66 @@ +#include + +using namespace std; + +bool comp(pair &a, pair &b) { + if (a.first != b.first) { + return a.first < b.first; + } + return a.second > b.second; +} + +bool intersecting(pair &a, pair &b) { + return a.second >= b.first; +} + +int main(){ + int n; cin >> n; + + vector> intervals(n); + for (int i = 0; i < n; i++) { + cin >> intervals[i].first; + cin >> intervals[i].second; + } + + sort(intervals.begin(), intervals.end(), comp); + + // first sort by start time, if two intervals start at the same time the one with the highest end time comes first + // always include the first interval + // keep track of the last included interval so far + // while there are intervals intersecting with the last included interval take the one with the highest ending value + // + // if there is some intersecting interval check if its end time is greater than the last included interval end time, if so include it and update count, else repeat process from the last intersecting interval + // if no intervals intersect with the last included interval then just process the next interval in order + + // in both cases update the count by 1 + + int curInterval = 1, lastIncludedInterval = 0, count = 1; + while (curInterval < n) { // curInterval holds the first interval thaat was not processed yet + int bestIntersectingInterval = lastIncludedInterval, j = curInterval; + while (j < n && intersecting(intervals[lastIncludedInterval], intervals[j])) { + int bestEndTime = intervals[bestIntersectingInterval].second; + int endTime = intervals[j].second; + if (endTime > bestEndTime) { + bestIntersectingInterval = j; + } + j++; + } + + if (bestIntersectingInterval != lastIncludedInterval) { + lastIncludedInterval = bestIntersectingInterval; + count++; + } else { + if (j == n) break; // this interval alredy covers every remaining interval + + lastIncludedInterval = j; + count++; + } + + if (curInterval == j) curInterval++; + else curInterval = j; + } + + cout << count << endl; + + return 0; +} \ No newline at end of file