Compare commits
3 Commits
| Author | SHA1 | Date |
|---|---|---|
 Johannes Kapfhammer
|
d0f6df5d24 | |
|
Johannes Kapfhammer
|
1cbaab8e46 | |
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Johannes Kapfhammer
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55a61a0050 |
18
README.md
18
README.md
|
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@ -1,16 +1,22 @@
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# Competitive Programmer's Handbook -- SOI Edition
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# Competitive Programmer's Handbook
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Competitive Programmer's Handbook is a modern introduction to competitive programming.
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The book discusses programming tricks and algorithm design techniques relevant in competitive programming.
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We adapted it to fit our SOI training.
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## CSES Problem Set
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## Credits
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The CSES Problem Set contains a collection of competitive programming problems.
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You can practice the techniques presented in the book by solving the problems.
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All credits belong to Antti Laaksonen.
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See original: https://github.com/pllk/cphb
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https://cses.fi/problemset/
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## License
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The license of the book is Creative Commons BY-NC-SA.
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## Other books
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Guide to Competitive Programming is a printed book, published by Springer, based on Competitive Programmer's Handbook.
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There is also a Russian edition Олимпиадное программирование (Olympiad Programming) and a Korean edition 알고리즘 트레이닝: 프로그래밍 대회 입문 가이드.
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https://cses.fi/book/
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66
book.tex
66
book.tex
|
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@ -8,7 +8,7 @@
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\usepackage[table]{xcolor}
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\usepackage{tikz}
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\usepackage{multicol}
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\usepackage{hyperref}
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\usepackage[colorlinks=true]{hyperref}
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\usepackage{array}
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\usepackage{microtype}
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@ -53,12 +53,12 @@
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stringstyle=\color{strings}
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}
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\date{Draft \today}
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\date{modified by Johannes Kapfhammer, February 2021}
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\usepackage[a4paper,vmargin=30mm,hmargin=33mm,footskip=15mm]{geometry}
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\title{\Huge Competitive Programmer's Handbook}
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\author{\Large Antti Laaksonen}
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\title{\Huge SOI Camp 2021 -- Graph Day}
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\author{\Large Competitive Programmer’s Handbook by Antti Laaksonen}
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\makeindex
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\usepackage[totoc]{idxlayout}
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@ -86,39 +86,47 @@
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\newcommand{\key}[1] {\textbf{#1}}
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\part{Basic techniques}
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\include{chapter01}
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\include{chapter02}
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\include{chapter03}
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\include{chapter04}
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\include{chapter05}
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\include{chapter06}
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\include{chapter07}
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\include{chapter08}
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\include{chapter09}
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\include{chapter10}
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\part{Graph algorithms}
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%\part{Basic techniques}
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%\include{chapter01}
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%\include{chapter02}
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%\include{chapter03}
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%\include{chapter04}
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%\include{chapter05}
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%\include{chapter06}
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%\include{chapter07}
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%\include{chapter08}
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%\include{chapter09}
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%\include{chapter10}
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\part{Main Topics}
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\include{chapter11}
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\include{chapter12}
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\include{chapter13}
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\include{chapter14}
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\include{chapter15}
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\include{chapter16}
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%\include{chapter20}
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\chapter{State Graphs}
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Please watch the video about state graphs.\\
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\url{https://www.youtube.com/watch?v=RdK3b9QWs94}
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\part{Advanced topics}
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\include{chapter15}
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\include{chapter17}
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\include{chapter18}
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\chapter{Bridges and Articulation Points}
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Please read our wiki: \url{https://soi.ch/wiki/bridges-articulationpoints/}
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\include{chapter19}
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\include{chapter20}
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\part{Advanced topics}
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\include{chapter21}
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\include{chapter22}
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\include{chapter23}
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\include{chapter24}
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\include{chapter25}
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\include{chapter26}
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\include{chapter27}
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\include{chapter28}
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\include{chapter29}
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\include{chapter30}
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% \part{Advanced topics}
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%\include{chapter21}
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%\include{chapter22}
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%\include{chapter23}
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%\include{chapter24}
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%\include{chapter25}
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%\include{chapter26}
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%\include{chapter27}
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%\include{chapter28}
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%\include{chapter29}
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%\include{chapter30}
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\cleardoublepage
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\phantomsection
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|
|
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117
chapter11.tex
117
chapter11.tex
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@ -29,6 +29,10 @@ the number of edges.
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The nodes are numbered
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using integers $1,2,\ldots,n$.
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Note: at SOI we usually say \key{vertex} (plural \key{vertices}) instead of \key{node}.
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Vertex and node can be used interchangeably.
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We also like to number the vertices 0-based as $0,1,\ldots,n-1$.
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For example, the following graph consists of 5 nodes and 7 edges:
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\begin{center}
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@ -51,9 +55,11 @@ For example, the following graph consists of 5 nodes and 7 edges:
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\index{path}
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A \key{path} leads from node $a$ to node $b$
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A \key{walk} leads from node $a$ to node $b$
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through edges of the graph.
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The \key{length} of a path is the number of
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A \key{path} is a walk where each node appears
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at most once in the path.
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The \key{length} of a path (or a walk) is the number of
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edges in it.
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For example, the above graph contains
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a path $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$
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@ -84,12 +90,10 @@ from node 1 to node 5:
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\index{cycle}
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A path is a \key{cycle} if the first and last
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node is the same.
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A \key{cycle} is a walk where the first and last
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node is the same, and every other vertex appears at most once.
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For example, the above graph contains
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a cycle $1 \rightarrow 3 \rightarrow 4 \rightarrow 1$.
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A path is \key{simple} if each node appears
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at most once in the path.
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%
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@ -478,21 +482,19 @@ way to represent graphs, and most algorithms can be
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efficiently implemented using them.
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A convenient way to store the adjacency lists is to declare
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an array of vectors as follows:
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a vector of vectors as follows:
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\begin{lstlisting}
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vector<int> adj[N];
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vector<vector<int>> g;
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\end{lstlisting}
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The constant $N$ is chosen so that all
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adjacency lists can be stored.
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For example, the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (3,3) {$2$};
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\node[draw, circle] (3) at (5,3) {$3$};
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\node[draw, circle] (4) at (3,1) {$4$};
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\node[draw, circle] (1) at (1,3) {$0$};
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\node[draw, circle] (2) at (3,3) {$1$};
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\node[draw, circle] (3) at (5,3) {$2$};
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\node[draw, circle] (4) at (3,1) {$3$};
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\path[draw,thick,->,>=latex] (1) -- (2);
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\path[draw,thick,->,>=latex] (2) -- (3);
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@ -503,11 +505,12 @@ For example, the graph
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\end{center}
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can be stored as follows:
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\begin{lstlisting}
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adj[1].push_back(2);
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adj[2].push_back(3);
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adj[2].push_back(4);
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adj[3].push_back(4);
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adj[4].push_back(1);
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g.assign(4, {}); // g now consists of 4 empty arrays
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g[0].push_back(1);
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g[1].push_back(2);
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g[1].push_back(3);
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g[2].push_back(3);
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g[3].push_back(0);
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\end{lstlisting}
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If the graph is undirected, it can be stored in a similar way,
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@ -517,7 +520,7 @@ For a weighted graph, the structure can be extended
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as follows:
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\begin{lstlisting}
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vector<pair<int,int>> adj[N];
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vector<vector<pair<int,int>>> g;
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\end{lstlisting}
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In this case, the adjacency list of node $a$
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@ -527,10 +530,10 @@ with weight $w$. For example, the graph
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|||
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (3,3) {$2$};
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\node[draw, circle] (3) at (5,3) {$3$};
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\node[draw, circle] (4) at (3,1) {$4$};
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\node[draw, circle] (1) at (1,3) {$0$};
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\node[draw, circle] (2) at (3,3) {$1$};
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\node[draw, circle] (3) at (5,3) {$2$};
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\node[draw, circle] (4) at (3,1) {$3$};
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\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
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\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
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|
@ -541,11 +544,12 @@ with weight $w$. For example, the graph
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\end{center}
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can be stored as follows:
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\begin{lstlisting}
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adj[1].push_back({2,5});
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adj[2].push_back({3,7});
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adj[2].push_back({4,6});
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adj[3].push_back({4,5});
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adj[4].push_back({1,2});
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g.assign(4, {});
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g[0].emplace_back(1,5);
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g[1].emplace_back(2,7);
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g[1].emplace_back(3,6);
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g[2].emplace_back(3,5);
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g[3].emplace_back(0,2);
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\end{lstlisting}
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||||
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||||
The benefit of using adjacency lists is that
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|
|
@ -555,7 +559,7 @@ For example, the following loop goes through all nodes
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to which we can move from node $s$:
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||||
|
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\begin{lstlisting}
|
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for (auto u : adj[s]) {
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for (auto u : g[s]) {
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// process node u
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||||
}
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\end{lstlisting}
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|
|
@ -570,7 +574,8 @@ We can efficiently check from an adjacency matrix
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if there is an edge between two nodes.
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The matrix can be stored as an array
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\begin{lstlisting}
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int adj[N][N];
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vector<vector<int>> adj;
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adj.assign(n, vector<int>(n, 0));
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\end{lstlisting}
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where each value $\texttt{adj}[a][b]$ indicates
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whether the graph contains an edge from
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|
|
@ -581,10 +586,10 @@ and otherwise $\texttt{adj}[a][b]=0$.
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For example, the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
|
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\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
\node[draw, circle] (1) at (1,3) {$0$};
|
||||
\node[draw, circle] (2) at (3,3) {$1$};
|
||||
\node[draw, circle] (3) at (5,3) {$2$};
|
||||
\node[draw, circle] (4) at (3,1) {$3$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
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\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
|
|
@ -704,10 +709,10 @@ Thus, the graph
|
|||
|
||||
\begin{center}
|
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
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\node[draw, circle] (4) at (3,1) {$4$};
|
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\node[draw, circle] (1) at (1,3) {$0$};
|
||||
\node[draw, circle] (2) at (3,3) {$1$};
|
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\node[draw, circle] (3) at (5,3) {$2$};
|
||||
\node[draw, circle] (4) at (3,1) {$3$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
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|
|
@ -718,11 +723,11 @@ Thus, the graph
|
|||
\end{center}
|
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can be represented as follows:
|
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\begin{lstlisting}
|
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edges.push_back({1,2});
|
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edges.push_back({2,3});
|
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edges.push_back({2,4});
|
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edges.push_back({3,4});
|
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edges.push_back({4,1});
|
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edges.emplace_back(0,2);
|
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edges.emplace_back(1,3);
|
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edges.emplace_back(1,4);
|
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edges.emplace_back(2,4);
|
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edges.emplace_back(3,1);
|
||||
\end{lstlisting}
|
||||
|
||||
\noindent
|
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|
|
@ -738,10 +743,10 @@ For example, the graph
|
|||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
\node[draw, circle] (1) at (1,3) {$0$};
|
||||
\node[draw, circle] (2) at (3,3) {$1$};
|
||||
\node[draw, circle] (3) at (5,3) {$2$};
|
||||
\node[draw, circle] (4) at (3,1) {$3$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
|
||||
|
|
@ -751,14 +756,14 @@ For example, the graph
|
|||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\begin{samepage}
|
||||
can be represented as follows\footnote{In some older compilers, the function
|
||||
\texttt{make\_tuple} must be used instead of the braces (for example,
|
||||
\texttt{make\_tuple(1,2,5)} instead of \texttt{\{1,2,5\}}).}:
|
||||
can be represented as follows\footnote{Instead of \texttt{emplace\_back(0,2,5)},
|
||||
one could also write \texttt{edges.push\_back(\{0,2,5\})} or
|
||||
\texttt{edges.push\_back(make\_tuple(0,2,5))}, however, using \texttt{emplace\_back} is generally preferred.}:
|
||||
\begin{lstlisting}
|
||||
edges.push_back({1,2,5});
|
||||
edges.push_back({2,3,7});
|
||||
edges.push_back({2,4,6});
|
||||
edges.push_back({3,4,5});
|
||||
edges.push_back({4,1,2});
|
||||
edges.emplace_back(0,2,5);
|
||||
edges.emplace_back(1,3,7);
|
||||
edges.emplace_back(1,4,6);
|
||||
edges.emplace_back(2,4,5);
|
||||
edges.emplace_back(3,1,2);
|
||||
\end{lstlisting}
|
||||
\end{samepage}
|
||||
|
|
|
|||
|
|
@ -129,11 +129,11 @@ a depth-first search at a given node.
|
|||
The function assumes that the graph is
|
||||
stored as adjacency lists in an array
|
||||
\begin{lstlisting}
|
||||
vector<int> adj[N];
|
||||
vector<vector<int>> g;
|
||||
\end{lstlisting}
|
||||
and also maintains an array
|
||||
\begin{lstlisting}
|
||||
bool visited[N];
|
||||
vector<bool> visited;
|
||||
\end{lstlisting}
|
||||
that keeps track of the visited nodes.
|
||||
Initially, each array value is \texttt{false},
|
||||
|
|
@ -145,7 +145,7 @@ void dfs(int s) {
|
|||
if (visited[s]) return;
|
||||
visited[s] = true;
|
||||
// process node s
|
||||
for (auto u: adj[s]) {
|
||||
for (auto u : g[s]) {
|
||||
dfs(u);
|
||||
}
|
||||
}
|
||||
|
|
@ -312,8 +312,8 @@ as adjacency lists and maintains the following
|
|||
data structures:
|
||||
\begin{lstlisting}
|
||||
queue<int> q;
|
||||
bool visited[N];
|
||||
int distance[N];
|
||||
vector<bool> visited(n);
|
||||
vector<int> distance(N);
|
||||
\end{lstlisting}
|
||||
|
||||
The queue \texttt{q}
|
||||
|
|
@ -336,7 +336,7 @@ q.push(x);
|
|||
while (!q.empty()) {
|
||||
int s = q.front(); q.pop();
|
||||
// process node s
|
||||
for (auto u : adj[s]) {
|
||||
for (auto u : g[s]) {
|
||||
if (visited[u]) continue;
|
||||
visited[u] = true;
|
||||
distance[u] = distance[s]+1;
|
||||
|
|
|
|||
578
chapter13.tex
578
chapter13.tex
|
|
@ -20,313 +20,6 @@ where more sophisticated algorithms
|
|||
are needed
|
||||
for finding shortest paths.
|
||||
|
||||
\section{Bellman–Ford algorithm}
|
||||
|
||||
\index{Bellman–Ford algorithm}
|
||||
|
||||
The \key{Bellman–Ford algorithm}\footnote{The algorithm is named after
|
||||
R. E. Bellman and L. R. Ford who published it independently
|
||||
in 1958 and 1956, respectively \cite{bel58,for56a}.} finds
|
||||
shortest paths from a starting node to all
|
||||
nodes of the graph.
|
||||
The algorithm can process all kinds of graphs,
|
||||
provided that the graph does not contain a
|
||||
cycle with negative length.
|
||||
If the graph contains a negative cycle,
|
||||
the algorithm can detect this.
|
||||
|
||||
The algorithm keeps track of distances
|
||||
from the starting node to all nodes of the graph.
|
||||
Initially, the distance to the starting node is 0
|
||||
and the distance to all other nodes in infinite.
|
||||
The algorithm reduces the distances by finding
|
||||
edges that shorten the paths until it is not
|
||||
possible to reduce any distance.
|
||||
|
||||
\subsubsection{Example}
|
||||
|
||||
Let us consider how the Bellman–Ford algorithm
|
||||
works in the following graph:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (1,3) {1};
|
||||
\node[draw, circle] (2) at (4,3) {2};
|
||||
\node[draw, circle] (3) at (1,1) {3};
|
||||
\node[draw, circle] (4) at (4,1) {4};
|
||||
\node[draw, circle] (5) at (6,2) {6};
|
||||
\node[color=red] at (1,3+0.55) {$0$};
|
||||
\node[color=red] at (4,3+0.55) {$\infty$};
|
||||
\node[color=red] at (1,1-0.55) {$\infty$};
|
||||
\node[color=red] at (4,1-0.55) {$\infty$};
|
||||
\node[color=red] at (6,2-0.55) {$\infty$};
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Each node of the graph is assigned a distance.
|
||||
Initially, the distance to the starting node is 0,
|
||||
and the distance to all other nodes is infinite.
|
||||
|
||||
The algorithm searches for edges that reduce distances.
|
||||
First, all edges from node 1 reduce distances:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (1,3) {1};
|
||||
\node[draw, circle] (2) at (4,3) {2};
|
||||
\node[draw, circle] (3) at (1,1) {3};
|
||||
\node[draw, circle] (4) at (4,1) {4};
|
||||
\node[draw, circle] (5) at (6,2) {5};
|
||||
\node[color=red] at (1,3+0.55) {$0$};
|
||||
\node[color=red] at (4,3+0.55) {$5$};
|
||||
\node[color=red] at (1,1-0.55) {$3$};
|
||||
\node[color=red] at (4,1-0.55) {$7$};
|
||||
\node[color=red] at (6,2-0.55) {$\infty$};
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
|
||||
|
||||
\path[draw=red,thick,->,line width=2pt] (1) -- (2);
|
||||
\path[draw=red,thick,->,line width=2pt] (1) -- (3);
|
||||
\path[draw=red,thick,->,line width=2pt] (1) -- (4);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
After this, edges
|
||||
$2 \rightarrow 5$ and $3 \rightarrow 4$
|
||||
reduce distances:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (1,3) {1};
|
||||
\node[draw, circle] (2) at (4,3) {2};
|
||||
\node[draw, circle] (3) at (1,1) {3};
|
||||
\node[draw, circle] (4) at (4,1) {4};
|
||||
\node[draw, circle] (5) at (6,2) {5};
|
||||
\node[color=red] at (1,3+0.55) {$0$};
|
||||
\node[color=red] at (4,3+0.55) {$5$};
|
||||
\node[color=red] at (1,1-0.55) {$3$};
|
||||
\node[color=red] at (4,1-0.55) {$4$};
|
||||
\node[color=red] at (6,2-0.55) {$7$};
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
|
||||
|
||||
\path[draw=red,thick,->,line width=2pt] (2) -- (5);
|
||||
\path[draw=red,thick,->,line width=2pt] (3) -- (4);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Finally, there is one more change:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (1,3) {1};
|
||||
\node[draw, circle] (2) at (4,3) {2};
|
||||
\node[draw, circle] (3) at (1,1) {3};
|
||||
\node[draw, circle] (4) at (4,1) {4};
|
||||
\node[draw, circle] (5) at (6,2) {5};
|
||||
\node[color=red] at (1,3+0.55) {$0$};
|
||||
\node[color=red] at (4,3+0.55) {$5$};
|
||||
\node[color=red] at (1,1-0.55) {$3$};
|
||||
\node[color=red] at (4,1-0.55) {$4$};
|
||||
\node[color=red] at (6,2-0.55) {$6$};
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
|
||||
|
||||
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
After this, no edge can reduce any distance.
|
||||
This means that the distances are final,
|
||||
and we have successfully
|
||||
calculated the shortest distances
|
||||
from the starting node to all nodes of the graph.
|
||||
|
||||
For example, the shortest distance 3
|
||||
from node 1 to node 5 corresponds to
|
||||
the following path:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (1,3) {1};
|
||||
\node[draw, circle] (2) at (4,3) {2};
|
||||
\node[draw, circle] (3) at (1,1) {3};
|
||||
\node[draw, circle] (4) at (4,1) {4};
|
||||
\node[draw, circle] (5) at (6,2) {5};
|
||||
\node[color=red] at (1,3+0.55) {$0$};
|
||||
\node[color=red] at (4,3+0.55) {$5$};
|
||||
\node[color=red] at (1,1-0.55) {$3$};
|
||||
\node[color=red] at (4,1-0.55) {$4$};
|
||||
\node[color=red] at (6,2-0.55) {$6$};
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
|
||||
|
||||
\path[draw=red,thick,->,line width=2pt] (1) -- (3);
|
||||
\path[draw=red,thick,->,line width=2pt] (3) -- (4);
|
||||
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\subsubsection{Implementation}
|
||||
|
||||
The following implementation of the
|
||||
Bellman–Ford algorithm determines the shortest distances
|
||||
from a node $x$ to all nodes of the graph.
|
||||
The code assumes that the graph is stored
|
||||
as an edge list \texttt{edges}
|
||||
that consists of tuples of the form $(a,b,w)$,
|
||||
meaning that there is an edge from node $a$ to node $b$
|
||||
with weight $w$.
|
||||
|
||||
The algorithm consists of $n-1$ rounds,
|
||||
and on each round the algorithm goes through
|
||||
all edges of the graph and tries to
|
||||
reduce the distances.
|
||||
The algorithm constructs an array \texttt{distance}
|
||||
that will contain the distances from $x$
|
||||
to all nodes of the graph.
|
||||
The constant \texttt{INF} denotes an infinite distance.
|
||||
|
||||
\begin{lstlisting}
|
||||
for (int i = 1; i <= n; i++) distance[i] = INF;
|
||||
distance[x] = 0;
|
||||
for (int i = 1; i <= n-1; i++) {
|
||||
for (auto e : edges) {
|
||||
int a, b, w;
|
||||
tie(a, b, w) = e;
|
||||
distance[b] = min(distance[b], distance[a]+w);
|
||||
}
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
The time complexity of the algorithm is $O(nm)$,
|
||||
because the algorithm consists of $n-1$ rounds and
|
||||
iterates through all $m$ edges during a round.
|
||||
If there are no negative cycles in the graph,
|
||||
all distances are final after $n-1$ rounds,
|
||||
because each shortest path can contain at most $n-1$ edges.
|
||||
|
||||
In practice, the final distances can usually
|
||||
be found faster than in $n-1$ rounds.
|
||||
Thus, a possible way to make the algorithm more efficient
|
||||
is to stop the algorithm if no distance
|
||||
can be reduced during a round.
|
||||
|
||||
\subsubsection{Negative cycles}
|
||||
|
||||
\index{negative cycle}
|
||||
|
||||
The Bellman–Ford algorithm can also be used to
|
||||
check if the graph contains a cycle with negative length.
|
||||
For example, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (2,1) {$2$};
|
||||
\node[draw, circle] (3) at (2,-1) {$3$};
|
||||
\node[draw, circle] (4) at (4,0) {$4$};
|
||||
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:$3$] {} (2);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:$1$] {} (4);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=below:$5$] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:$-7$] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=right:$2$] {} (3);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\noindent
|
||||
contains a negative cycle
|
||||
$2 \rightarrow 3 \rightarrow 4 \rightarrow 2$
|
||||
with length $-4$.
|
||||
|
||||
If the graph contains a negative cycle,
|
||||
we can shorten infinitely many times
|
||||
any path that contains the cycle by repeating the cycle
|
||||
again and again.
|
||||
Thus, the concept of a shortest path
|
||||
is not meaningful in this situation.
|
||||
|
||||
A negative cycle can be detected
|
||||
using the Bellman–Ford algorithm by
|
||||
running the algorithm for $n$ rounds.
|
||||
If the last round reduces any distance,
|
||||
the graph contains a negative cycle.
|
||||
Note that this algorithm can be used to
|
||||
search for
|
||||
a negative cycle in the whole graph
|
||||
regardless of the starting node.
|
||||
|
||||
\subsubsection{SPFA algorithm}
|
||||
|
||||
\index{SPFA algorithm}
|
||||
|
||||
The \key{SPFA algorithm} (''Shortest Path Faster Algorithm'') \cite{fan94}
|
||||
is a variant of the Bellman–Ford algorithm,
|
||||
that is often more efficient than the original algorithm.
|
||||
The SPFA algorithm does not go through all the edges on each round,
|
||||
but instead, it chooses the edges to be examined
|
||||
in a more intelligent way.
|
||||
|
||||
The algorithm maintains a queue of nodes that might
|
||||
be used for reducing the distances.
|
||||
First, the algorithm adds the starting node $x$
|
||||
to the queue.
|
||||
Then, the algorithm always processes the
|
||||
first node in the queue, and when an edge
|
||||
$a \rightarrow b$ reduces a distance,
|
||||
node $b$ is added to the queue.
|
||||
%
|
||||
% The following implementation uses a
|
||||
% \texttt{queue} \texttt{q}.
|
||||
% In addition, an array \texttt{inqueue} indicates
|
||||
% if a node is already in the queue,
|
||||
% in which case the algorithm does not add
|
||||
% the node to the queue again.
|
||||
%
|
||||
% \begin{lstlisting}
|
||||
% for (int i = 1; i <= n; i++) distance[i] = INF;
|
||||
% distance[x] = 0;
|
||||
% q.push(x);
|
||||
% while (!q.empty()) {
|
||||
% int a = q.front(); q.pop();
|
||||
% inqueue[a] = false;
|
||||
% for (auto b : v[a]) {
|
||||
% if (distance[a]+b.second < distance[b.first]) {
|
||||
% distance[b.first] = distance[a]+b.second;
|
||||
% if (!inqueue[b]) {q.push(b); inqueue[b] = true;}
|
||||
% }
|
||||
% }
|
||||
% }
|
||||
% \end{lstlisting}
|
||||
|
||||
The efficiency of the SPFA algorithm depends
|
||||
on the structure of the graph:
|
||||
the algorithm is often efficient,
|
||||
but its worst case time complexity is still
|
||||
$O(nm)$ and it is possible to create inputs
|
||||
that make the algorithm as slow as the
|
||||
original Bellman–Ford algorithm.
|
||||
|
||||
\section{Dijkstra's algorithm}
|
||||
|
||||
\index{Dijkstra's algorithm}
|
||||
|
|
@ -334,15 +27,12 @@ original Bellman–Ford algorithm.
|
|||
\key{Dijkstra's algorithm}\footnote{E. W. Dijkstra published the algorithm in 1959 \cite{dij59};
|
||||
however, his original paper does not mention how to implement the algorithm efficiently.}
|
||||
finds shortest
|
||||
paths from the starting node to all nodes of the graph,
|
||||
like the Bellman–Ford algorithm.
|
||||
The benefit of Dijsktra's algorithm is that
|
||||
it is more efficient and can be used for
|
||||
paths from the starting node to all nodes of the graph.
|
||||
Dijkstra's algorithm is very efficient and can be used for
|
||||
processing large graphs.
|
||||
However, the algorithm requires that there
|
||||
are no negative weight edges in the graph.
|
||||
|
||||
Like the Bellman–Ford algorithm,
|
||||
Dijkstra's algorithm maintains distances
|
||||
to the nodes and reduces them during the search.
|
||||
Dijkstra's algorithm is efficient, because
|
||||
|
|
@ -543,9 +233,9 @@ The following implementation of Dijkstra's algorithm
|
|||
calculates the minimum distances from a node $x$
|
||||
to other nodes of the graph.
|
||||
The graph is stored as adjacency lists
|
||||
so that \texttt{adj[$a$]} contains a pair $(b,w)$
|
||||
always when there is an edge from node $a$ to node $b$
|
||||
with weight $w$.
|
||||
so that \texttt{g[$v$]} contains a pair $(w,\text{cost})$
|
||||
always when there is an edge from node $v$ to node $w$
|
||||
with weight $\text{cost}$.
|
||||
|
||||
An efficient implementation of Dijkstra's algorithm
|
||||
requires that it is possible to efficiently find the
|
||||
|
|
@ -556,40 +246,42 @@ Using a priority queue, the next node to be processed
|
|||
can be retrieved in logarithmic time.
|
||||
|
||||
In the following code, the priority queue
|
||||
\texttt{q} contains pairs of the form $(-d,x)$,
|
||||
\texttt{pq} contains pairs of the form $(d,x)$,
|
||||
meaning that the current distance to node $x$ is $d$.
|
||||
|
||||
The array $\texttt{distance}$ contains the distance to
|
||||
each node, and the array $\texttt{processed}$ indicates
|
||||
whether a node has been processed.
|
||||
Initially the distance is $0$ to $x$ and $\infty$ to all other nodes.
|
||||
each node. Initially, the distance is $0$ to $\text{start}$ and $-1$ to all
|
||||
other nodes. We use $-1$ as invalid value to denote that the node
|
||||
has not been reached yet.
|
||||
|
||||
\begin{lstlisting}
|
||||
for (int i = 1; i <= n; i++) distance[i] = INF;
|
||||
distance[x] = 0;
|
||||
q.push({0,x});
|
||||
while (!q.empty()) {
|
||||
int a = q.top().second; q.pop();
|
||||
if (processed[a]) continue;
|
||||
processed[a] = true;
|
||||
for (auto u : adj[a]) {
|
||||
int b = u.first, w = u.second;
|
||||
if (distance[a]+w < distance[b]) {
|
||||
distance[b] = distance[a]+w;
|
||||
q.push({-distance[b],b});
|
||||
}
|
||||
}
|
||||
vector<int> distance(n, -1);
|
||||
priority_queue<pair<int, int>,
|
||||
vector<pair<int, int>>,
|
||||
greater<pair<int, int>>> pq;
|
||||
distance[start] = 0;
|
||||
pq.emplace(0, start);
|
||||
while (!pq.empty()) {
|
||||
auto [d, v] = q.top();
|
||||
q.pop();
|
||||
if (distance[v] != -1) continue;
|
||||
distance[v] = d;
|
||||
for (auto [w, cost] : g[v])
|
||||
q.emplace(d + cost, w);
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
Note that the priority queue contains \emph{negative}
|
||||
distances to nodes.
|
||||
The reason for this is that the
|
||||
default version of the C++ priority queue finds maximum
|
||||
elements, while we want to find minimum elements.
|
||||
By using negative distances,
|
||||
we can directly use the default priority queue\footnote{Of
|
||||
course, we could also declare the priority queue as in Chapter 4.5
|
||||
and use positive distances, but the implementation would be a bit longer.}.
|
||||
Note that the type of the priority queue is not
|
||||
\verb|priority_queue<pair<int, int>| but instead
|
||||
\verb|priority_queue<pair<int,int>,vector<pair<int,int>>,greater<pair<int,int>>>|.
|
||||
This is because in C++, a priority queue by default puts the
|
||||
\emph{largest} element on top, so we reverse the ordering by changing
|
||||
the comparison operator from \verb|less| (the default) to
|
||||
\verb|greater| (which does the opposite).
|
||||
|
||||
In case you forget, you can look up the syntax for the priority queue
|
||||
in the C++ cheatsheet linked on the camp page.
|
||||
|
||||
Also note that there may be several instances of the same
|
||||
node in the priority queue; however, only the instance with the
|
||||
minimum distance will be processed.
|
||||
|
|
@ -598,205 +290,3 @@ The time complexity of the above implementation is
|
|||
$O(n+m \log m)$, because the algorithm goes through
|
||||
all nodes of the graph and adds for each edge
|
||||
at most one distance to the priority queue.
|
||||
|
||||
\section{Floyd–Warshall algorithm}
|
||||
|
||||
\index{Floyd–Warshall algorithm}
|
||||
|
||||
The \key{Floyd–Warshall algorithm}\footnote{The algorithm
|
||||
is named after R. W. Floyd and S. Warshall
|
||||
who published it independently in 1962 \cite{flo62,war62}.}
|
||||
provides an alternative way to approach the problem
|
||||
of finding shortest paths.
|
||||
Unlike the other algorithms of this chapter,
|
||||
it finds all shortest paths between the nodes
|
||||
in a single run.
|
||||
|
||||
The algorithm maintains a two-dimensional array
|
||||
that contains distances between the nodes.
|
||||
First, distances are calculated only using
|
||||
direct edges between the nodes,
|
||||
and after this, the algorithm reduces distances
|
||||
by using intermediate nodes in paths.
|
||||
|
||||
\subsubsection{Example}
|
||||
|
||||
Let us consider how the Floyd–Warshall algorithm
|
||||
works in the following graph:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$3$};
|
||||
\node[draw, circle] (2) at (4,3) {$4$};
|
||||
\node[draw, circle] (3) at (1,1) {$2$};
|
||||
\node[draw, circle] (4) at (4,1) {$1$};
|
||||
\node[draw, circle] (5) at (6,2) {$5$};
|
||||
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
Initially, the distance from each node to itself is $0$,
|
||||
and the distance between nodes $a$ and $b$ is $x$
|
||||
if there is an edge between nodes $a$ and $b$ with weight $x$.
|
||||
All other distances are infinite.
|
||||
|
||||
In this graph, the initial array is as follows:
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrr}
|
||||
& 1 & 2 & 3 & 4 & 5 \\
|
||||
\hline
|
||||
1 & 0 & 5 & $\infty$ & 9 & 1 \\
|
||||
2 & 5 & 0 & 2 & $\infty$ & $\infty$ \\
|
||||
3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\
|
||||
4 & 9 & $\infty$ & 7 & 0 & 2 \\
|
||||
5 & 1 & $\infty$ & $\infty$ & 2 & 0 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\vspace{10pt}
|
||||
The algorithm consists of consecutive rounds.
|
||||
On each round, the algorithm selects a new node
|
||||
that can act as an intermediate node in paths from now on,
|
||||
and distances are reduced using this node.
|
||||
|
||||
On the first round, node 1 is the new intermediate node.
|
||||
There is a new path between nodes 2 and 4
|
||||
with length 14, because node 1 connects them.
|
||||
There is also a new path
|
||||
between nodes 2 and 5 with length 6.
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrr}
|
||||
& 1 & 2 & 3 & 4 & 5 \\
|
||||
\hline
|
||||
1 & 0 & 5 & $\infty$ & 9 & 1 \\
|
||||
2 & 5 & 0 & 2 & \textbf{14} & \textbf{6} \\
|
||||
3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\
|
||||
4 & 9 & \textbf{14} & 7 & 0 & 2 \\
|
||||
5 & 1 & \textbf{6} & $\infty$ & 2 & 0 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\vspace{10pt}
|
||||
|
||||
On the second round, node 2 is the new intermediate node.
|
||||
This creates new paths between nodes 1 and 3
|
||||
and between nodes 3 and 5:
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrr}
|
||||
& 1 & 2 & 3 & 4 & 5 \\
|
||||
\hline
|
||||
1 & 0 & 5 & \textbf{7} & 9 & 1 \\
|
||||
2 & 5 & 0 & 2 & 14 & 6 \\
|
||||
3 & \textbf{7} & 2 & 0 & 7 & \textbf{8} \\
|
||||
4 & 9 & 14 & 7 & 0 & 2 \\
|
||||
5 & 1 & 6 & \textbf{8} & 2 & 0 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\vspace{10pt}
|
||||
|
||||
On the third round, node 3 is the new intermediate round.
|
||||
There is a new path between nodes 2 and 4:
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrr}
|
||||
& 1 & 2 & 3 & 4 & 5 \\
|
||||
\hline
|
||||
1 & 0 & 5 & 7 & 9 & 1 \\
|
||||
2 & 5 & 0 & 2 & \textbf{9} & 6 \\
|
||||
3 & 7 & 2 & 0 & 7 & 8 \\
|
||||
4 & 9 & \textbf{9} & 7 & 0 & 2 \\
|
||||
5 & 1 & 6 & 8 & 2 & 0 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\vspace{10pt}
|
||||
|
||||
The algorithm continues like this,
|
||||
until all nodes have been appointed intermediate nodes.
|
||||
After the algorithm has finished, the array contains
|
||||
the minimum distances between any two nodes:
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrr}
|
||||
& 1 & 2 & 3 & 4 & 5 \\
|
||||
\hline
|
||||
1 & 0 & 5 & 7 & 3 & 1 \\
|
||||
2 & 5 & 0 & 2 & 8 & 6 \\
|
||||
3 & 7 & 2 & 0 & 7 & 8 \\
|
||||
4 & 3 & 8 & 7 & 0 & 2 \\
|
||||
5 & 1 & 6 & 8 & 2 & 0 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
For example, the array tells us that the
|
||||
shortest distance between nodes 2 and 4 is 8.
|
||||
This corresponds to the following path:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$3$};
|
||||
\node[draw, circle] (2) at (4,3) {$4$};
|
||||
\node[draw, circle] (3) at (1,1) {$2$};
|
||||
\node[draw, circle] (4) at (4,1) {$1$};
|
||||
\node[draw, circle] (5) at (6,2) {$5$};
|
||||
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||||
|
||||
\path[draw=red,thick,->,line width=2pt] (3) -- (4);
|
||||
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
|
||||
\path[draw=red,thick,->,line width=2pt] (5) -- (2);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\subsubsection{Implementation}
|
||||
|
||||
The advantage of the
|
||||
Floyd–Warshall algorithm that it is
|
||||
easy to implement.
|
||||
The following code constructs a
|
||||
distance matrix where $\texttt{distance}[a][b]$
|
||||
is the shortest distance between nodes $a$ and $b$.
|
||||
First, the algorithm initializes \texttt{distance}
|
||||
using the adjacency matrix \texttt{adj} of the graph:
|
||||
|
||||
\begin{lstlisting}
|
||||
for (int i = 1; i <= n; i++) {
|
||||
for (int j = 1; j <= n; j++) {
|
||||
if (i == j) distance[i][j] = 0;
|
||||
else if (adj[i][j]) distance[i][j] = adj[i][j];
|
||||
else distance[i][j] = INF;
|
||||
}
|
||||
}
|
||||
\end{lstlisting}
|
||||
After this, the shortest distances can be found as follows:
|
||||
\begin{lstlisting}
|
||||
for (int k = 1; k <= n; k++) {
|
||||
for (int i = 1; i <= n; i++) {
|
||||
for (int j = 1; j <= n; j++) {
|
||||
distance[i][j] = min(distance[i][j],
|
||||
distance[i][k]+distance[k][j]);
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
The time complexity of the algorithm is $O(n^3)$,
|
||||
because it contains three nested loops
|
||||
that go through the nodes of the graph.
|
||||
|
||||
Since the implementation of the Floyd–Warshall
|
||||
algorithm is simple, the algorithm can be
|
||||
a good choice even if it is only needed to find a
|
||||
single shortest path in the graph.
|
||||
However, the algorithm can only be used when the graph
|
||||
is so small that a cubic time complexity is fast enough.
|
||||
|
|
|
|||
|
|
@ -113,17 +113,16 @@ a depth-first search at an arbitrary node.
|
|||
The following recursive function can be used:
|
||||
|
||||
\begin{lstlisting}
|
||||
void dfs(int s, int e) {
|
||||
// process node s
|
||||
for (auto u : adj[s]) {
|
||||
if (u != e) dfs(u, s);
|
||||
}
|
||||
void dfs(int v, int p) {
|
||||
for (auto w : g[v])
|
||||
if (w != p)
|
||||
dfs(w, v);
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
The function is given two parameters: the current node $s$
|
||||
and the previous node $e$.
|
||||
The purpose of the parameter $e$ is to make sure
|
||||
The function is given two parameters: the current node $v$
|
||||
and the previous node $p$.
|
||||
The purpose of the parameter $p$ is to make sure
|
||||
that the search only moves to nodes
|
||||
that have not been visited yet.
|
||||
|
||||
|
|
@ -131,37 +130,36 @@ The following function call starts the search
|
|||
at node $x$:
|
||||
|
||||
\begin{lstlisting}
|
||||
dfs(x, 0);
|
||||
dfs(x, -1);
|
||||
\end{lstlisting}
|
||||
|
||||
In the first call $e=0$, because there is no
|
||||
In the first call $p=-1$, because there is no
|
||||
previous node, and it is allowed
|
||||
to proceed to any direction in the tree.
|
||||
|
||||
\subsubsection{Dynamic programming}
|
||||
|
||||
Dynamic programming can be used to calculate
|
||||
some information during a tree traversal.
|
||||
Using dynamic programming, we can, for example,
|
||||
\subsubsection{Storing Information}
|
||||
We can calculate
|
||||
some information during a tree traversal and store that for later use.
|
||||
We can, for example,
|
||||
calculate in $O(n)$ time for each node of a rooted tree the
|
||||
number of nodes in its subtree
|
||||
or the length of the longest path from the node
|
||||
to a leaf.
|
||||
|
||||
As an example, let us calculate for each node $s$
|
||||
a value $\texttt{count}[s]$: the number of nodes in its subtree.
|
||||
As an example, let us calculate for each node $v$
|
||||
a value $\texttt{subtreesize}[v]$: the number of nodes in its subtree.
|
||||
The subtree contains the node itself and
|
||||
all nodes in the subtrees of its children,
|
||||
so we can calculate the number of nodes
|
||||
recursively using the following code:
|
||||
|
||||
\begin{lstlisting}
|
||||
void dfs(int s, int e) {
|
||||
count[s] = 1;
|
||||
for (auto u : adj[s]) {
|
||||
if (u == e) continue;
|
||||
dfs(u, s);
|
||||
count[s] += count[u];
|
||||
void dfs(int v, int p) {
|
||||
subtreesize[s] = 1;
|
||||
for (auto w : g[v]) {
|
||||
if (w == p) continue;
|
||||
dfs(w, v);
|
||||
subtreesize[s] += subtreesize[u];
|
||||
}
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
|
@ -220,7 +218,7 @@ to obtain another path with length 4.
|
|||
|
||||
Next we will discuss two $O(n)$ time algorithms
|
||||
for calculating the diameter of a tree.
|
||||
The first algorithm is based on dynamic programming,
|
||||
The first algorithm is based on the previous idea of storing information,
|
||||
and the second algorithm uses two depth-first searches.
|
||||
|
||||
\subsubsection{Algorithm 1}
|
||||
|
|
@ -279,7 +277,7 @@ because there is a path
|
|||
$6 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow 7$.
|
||||
In this case, $\texttt{maxLength}(1)$ equals the diameter.
|
||||
|
||||
Dynamic programming can be used to calculate the above
|
||||
We can calculate the above
|
||||
values for all nodes in $O(n)$ time.
|
||||
First, to calculate $\texttt{toLeaf}(x)$,
|
||||
we go through the children of $x$,
|
||||
|
|
@ -448,8 +446,8 @@ goes through its child 2:
|
|||
\path[draw,thick,->,color=red,line width=2pt] (3) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
This part is easy to solve in $O(n)$ time, because we can use
|
||||
dynamic programming as we have done previously.
|
||||
This part is easy to solve in $O(n)$ time, because we can use a
|
||||
similar technique to what we have done previously.
|
||||
|
||||
Then, the second part of the problem is to calculate
|
||||
for every node $x$ the maximum length of a path
|
||||
|
|
|
|||
475
chapter16.tex
475
chapter16.tex
|
|
@ -1,20 +1,4 @@
|
|||
\chapter{Directed graphs}
|
||||
|
||||
In this chapter, we focus on two classes of directed graphs:
|
||||
\begin{itemize}
|
||||
\item \key{Acyclic graphs}:
|
||||
There are no cycles in the graph,
|
||||
so there is no path from any node to itself\footnote{Directed acyclic
|
||||
graphs are sometimes called DAGs.}.
|
||||
\item \key{Successor graphs}:
|
||||
The outdegree of each node is 1,
|
||||
so each node has a unique successor.
|
||||
\end{itemize}
|
||||
It turns out that in both cases,
|
||||
we can design efficient algorithms that are based
|
||||
on the special properties of the graphs.
|
||||
|
||||
\section{Topological sorting}
|
||||
\chapter{Topological sorting}
|
||||
|
||||
\index{topological sorting}
|
||||
\index{cycle}
|
||||
|
|
@ -249,460 +233,3 @@ The search reaches node 2 whose state is 1,
|
|||
which means that the graph contains a cycle.
|
||||
In this example, there is a cycle
|
||||
$2 \rightarrow 3 \rightarrow 5 \rightarrow 2$.
|
||||
|
||||
\section{Dynamic programming}
|
||||
|
||||
If a directed graph is acyclic,
|
||||
dynamic programming can be applied to it.
|
||||
For example, we can efficiently solve the following
|
||||
problems concerning paths from a starting node
|
||||
to an ending node:
|
||||
|
||||
\begin{itemize}
|
||||
\item how many different paths are there?
|
||||
\item what is the shortest/longest path?
|
||||
\item what is the minimum/maximum number of edges in a path?
|
||||
\item which nodes certainly appear in any path?
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{Counting the number of paths}
|
||||
|
||||
As an example, let us calculate the number of paths
|
||||
from node 1 to node 6 in the following graph:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,5) {$1$};
|
||||
\node[draw, circle] (2) at (3,5) {$2$};
|
||||
\node[draw, circle] (3) at (5,5) {$3$};
|
||||
\node[draw, circle] (4) at (1,3) {$4$};
|
||||
\node[draw, circle] (5) at (3,3) {$5$};
|
||||
\node[draw, circle] (6) at (5,3) {$6$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (1) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (5);
|
||||
\path[draw,thick,->,>=latex] (5) -- (2);
|
||||
\path[draw,thick,->,>=latex] (5) -- (3);
|
||||
\path[draw,thick,->,>=latex] (3) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
There are a total of three such paths:
|
||||
\begin{itemize}
|
||||
\item $1 \rightarrow 2 \rightarrow 3 \rightarrow 6$
|
||||
\item $1 \rightarrow 4 \rightarrow 5 \rightarrow 2 \rightarrow 3 \rightarrow 6$
|
||||
\item $1 \rightarrow 4 \rightarrow 5 \rightarrow 3 \rightarrow 6$
|
||||
\end{itemize}
|
||||
|
||||
Let $\texttt{paths}(x)$ denote the number of paths from
|
||||
node 1 to node $x$.
|
||||
As a base case, $\texttt{paths}(1)=1$.
|
||||
Then, to calculate other values of $\texttt{paths}(x)$,
|
||||
we may use the recursion
|
||||
\[\texttt{paths}(x) = \texttt{paths}(a_1)+\texttt{paths}(a_2)+\cdots+\texttt{paths}(a_k)\]
|
||||
where $a_1,a_2,\ldots,a_k$ are the nodes from which there
|
||||
is an edge to $x$.
|
||||
Since the graph is acyclic, the values of $\texttt{paths}(x)$
|
||||
can be calculated in the order of a topological sort.
|
||||
A topological sort for the above graph is as follows:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (4.5,0) {$2$};
|
||||
\node[draw, circle] (3) at (6,0) {$3$};
|
||||
\node[draw, circle] (4) at (1.5,0) {$4$};
|
||||
\node[draw, circle] (5) at (3,0) {$5$};
|
||||
\node[draw, circle] (6) at (7.5,0) {$6$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) edge [bend left=30] (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (1) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (5);
|
||||
\path[draw,thick,->,>=latex] (5) -- (2);
|
||||
\path[draw,thick,->,>=latex] (5) edge [bend right=30] (3);
|
||||
\path[draw,thick,->,>=latex] (3) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Hence, the numbers of paths are as follows:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,5) {$1$};
|
||||
\node[draw, circle] (2) at (3,5) {$2$};
|
||||
\node[draw, circle] (3) at (5,5) {$3$};
|
||||
\node[draw, circle] (4) at (1,3) {$4$};
|
||||
\node[draw, circle] (5) at (3,3) {$5$};
|
||||
\node[draw, circle] (6) at (5,3) {$6$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (1) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (5);
|
||||
\path[draw,thick,->,>=latex] (5) -- (2);
|
||||
\path[draw,thick,->,>=latex] (5) -- (3);
|
||||
\path[draw,thick,->,>=latex] (3) -- (6);
|
||||
|
||||
\node[color=red] at (1,2.3) {$1$};
|
||||
\node[color=red] at (3,2.3) {$1$};
|
||||
\node[color=red] at (5,2.3) {$3$};
|
||||
\node[color=red] at (1,5.7) {$1$};
|
||||
\node[color=red] at (3,5.7) {$2$};
|
||||
\node[color=red] at (5,5.7) {$3$};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
For example, to calculate the value of $\texttt{paths}(3)$,
|
||||
we can use the formula $\texttt{paths}(2)+\texttt{paths}(5)$,
|
||||
because there are edges from nodes 2 and 5
|
||||
to node 3.
|
||||
Since $\texttt{paths}(2)=2$ and $\texttt{paths}(5)=1$, we conclude that $\texttt{paths}(3)=3$.
|
||||
|
||||
\subsubsection{Extending Dijkstra's algorithm}
|
||||
|
||||
\index{Dijkstra's algorithm}
|
||||
|
||||
A by-product of Dijkstra's algorithm is a directed, acyclic
|
||||
graph that indicates for each node of the original graph
|
||||
the possible ways to reach the node using a shortest path
|
||||
from the starting node.
|
||||
Dynamic programming can be applied to that graph.
|
||||
For example, in the graph
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (2,0) {$2$};
|
||||
\node[draw, circle] (3) at (0,-2) {$3$};
|
||||
\node[draw, circle] (4) at (2,-2) {$4$};
|
||||
\node[draw, circle] (5) at (4,-1) {$5$};
|
||||
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
|
||||
\path[draw,thick,-] (1) -- node[font=\small,label=left:5] {} (3);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=right:4] {} (4);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:8] {} (5);
|
||||
\path[draw,thick,-] (3) -- node[font=\small,label=below:2] {} (4);
|
||||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (3);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
the shortest paths from node 1 may use the following edges:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (2,0) {$2$};
|
||||
\node[draw, circle] (3) at (0,-2) {$3$};
|
||||
\node[draw, circle] (4) at (2,-2) {$4$};
|
||||
\node[draw, circle] (5) at (4,-1) {$5$};
|
||||
|
||||
\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2);
|
||||
\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3);
|
||||
\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4);
|
||||
\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4);
|
||||
\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5);
|
||||
\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
Now we can, for example, calculate the number of
|
||||
shortest paths from node 1 to node 5
|
||||
using dynamic programming:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (2,0) {$2$};
|
||||
\node[draw, circle] (3) at (0,-2) {$3$};
|
||||
\node[draw, circle] (4) at (2,-2) {$4$};
|
||||
\node[draw, circle] (5) at (4,-1) {$5$};
|
||||
|
||||
\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2);
|
||||
\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3);
|
||||
\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4);
|
||||
\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4);
|
||||
\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5);
|
||||
\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3);
|
||||
|
||||
\node[color=red] at (0,0.7) {$1$};
|
||||
\node[color=red] at (2,0.7) {$1$};
|
||||
\node[color=red] at (0,-2.7) {$2$};
|
||||
\node[color=red] at (2,-2.7) {$3$};
|
||||
\node[color=red] at (4,-1.7) {$3$};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\subsubsection{Representing problems as graphs}
|
||||
|
||||
Actually, any dynamic programming problem
|
||||
can be represented as a directed, acyclic graph.
|
||||
In such a graph, each node corresponds to a dynamic programming state
|
||||
and the edges indicate how the states depend on each other.
|
||||
|
||||
As an example, consider the problem
|
||||
of forming a sum of money $n$
|
||||
using coins
|
||||
$\{c_1,c_2,\ldots,c_k\}$.
|
||||
In this problem, we can construct a graph where
|
||||
each node corresponds to a sum of money,
|
||||
and the edges show how the coins can be chosen.
|
||||
For example, for coins $\{1,3,4\}$ and $n=6$,
|
||||
the graph is as follows:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (0) at (0,0) {$0$};
|
||||
\node[draw, circle] (1) at (2,0) {$1$};
|
||||
\node[draw, circle] (2) at (4,0) {$2$};
|
||||
\node[draw, circle] (3) at (6,0) {$3$};
|
||||
\node[draw, circle] (4) at (8,0) {$4$};
|
||||
\node[draw, circle] (5) at (10,0) {$5$};
|
||||
\node[draw, circle] (6) at (12,0) {$6$};
|
||||
|
||||
\path[draw,thick,->] (0) -- (1);
|
||||
\path[draw,thick,->] (1) -- (2);
|
||||
\path[draw,thick,->] (2) -- (3);
|
||||
\path[draw,thick,->] (3) -- (4);
|
||||
\path[draw,thick,->] (4) -- (5);
|
||||
\path[draw,thick,->] (5) -- (6);
|
||||
|
||||
\path[draw,thick,->] (0) edge [bend right=30] (3);
|
||||
\path[draw,thick,->] (1) edge [bend right=30] (4);
|
||||
\path[draw,thick,->] (2) edge [bend right=30] (5);
|
||||
\path[draw,thick,->] (3) edge [bend right=30] (6);
|
||||
|
||||
\path[draw,thick,->] (0) edge [bend left=30] (4);
|
||||
\path[draw,thick,->] (1) edge [bend left=30] (5);
|
||||
\path[draw,thick,->] (2) edge [bend left=30] (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
Using this representation,
|
||||
the shortest path from node 0 to node $n$
|
||||
corresponds to a solution with the minimum number of coins,
|
||||
and the total number of paths from node 0 to node $n$
|
||||
equals the total number of solutions.
|
||||
|
||||
\section{Successor paths}
|
||||
|
||||
\index{successor graph}
|
||||
\index{functional graph}
|
||||
|
||||
For the rest of the chapter,
|
||||
we will focus on \key{successor graphs}.
|
||||
In those graphs,
|
||||
the outdegree of each node is 1, i.e.,
|
||||
exactly one edge starts at each node.
|
||||
A successor graph consists of one or more
|
||||
components, each of which contains
|
||||
one cycle and some paths that lead to it.
|
||||
|
||||
Successor graphs are sometimes called
|
||||
\key{functional graphs}.
|
||||
The reason for this is that any successor graph
|
||||
corresponds to a function that defines
|
||||
the edges of the graph.
|
||||
The parameter for the function is a node of the graph,
|
||||
and the function gives the successor of that node.
|
||||
|
||||
For example, the function
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrrrrrr}
|
||||
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
|
||||
\hline
|
||||
$\texttt{succ}(x)$ & 3 & 5 & 7 & 6 & 2 & 2 & 1 & 6 & 3 \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
defines the following graph:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (0,0) {$1$};
|
||||
\node[draw, circle] (2) at (2,0) {$2$};
|
||||
\node[draw, circle] (3) at (-2,0) {$3$};
|
||||
\node[draw, circle] (4) at (1,-3) {$4$};
|
||||
\node[draw, circle] (5) at (4,0) {$5$};
|
||||
\node[draw, circle] (6) at (2,-1.5) {$6$};
|
||||
\node[draw, circle] (7) at (-2,-1.5) {$7$};
|
||||
\node[draw, circle] (8) at (3,-3) {$8$};
|
||||
\node[draw, circle] (9) at (-4,0) {$9$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (3);
|
||||
\path[draw,thick,->] (2) edge [bend left=40] (5);
|
||||
\path[draw,thick,->] (3) -- (7);
|
||||
\path[draw,thick,->] (4) -- (6);
|
||||
\path[draw,thick,->] (5) edge [bend left=40] (2);
|
||||
\path[draw,thick,->] (6) -- (2);
|
||||
\path[draw,thick,->] (7) -- (1);
|
||||
\path[draw,thick,->] (8) -- (6);
|
||||
\path[draw,thick,->] (9) -- (3);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
Since each node of a successor graph has a
|
||||
unique successor, we can also define a function $\texttt{succ}(x,k)$
|
||||
that gives the node that we will reach if
|
||||
we begin at node $x$ and walk $k$ steps forward.
|
||||
For example, in the above graph $\texttt{succ}(4,6)=2$,
|
||||
because we will reach node 2 by walking 6 steps from node 4:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (0,0) {$4$};
|
||||
\node[draw, circle] (2) at (1.5,0) {$6$};
|
||||
\node[draw, circle] (3) at (3,0) {$2$};
|
||||
\node[draw, circle] (4) at (4.5,0) {$5$};
|
||||
\node[draw, circle] (5) at (6,0) {$2$};
|
||||
\node[draw, circle] (6) at (7.5,0) {$5$};
|
||||
\node[draw, circle] (7) at (9,0) {$2$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (2);
|
||||
\path[draw,thick,->] (2) -- (3);
|
||||
\path[draw,thick,->] (3) -- (4);
|
||||
\path[draw,thick,->] (4) -- (5);
|
||||
\path[draw,thick,->] (5) -- (6);
|
||||
\path[draw,thick,->] (6) -- (7);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
A straightforward way to calculate a value of $\texttt{succ}(x,k)$
|
||||
is to start at node $x$ and walk $k$ steps forward, which takes $O(k)$ time.
|
||||
However, using preprocessing, any value of $\texttt{succ}(x,k)$
|
||||
can be calculated in only $O(\log k)$ time.
|
||||
|
||||
The idea is to precalculate all values of $\texttt{succ}(x,k)$ where
|
||||
$k$ is a power of two and at most $u$, where $u$ is
|
||||
the maximum number of steps we will ever walk.
|
||||
This can be efficiently done, because
|
||||
we can use the following recursion:
|
||||
|
||||
\begin{equation*}
|
||||
\texttt{succ}(x,k) = \begin{cases}
|
||||
\texttt{succ}(x) & k = 1\\
|
||||
\texttt{succ}(\texttt{succ}(x,k/2),k/2) & k > 1\\
|
||||
\end{cases}
|
||||
\end{equation*}
|
||||
|
||||
Precalculating the values takes $O(n \log u)$ time,
|
||||
because $O(\log u)$ values are calculated for each node.
|
||||
In the above graph, the first values are as follows:
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{r|rrrrrrrrr}
|
||||
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
|
||||
\hline
|
||||
$\texttt{succ}(x,1)$ & 3 & 5 & 7 & 6 & 2 & 2 & 1 & 6 & 3 \\
|
||||
$\texttt{succ}(x,2)$ & 7 & 2 & 1 & 2 & 5 & 5 & 3 & 2 & 7 \\
|
||||
$\texttt{succ}(x,4)$ & 3 & 2 & 7 & 2 & 5 & 5 & 1 & 2 & 3 \\
|
||||
$\texttt{succ}(x,8)$ & 7 & 2 & 1 & 2 & 5 & 5 & 3 & 2 & 7 \\
|
||||
$\cdots$ \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
After this, any value of $\texttt{succ}(x,k)$ can be calculated
|
||||
by presenting the number of steps $k$ as a sum of powers of two.
|
||||
For example, if we want to calculate the value of $\texttt{succ}(x,11)$,
|
||||
we first form the representation $11=8+2+1$.
|
||||
Using that,
|
||||
\[\texttt{succ}(x,11)=\texttt{succ}(\texttt{succ}(\texttt{succ}(x,8),2),1).\]
|
||||
For example, in the previous graph
|
||||
\[\texttt{succ}(4,11)=\texttt{succ}(\texttt{succ}(\texttt{succ}(4,8),2),1)=5.\]
|
||||
|
||||
Such a representation always consists of
|
||||
$O(\log k)$ parts, so calculating a value of $\texttt{succ}(x,k)$
|
||||
takes $O(\log k)$ time.
|
||||
|
||||
\section{Cycle detection}
|
||||
|
||||
\index{cycle}
|
||||
\index{cycle detection}
|
||||
|
||||
Consider a successor graph that only contains
|
||||
a path that ends in a cycle.
|
||||
We may ask the following questions:
|
||||
if we begin our walk at the starting node,
|
||||
what is the first node in the cycle
|
||||
and how many nodes does the cycle contain?
|
||||
|
||||
For example, in the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (5) at (0,0) {$5$};
|
||||
\node[draw, circle] (4) at (-2,0) {$4$};
|
||||
\node[draw, circle] (6) at (-1,1.5) {$6$};
|
||||
\node[draw, circle] (3) at (-4,0) {$3$};
|
||||
\node[draw, circle] (2) at (-6,0) {$2$};
|
||||
\node[draw, circle] (1) at (-8,0) {$1$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (2);
|
||||
\path[draw,thick,->] (2) -- (3);
|
||||
\path[draw,thick,->] (3) -- (4);
|
||||
\path[draw,thick,->] (4) -- (5);
|
||||
\path[draw,thick,->] (5) -- (6);
|
||||
\path[draw,thick,->] (6) -- (4);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
we begin our walk at node 1,
|
||||
the first node that belongs to the cycle is node 4, and the cycle consists
|
||||
of three nodes (4, 5 and 6).
|
||||
|
||||
A simple way to detect the cycle is to walk in the
|
||||
graph and keep track of
|
||||
all nodes that have been visited. Once a node is visited
|
||||
for the second time, we can conclude
|
||||
that the node is the first node in the cycle.
|
||||
This method works in $O(n)$ time and also uses
|
||||
$O(n)$ memory.
|
||||
|
||||
However, there are better algorithms for cycle detection.
|
||||
The time complexity of such algorithms is still $O(n)$,
|
||||
but they only use $O(1)$ memory.
|
||||
This is an important improvement if $n$ is large.
|
||||
Next we will discuss Floyd's algorithm that
|
||||
achieves these properties.
|
||||
|
||||
\subsubsection{Floyd's algorithm}
|
||||
|
||||
\index{Floyd's algorithm}
|
||||
|
||||
\key{Floyd's algorithm}\footnote{The idea of the algorithm is mentioned in \cite{knu982}
|
||||
and attributed to R. W. Floyd; however, it is not known if Floyd actually
|
||||
discovered the algorithm.} walks forward
|
||||
in the graph using two pointers $a$ and $b$.
|
||||
Both pointers begin at a node $x$ that
|
||||
is the starting node of the graph.
|
||||
Then, on each turn, the pointer $a$ walks
|
||||
one step forward and the pointer $b$
|
||||
walks two steps forward.
|
||||
The process continues until
|
||||
the pointers meet each other:
|
||||
\begin{lstlisting}
|
||||
a = succ(x);
|
||||
b = succ(succ(x));
|
||||
while (a != b) {
|
||||
a = succ(a);
|
||||
b = succ(succ(b));
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
At this point, the pointer $a$ has walked $k$ steps
|
||||
and the pointer $b$ has walked $2k$ steps,
|
||||
so the length of the cycle divides $k$.
|
||||
Thus, the first node that belongs to the cycle
|
||||
can be found by moving the pointer $a$ to node $x$
|
||||
and advancing the pointers
|
||||
step by step until they meet again.
|
||||
\begin{lstlisting}
|
||||
a = x;
|
||||
while (a != b) {
|
||||
a = succ(a);
|
||||
b = succ(b);
|
||||
}
|
||||
first = a;
|
||||
\end{lstlisting}
|
||||
|
||||
After this, the length of the cycle
|
||||
can be calculated as follows:
|
||||
\begin{lstlisting}
|
||||
b = succ(a);
|
||||
length = 1;
|
||||
while (a != b) {
|
||||
b = succ(b);
|
||||
length++;
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
|
|
|||
202
chapter17.tex
202
chapter17.tex
|
|
@ -359,205 +359,3 @@ that create the remaining strongly connected components:
|
|||
The time complexity of the algorithm is $O(n+m)$,
|
||||
because the algorithm
|
||||
performs two depth-first searches.
|
||||
|
||||
\section{2SAT problem}
|
||||
|
||||
\index{2SAT problem}
|
||||
|
||||
Strong connectivity is also linked with the
|
||||
\key{2SAT problem}\footnote{The algorithm presented here was
|
||||
introduced in \cite{asp79}.
|
||||
There is also another well-known linear-time algorithm \cite{eve75}
|
||||
that is based on backtracking.}.
|
||||
In this problem, we are given a logical formula
|
||||
\[
|
||||
(a_1 \lor b_1) \land (a_2 \lor b_2) \land \cdots \land (a_m \lor b_m),
|
||||
\]
|
||||
where each $a_i$ and $b_i$ is either a logical variable
|
||||
($x_1,x_2,\ldots,x_n$)
|
||||
or a negation of a logical variable
|
||||
($\lnot x_1, \lnot x_2, \ldots, \lnot x_n$).
|
||||
The symbols ''$\land$'' and ''$\lor$'' denote
|
||||
logical operators ''and'' and ''or''.
|
||||
Our task is to assign each variable a value
|
||||
so that the formula is true, or state
|
||||
that this is not possible.
|
||||
|
||||
For example, the formula
|
||||
\[
|
||||
L_1 = (x_2 \lor \lnot x_1) \land
|
||||
(\lnot x_1 \lor \lnot x_2) \land
|
||||
(x_1 \lor x_3) \land
|
||||
(\lnot x_2 \lor \lnot x_3) \land
|
||||
(x_1 \lor x_4)
|
||||
\]
|
||||
is true when the variables are assigned as follows:
|
||||
|
||||
\[
|
||||
\begin{cases}
|
||||
x_1 = \textrm{false} \\
|
||||
x_2 = \textrm{false} \\
|
||||
x_3 = \textrm{true} \\
|
||||
x_4 = \textrm{true} \\
|
||||
\end{cases}
|
||||
\]
|
||||
|
||||
However, the formula
|
||||
\[
|
||||
L_2 = (x_1 \lor x_2) \land
|
||||
(x_1 \lor \lnot x_2) \land
|
||||
(\lnot x_1 \lor x_3) \land
|
||||
(\lnot x_1 \lor \lnot x_3)
|
||||
\]
|
||||
is always false, regardless of how we
|
||||
assign the values.
|
||||
The reason for this is that we cannot
|
||||
choose a value for $x_1$
|
||||
without creating a contradiction.
|
||||
If $x_1$ is false, both $x_2$ and $\lnot x_2$
|
||||
should be true which is impossible,
|
||||
and if $x_1$ is true, both $x_3$ and $\lnot x_3$
|
||||
should be true which is also impossible.
|
||||
|
||||
The 2SAT problem can be represented as a graph
|
||||
whose nodes correspond to
|
||||
variables $x_i$ and negations $\lnot x_i$,
|
||||
and edges determine the connections
|
||||
between the variables.
|
||||
Each pair $(a_i \lor b_i)$ generates two edges:
|
||||
$\lnot a_i \to b_i$ and $\lnot b_i \to a_i$.
|
||||
This means that if $a_i$ does not hold,
|
||||
$b_i$ must hold, and vice versa.
|
||||
|
||||
The graph for the formula $L_1$ is:
|
||||
\\
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=1.0,minimum size=2pt]
|
||||
\node[draw, circle, inner sep=1.3pt] (1) at (1,2) {$\lnot x_3$};
|
||||
\node[draw, circle] (2) at (3,2) {$x_2$};
|
||||
\node[draw, circle, inner sep=1.3pt] (3) at (1,0) {$\lnot x_4$};
|
||||
\node[draw, circle] (4) at (3,0) {$x_1$};
|
||||
\node[draw, circle, inner sep=1.3pt] (5) at (5,2) {$\lnot x_1$};
|
||||
\node[draw, circle] (6) at (7,2) {$x_4$};
|
||||
\node[draw, circle, inner sep=1.3pt] (7) at (5,0) {$\lnot x_2$};
|
||||
\node[draw, circle] (8) at (7,0) {$x_3$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (4);
|
||||
\path[draw,thick,->] (4) -- (2);
|
||||
\path[draw,thick,->] (2) -- (1);
|
||||
\path[draw,thick,->] (3) -- (4);
|
||||
\path[draw,thick,->] (2) -- (5);
|
||||
\path[draw,thick,->] (4) -- (7);
|
||||
\path[draw,thick,->] (5) -- (6);
|
||||
\path[draw,thick,->] (5) -- (8);
|
||||
\path[draw,thick,->] (8) -- (7);
|
||||
\path[draw,thick,->] (7) -- (5);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
And the graph for the formula $L_2$ is:
|
||||
\\
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=1.0,minimum size=2pt]
|
||||
\node[draw, circle] (1) at (1,2) {$x_3$};
|
||||
\node[draw, circle] (2) at (3,2) {$x_2$};
|
||||
\node[draw, circle, inner sep=1.3pt] (3) at (5,2) {$\lnot x_2$};
|
||||
\node[draw, circle, inner sep=1.3pt] (4) at (7,2) {$\lnot x_3$};
|
||||
\node[draw, circle, inner sep=1.3pt] (5) at (4,3.5) {$\lnot x_1$};
|
||||
\node[draw, circle] (6) at (4,0.5) {$x_1$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (5);
|
||||
\path[draw,thick,->] (4) -- (5);
|
||||
\path[draw,thick,->] (6) -- (1);
|
||||
\path[draw,thick,->] (6) -- (4);
|
||||
\path[draw,thick,->] (5) -- (2);
|
||||
\path[draw,thick,->] (5) -- (3);
|
||||
\path[draw,thick,->] (2) -- (6);
|
||||
\path[draw,thick,->] (3) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
The structure of the graph tells us whether
|
||||
it is possible to assign the values
|
||||
of the variables so
|
||||
that the formula is true.
|
||||
It turns out that this can be done
|
||||
exactly when there are no nodes
|
||||
$x_i$ and $\lnot x_i$ such that
|
||||
both nodes belong to the
|
||||
same strongly connected component.
|
||||
If there are such nodes,
|
||||
the graph contains
|
||||
a path from $x_i$ to $\lnot x_i$
|
||||
and also a path from $\lnot x_i$ to $x_i$,
|
||||
so both $x_i$ and $\lnot x_i$ should be true
|
||||
which is not possible.
|
||||
|
||||
In the graph of the formula $L_1$
|
||||
there are no nodes $x_i$ and $\lnot x_i$
|
||||
such that both nodes
|
||||
belong to the same strongly connected component,
|
||||
so a solution exists.
|
||||
In the graph of the formula $L_2$
|
||||
all nodes belong to the same strongly connected component,
|
||||
so a solution does not exist.
|
||||
|
||||
If a solution exists, the values for the variables
|
||||
can be found by going through the nodes of the
|
||||
component graph in a reverse topological sort order.
|
||||
At each step, we process a component
|
||||
that does not contain edges that lead to an
|
||||
unprocessed component.
|
||||
If the variables in the component
|
||||
have not been assigned values,
|
||||
their values will be determined
|
||||
according to the values in the component,
|
||||
and if they already have values,
|
||||
they remain unchanged.
|
||||
The process continues until each variable
|
||||
has been assigned a value.
|
||||
|
||||
The component graph for the formula $L_1$ is as follows:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=1.0]
|
||||
\node[draw, circle] (1) at (0,0) {$A$};
|
||||
\node[draw, circle] (2) at (2,0) {$B$};
|
||||
\node[draw, circle] (3) at (4,0) {$C$};
|
||||
\node[draw, circle] (4) at (6,0) {$D$};
|
||||
|
||||
\path[draw,thick,->] (1) -- (2);
|
||||
\path[draw,thick,->] (2) -- (3);
|
||||
\path[draw,thick,->] (3) -- (4);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
The components are
|
||||
$A = \{\lnot x_4\}$,
|
||||
$B = \{x_1, x_2, \lnot x_3\}$,
|
||||
$C = \{\lnot x_1, \lnot x_2, x_3\}$ and
|
||||
$D = \{x_4\}$.
|
||||
When constructing the solution,
|
||||
we first process the component $D$
|
||||
where $x_4$ becomes true.
|
||||
After this, we process the component $C$
|
||||
where $x_1$ and $x_2$ become false
|
||||
and $x_3$ becomes true.
|
||||
All variables have been assigned values,
|
||||
so the remaining components $A$ and $B$
|
||||
do not change the variables.
|
||||
|
||||
Note that this method works, because the
|
||||
graph has a special structure:
|
||||
if there are paths from node $x_i$ to node $x_j$
|
||||
and from node $x_j$ to node $\lnot x_j$,
|
||||
then node $x_i$ never becomes true.
|
||||
The reason for this is that there is also
|
||||
a path from node $\lnot x_j$ to node $\lnot x_i$,
|
||||
and both $x_i$ and $x_j$ become false.
|
||||
|
||||
\index{3SAT problem}
|
||||
|
||||
A more difficult problem is the \key{3SAT problem},
|
||||
where each part of the formula is of the form
|
||||
$(a_i \lor b_i \lor c_i)$.
|
||||
This problem is NP-hard, so no efficient algorithm
|
||||
for solving the problem is known.
|
||||
|
|
|
|||
|
|
@ -55,7 +55,7 @@ is $O(k)$, which may be slow, because a tree of $n$
|
|||
nodes may have a chain of $n$ nodes.
|
||||
|
||||
Fortunately, using a technique similar to that
|
||||
used in Chapter 16.3, any value of $\texttt{ancestor}(x,k)$
|
||||
used in Chapter 16.3 (of the full book), any value of $\texttt{ancestor}(x,k)$
|
||||
can be efficiently calculated in $O(\log k)$ time
|
||||
after preprocessing.
|
||||
The idea is to precalculate all values $\texttt{ancestor}(x,k)$
|
||||
|
|
|
|||
|
|
@ -1,58 +0,0 @@
|
|||
\relax
|
||||
\providecommand\hyper@newdestlabel[2]{}
|
||||
\providecommand\babel@aux[2]{}
|
||||
\@nameuse{bbl@beforestart}
|
||||
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
|
||||
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
|
||||
\global\let\oldcontentsline\contentsline
|
||||
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
|
||||
\global\let\oldnewlabel\newlabel
|
||||
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
|
||||
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
|
||||
\AtEndDocument{\ifx\hyper@anchor\@undefined
|
||||
\let\contentsline\oldcontentsline
|
||||
\let\newlabel\oldnewlabel
|
||||
\fi}
|
||||
\fi}
|
||||
\global\let\hyper@last\relax
|
||||
\gdef\HyperFirstAtBeginDocument#1{#1}
|
||||
\providecommand\HyField@AuxAddToFields[1]{}
|
||||
\providecommand\HyField@AuxAddToCoFields[2]{}
|
||||
\babel@aux{english}{}
|
||||
\@input{preface.aux}
|
||||
\@writefile{toc}{\contentsline {part}{I\hspace {1em}Basic techniques}{1}{part.1}\protected@file@percent }
|
||||
\@input{chapter01.aux}
|
||||
\@input{chapter02.aux}
|
||||
\@input{chapter03.aux}
|
||||
\@input{chapter04.aux}
|
||||
\@input{chapter05.aux}
|
||||
\@input{chapter06.aux}
|
||||
\@input{chapter07.aux}
|
||||
\@input{chapter08.aux}
|
||||
\@input{chapter09.aux}
|
||||
\@input{chapter10.aux}
|
||||
\@writefile{toc}{\contentsline {part}{II\hspace {1em}Graph algorithms}{107}{part.2}\protected@file@percent }
|
||||
\@input{chapter11.aux}
|
||||
\@input{chapter12.aux}
|
||||
\@input{chapter13.aux}
|
||||
\@input{chapter14.aux}
|
||||
\@input{chapter15.aux}
|
||||
\@input{chapter16.aux}
|
||||
\@input{chapter17.aux}
|
||||
\@input{chapter18.aux}
|
||||
\@input{chapter19.aux}
|
||||
\@input{chapter20.aux}
|
||||
\@writefile{toc}{\contentsline {part}{III\hspace {1em}Advanced topics}{195}{part.3}\protected@file@percent }
|
||||
\@input{chapter21.aux}
|
||||
\@input{chapter22.aux}
|
||||
\@input{chapter23.aux}
|
||||
\@input{chapter24.aux}
|
||||
\@input{chapter25.aux}
|
||||
\@input{chapter26.aux}
|
||||
\@input{chapter27.aux}
|
||||
\@input{chapter28.aux}
|
||||
\@input{chapter29.aux}
|
||||
\@input{chapter30.aux}
|
||||
\@input{list.aux}
|
||||
\@writefile{toc}{\contentsline {chapter}{Bibliography}{281}{section*.3}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {chapter}{Index}{287}{chapter*.5}\protected@file@percent }
|
||||
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|
@ -1,355 +0,0 @@
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345
output/book.idx
345
output/book.idx
|
|
@ -1,345 +0,0 @@
|
|||
\indexentry{programming language|hyperpage}{3}
|
||||
\indexentry{input and output|hyperpage}{4}
|
||||
\indexentry{integer|hyperpage}{6}
|
||||
\indexentry{remainder|hyperpage}{6}
|
||||
\indexentry{modular arithmetic|hyperpage}{6}
|
||||
\indexentry{floating point number|hyperpage}{7}
|
||||
\indexentry{tuppdef@\texttt{typedef}|hyperpage}{8}
|
||||
\indexentry{macro|hyperpage}{9}
|
||||
\indexentry{arithmetic progression|hyperpage}{10}
|
||||
\indexentry{Faulhaber's formula|hyperpage}{10}
|
||||
\indexentry{geometric progression|hyperpage}{11}
|
||||
\indexentry{harmonic sum|hyperpage}{11}
|
||||
\indexentry{set theory|hyperpage}{12}
|
||||
\indexentry{set|hyperpage}{12}
|
||||
\indexentry{intersection|hyperpage}{12}
|
||||
\indexentry{union|hyperpage}{12}
|
||||
\indexentry{difference|hyperpage}{12}
|
||||
\indexentry{subset|hyperpage}{12}
|
||||
\indexentry{universal set|hyperpage}{12}
|
||||
\indexentry{complement|hyperpage}{12}
|
||||
\indexentry{logic|hyperpage}{13}
|
||||
\indexentry{negation|hyperpage}{13}
|
||||
\indexentry{conjuction|hyperpage}{13}
|
||||
\indexentry{disjunction|hyperpage}{13}
|
||||
\indexentry{implication|hyperpage}{13}
|
||||
\indexentry{equivalence|hyperpage}{13}
|
||||
\indexentry{predicate|hyperpage}{13}
|
||||
\indexentry{quantifier|hyperpage}{13}
|
||||
\indexentry{factorial|hyperpage}{14}
|
||||
\indexentry{Fibonacci number|hyperpage}{14}
|
||||
\indexentry{Binet's formula|hyperpage}{14}
|
||||
\indexentry{logarithm|hyperpage}{14}
|
||||
\indexentry{natural logarithm|hyperpage}{15}
|
||||
\indexentry{time complexity|hyperpage}{17}
|
||||
\indexentry{complexity classes|hyperpage}{20}
|
||||
\indexentry{constant-time algorithm|hyperpage}{20}
|
||||
\indexentry{logarithmic algorithm|hyperpage}{20}
|
||||
\indexentry{linear algorithm|hyperpage}{20}
|
||||
\indexentry{quadratic algorithm|hyperpage}{20}
|
||||
\indexentry{cubic algorithm|hyperpage}{20}
|
||||
\indexentry{polynomial algorithm|hyperpage}{20}
|
||||
\indexentry{NP-hard problem|hyperpage}{20}
|
||||
\indexentry{constant factor|hyperpage}{21}
|
||||
\indexentry{maximum subarray sum|hyperpage}{21}
|
||||
\indexentry{Kadane's algorithm|hyperpage}{23}
|
||||
\indexentry{sorting|hyperpage}{25}
|
||||
\indexentry{bubble sort|hyperpage}{25}
|
||||
\indexentry{inversion|hyperpage}{26}
|
||||
\indexentry{merge sort|hyperpage}{27}
|
||||
\indexentry{counting sort|hyperpage}{28}
|
||||
\indexentry{sort@\texttt{sort}|hyperpage}{29}
|
||||
\indexentry{comparison operator|hyperpage}{30}
|
||||
\indexentry{pair@\texttt{pair}|hyperpage}{30}
|
||||
\indexentry{tuple@\texttt{tuple}|hyperpage}{30}
|
||||
\indexentry{comparison function|hyperpage}{31}
|
||||
\indexentry{binary search|hyperpage}{31}
|
||||
\indexentry{data structure|hyperpage}{35}
|
||||
\indexentry{dynamic array|hyperpage}{35}
|
||||
\indexentry{vector|hyperpage}{35}
|
||||
\indexentry{string|hyperpage}{36}
|
||||
\indexentry{set|hyperpage}{37}
|
||||
\indexentry{map|hyperpage}{38}
|
||||
\indexentry{iterator|hyperpage}{39}
|
||||
\indexentry{sort@\texttt{sort}|hyperpage}{39}
|
||||
\indexentry{reverse@\texttt{reverse}|hyperpage}{39}
|
||||
\indexentry{random\_shuffle@\texttt{random\_shuffle}|hyperpage}{39}
|
||||
\indexentry{bitset|hyperpage}{41}
|
||||
\indexentry{deque|hyperpage}{42}
|
||||
\indexentry{stack|hyperpage}{42}
|
||||
\indexentry{queue|hyperpage}{43}
|
||||
\indexentry{priority queue|hyperpage}{43}
|
||||
\indexentry{heap|hyperpage}{43}
|
||||
\indexentry{subset|hyperpage}{47}
|
||||
\indexentry{permutation|hyperpage}{49}
|
||||
\indexentry{next\_permutation@\texttt{next\_permutation}|hyperpage}{49}
|
||||
\indexentry{backtracking|hyperpage}{50}
|
||||
\indexentry{queen problem|hyperpage}{50}
|
||||
\indexentry{meet in the middle|hyperpage}{54}
|
||||
\indexentry{greedy algorithm|hyperpage}{57}
|
||||
\indexentry{data compression|hyperpage}{62}
|
||||
\indexentry{binary code|hyperpage}{62}
|
||||
\indexentry{codeword|hyperpage}{62}
|
||||
\indexentry{Huffman coding|hyperpage}{63}
|
||||
\indexentry{dynamic programming|hyperpage}{65}
|
||||
\indexentry{memoization|hyperpage}{67}
|
||||
\indexentry{longest increasing subsequence|hyperpage}{70}
|
||||
\indexentry{knapsack|hyperpage}{72}
|
||||
\indexentry{edit distance|hyperpage}{74}
|
||||
\indexentry{Levenshtein distance|hyperpage}{74}
|
||||
\indexentry{amortized analysis|hyperpage}{77}
|
||||
\indexentry{two pointers method|hyperpage}{77}
|
||||
\indexentry{2SUM problem|hyperpage}{78}
|
||||
\indexentry{3SUM problem|hyperpage}{79}
|
||||
\indexentry{nearest smaller elements|hyperpage}{79}
|
||||
\indexentry{sliding window|hyperpage}{81}
|
||||
\indexentry{sliding window minimum|hyperpage}{81}
|
||||
\indexentry{range query|hyperpage}{83}
|
||||
\indexentry{sum query|hyperpage}{83}
|
||||
\indexentry{minimum query|hyperpage}{83}
|
||||
\indexentry{maximum query|hyperpage}{83}
|
||||
\indexentry{prefix sum array|hyperpage}{84}
|
||||
\indexentry{sparse table|hyperpage}{85}
|
||||
\indexentry{binary indexed tree|hyperpage}{86}
|
||||
\indexentry{Fenwick tree|hyperpage}{86}
|
||||
\indexentry{segment tree|hyperpage}{89}
|
||||
\indexentry{index compression|hyperpage}{93}
|
||||
\indexentry{difference array|hyperpage}{93}
|
||||
\indexentry{bit representation|hyperpage}{95}
|
||||
\indexentry{and operation|hyperpage}{96}
|
||||
\indexentry{or operation|hyperpage}{96}
|
||||
\indexentry{xor operation|hyperpage}{97}
|
||||
\indexentry{not operation|hyperpage}{97}
|
||||
\indexentry{bit shift|hyperpage}{97}
|
||||
\indexentry{Hamming distance|hyperpage}{100}
|
||||
\indexentry{graph|hyperpage}{109}
|
||||
\indexentry{node|hyperpage}{109}
|
||||
\indexentry{edge|hyperpage}{109}
|
||||
\indexentry{path|hyperpage}{109}
|
||||
\indexentry{cycle|hyperpage}{109}
|
||||
\indexentry{connected graph|hyperpage}{110}
|
||||
\indexentry{component|hyperpage}{110}
|
||||
\indexentry{tree|hyperpage}{110}
|
||||
\indexentry{directed graph|hyperpage}{110}
|
||||
\indexentry{weighted graph|hyperpage}{111}
|
||||
\indexentry{neighbor|hyperpage}{111}
|
||||
\indexentry{degree|hyperpage}{111}
|
||||
\indexentry{regular graph|hyperpage}{111}
|
||||
\indexentry{complete graph|hyperpage}{111}
|
||||
\indexentry{indegree|hyperpage}{111}
|
||||
\indexentry{outdegree|hyperpage}{111}
|
||||
\indexentry{coloring|hyperpage}{112}
|
||||
\indexentry{bipartite graph|hyperpage}{112}
|
||||
\indexentry{simple graph|hyperpage}{112}
|
||||
\indexentry{adjacency list|hyperpage}{113}
|
||||
\indexentry{adjacency matrix|hyperpage}{114}
|
||||
\indexentry{edge list|hyperpage}{115}
|
||||
\indexentry{depth-first search|hyperpage}{117}
|
||||
\indexentry{breadth-first search|hyperpage}{119}
|
||||
\indexentry{connected graph|hyperpage}{121}
|
||||
\indexentry{cycle|hyperpage}{121}
|
||||
\indexentry{bipartite graph|hyperpage}{122}
|
||||
\indexentry{shortest path|hyperpage}{123}
|
||||
\indexentry{Bellman–Ford algorithm|hyperpage}{123}
|
||||
\indexentry{negative cycle|hyperpage}{125}
|
||||
\indexentry{SPFA algorithm|hyperpage}{126}
|
||||
\indexentry{Dijkstra's algorithm|hyperpage}{126}
|
||||
\indexentry{Floyd–Warshall algorithm|hyperpage}{129}
|
||||
\indexentry{tree|hyperpage}{133}
|
||||
\indexentry{leaf|hyperpage}{133}
|
||||
\indexentry{root|hyperpage}{133}
|
||||
\indexentry{rooted tree|hyperpage}{133}
|
||||
\indexentry{child|hyperpage}{133}
|
||||
\indexentry{parent|hyperpage}{133}
|
||||
\indexentry{subtree|hyperpage}{133}
|
||||
\indexentry{diameter|hyperpage}{135}
|
||||
\indexentry{binary tree|hyperpage}{139}
|
||||
\indexentry{pre-order|hyperpage}{139}
|
||||
\indexentry{in-order|hyperpage}{139}
|
||||
\indexentry{post-order|hyperpage}{139}
|
||||
\indexentry{spanning tree|hyperpage}{141}
|
||||
\indexentry{minimum spanning tree|hyperpage}{141}
|
||||
\indexentry{maximum spanning tree|hyperpage}{142}
|
||||
\indexentry{Kruskal's algorithm|hyperpage}{142}
|
||||
\indexentry{union-find structure|hyperpage}{145}
|
||||
\indexentry{Prim's algorithm|hyperpage}{147}
|
||||
\indexentry{topological sorting|hyperpage}{149}
|
||||
\indexentry{cycle|hyperpage}{149}
|
||||
\indexentry{Dijkstra's algorithm|hyperpage}{153}
|
||||
\indexentry{successor graph|hyperpage}{154}
|
||||
\indexentry{functional graph|hyperpage}{154}
|
||||
\indexentry{cycle|hyperpage}{155}
|
||||
\indexentry{cycle detection|hyperpage}{155}
|
||||
\indexentry{Floyd's algorithm|hyperpage}{156}
|
||||
\indexentry{strongly connected graph|hyperpage}{157}
|
||||
\indexentry{strongly connected component|hyperpage}{157}
|
||||
\indexentry{component graph|hyperpage}{157}
|
||||
\indexentry{Kosaraju's algorithm|hyperpage}{158}
|
||||
\indexentry{2SAT problem|hyperpage}{160}
|
||||
\indexentry{3SAT problem|hyperpage}{162}
|
||||
\indexentry{tree query|hyperpage}{163}
|
||||
\indexentry{ancestor|hyperpage}{163}
|
||||
\indexentry{tree traversal array|hyperpage}{164}
|
||||
\indexentry{lowest common ancestor|hyperpage}{167}
|
||||
\indexentry{Euler tour technique|hyperpage}{168}
|
||||
\indexentry{Eulerian path|hyperpage}{173}
|
||||
\indexentry{Eulerian circuit|hyperpage}{174}
|
||||
\indexentry{Hierholzer's algorithm|hyperpage}{175}
|
||||
\indexentry{Hamiltonian path|hyperpage}{177}
|
||||
\indexentry{Hamiltonian circuit|hyperpage}{177}
|
||||
\indexentry{Dirac's theorem|hyperpage}{177}
|
||||
\indexentry{Ore's theorem|hyperpage}{177}
|
||||
\indexentry{De Bruijn sequence|hyperpage}{178}
|
||||
\indexentry{knight's tour|hyperpage}{179}
|
||||
\indexentry{heuristic|hyperpage}{179}
|
||||
\indexentry{Warnsdorf's rule|hyperpage}{179}
|
||||
\indexentry{flow|hyperpage}{181}
|
||||
\indexentry{maximum flow|hyperpage}{181}
|
||||
\indexentry{cut|hyperpage}{182}
|
||||
\indexentry{minimum cut|hyperpage}{182}
|
||||
\indexentry{Ford–Fulkerson algorithm|hyperpage}{182}
|
||||
\indexentry{Edmonds–Karp algorithm|hyperpage}{184}
|
||||
\indexentry{scaling algorithm|hyperpage}{185}
|
||||
\indexentry{minimum cut|hyperpage}{185}
|
||||
\indexentry{matching|hyperpage}{187}
|
||||
\indexentry{maximum matching|hyperpage}{187}
|
||||
\indexentry{Hall's theorem|hyperpage}{189}
|
||||
\indexentry{perfect matching|hyperpage}{189}
|
||||
\indexentry{Kőnig's theorem|hyperpage}{189}
|
||||
\indexentry{node cover|hyperpage}{189}
|
||||
\indexentry{minimum node cover|hyperpage}{189}
|
||||
\indexentry{independent set|hyperpage}{190}
|
||||
\indexentry{maximum independent set|hyperpage}{190}
|
||||
\indexentry{path cover|hyperpage}{190}
|
||||
\indexentry{Dilworth's theorem|hyperpage}{193}
|
||||
\indexentry{antichain|hyperpage}{193}
|
||||
\indexentry{number theory|hyperpage}{197}
|
||||
\indexentry{divisibility|hyperpage}{197}
|
||||
\indexentry{factor|hyperpage}{197}
|
||||
\indexentry{divisor|hyperpage}{197}
|
||||
\indexentry{prime|hyperpage}{197}
|
||||
\indexentry{prime decomposition|hyperpage}{197}
|
||||
\indexentry{perfect number|hyperpage}{198}
|
||||
\indexentry{Goldbach's conjecture|hyperpage}{199}
|
||||
\indexentry{twin prime|hyperpage}{199}
|
||||
\indexentry{Legendre's conjecture|hyperpage}{199}
|
||||
\indexentry{sieve of Eratosthenes|hyperpage}{200}
|
||||
\indexentry{harmonic sum|hyperpage}{200}
|
||||
\indexentry{greatest common divisor|hyperpage}{200}
|
||||
\indexentry{least common multiple|hyperpage}{200}
|
||||
\indexentry{Euclid's algorithm|hyperpage}{200}
|
||||
\indexentry{coprime|hyperpage}{201}
|
||||
\indexentry{Euler's totient function|hyperpage}{201}
|
||||
\indexentry{modular arithmetic|hyperpage}{201}
|
||||
\indexentry{Fermat's theorem|hyperpage}{202}
|
||||
\indexentry{Euler's theorem|hyperpage}{202}
|
||||
\indexentry{modular inverse|hyperpage}{202}
|
||||
\indexentry{Diophantine equation|hyperpage}{204}
|
||||
\indexentry{extended Euclid's algorithm|hyperpage}{204}
|
||||
\indexentry{Chinese remainder theorem|hyperpage}{205}
|
||||
\indexentry{Lagrange's theorem|hyperpage}{205}
|
||||
\indexentry{Zeckendorf's theorem|hyperpage}{206}
|
||||
\indexentry{Fibonacci number|hyperpage}{206}
|
||||
\indexentry{Pythagorean triple|hyperpage}{206}
|
||||
\indexentry{Euclid's formula|hyperpage}{206}
|
||||
\indexentry{Wilson's theorem|hyperpage}{206}
|
||||
\indexentry{combinatorics|hyperpage}{207}
|
||||
\indexentry{binomial coefficient|hyperpage}{208}
|
||||
\indexentry{Pascal's triangle|hyperpage}{209}
|
||||
\indexentry{multinomial coefficient|hyperpage}{210}
|
||||
\indexentry{Catalan number|hyperpage}{210}
|
||||
\indexentry{parenthesis expression|hyperpage}{211}
|
||||
\indexentry{inclusion-exclusion|hyperpage}{212}
|
||||
\indexentry{derangement|hyperpage}{213}
|
||||
\indexentry{Burnside's lemma|hyperpage}{214}
|
||||
\indexentry{Cayley's formula|hyperpage}{215}
|
||||
\indexentry{Prüfer code|hyperpage}{216}
|
||||
\indexentry{matrix|hyperpage}{217}
|
||||
\indexentry{vector|hyperpage}{217}
|
||||
\indexentry{transpose|hyperpage}{217}
|
||||
\indexentry{square matrix|hyperpage}{217}
|
||||
\indexentry{matrix multiplication|hyperpage}{218}
|
||||
\indexentry{identity matrix|hyperpage}{218}
|
||||
\indexentry{matrix power|hyperpage}{219}
|
||||
\indexentry{determinant|hyperpage}{219}
|
||||
\indexentry{cofactor|hyperpage}{219}
|
||||
\indexentry{inverse matrix|hyperpage}{220}
|
||||
\indexentry{linear recurrence|hyperpage}{220}
|
||||
\indexentry{Fibonacci number|hyperpage}{220}
|
||||
\indexentry{Kirchhoff's theorem|hyperpage}{223}
|
||||
\indexentry{spanning tree|hyperpage}{223}
|
||||
\indexentry{Laplacean matrix|hyperpage}{224}
|
||||
\indexentry{probability|hyperpage}{225}
|
||||
\indexentry{conditional probability|hyperpage}{227}
|
||||
\indexentry{independence|hyperpage}{228}
|
||||
\indexentry{random variable|hyperpage}{228}
|
||||
\indexentry{expected value|hyperpage}{229}
|
||||
\indexentry{distribution|hyperpage}{229}
|
||||
\indexentry{uniform distribution|hyperpage}{230}
|
||||
\indexentry{binomial distribution|hyperpage}{230}
|
||||
\indexentry{geometric distribution|hyperpage}{230}
|
||||
\indexentry{Markov chain|hyperpage}{230}
|
||||
\indexentry{randomized algorithm|hyperpage}{231}
|
||||
\indexentry{Monte Carlo algorithm|hyperpage}{231}
|
||||
\indexentry{Las Vegas algorithm|hyperpage}{231}
|
||||
\indexentry{order statistic|hyperpage}{232}
|
||||
\indexentry{matrix multiplication|hyperpage}{232}
|
||||
\indexentry{quicksort|hyperpage}{232}
|
||||
\indexentry{quickselect|hyperpage}{232}
|
||||
\indexentry{Freivalds' algoritm|hyperpage}{232}
|
||||
\indexentry{coloring|hyperpage}{233}
|
||||
\indexentry{winning state|hyperpage}{235}
|
||||
\indexentry{losing state|hyperpage}{235}
|
||||
\indexentry{nim game|hyperpage}{237}
|
||||
\indexentry{nim sum|hyperpage}{237}
|
||||
\indexentry{misère game|hyperpage}{238}
|
||||
\indexentry{Sprague–Grundy theorem|hyperpage}{238}
|
||||
\indexentry{Grundy number|hyperpage}{238}
|
||||
\indexentry{mex function|hyperpage}{238}
|
||||
\indexentry{Grundy's game|hyperpage}{241}
|
||||
\indexentry{pattern matching|hyperpage}{243}
|
||||
\indexentry{string|hyperpage}{243}
|
||||
\indexentry{alphabet|hyperpage}{243}
|
||||
\indexentry{substring|hyperpage}{243}
|
||||
\indexentry{subsequence|hyperpage}{243}
|
||||
\indexentry{prefix|hyperpage}{243}
|
||||
\indexentry{suffix|hyperpage}{243}
|
||||
\indexentry{rotation|hyperpage}{243}
|
||||
\indexentry{period|hyperpage}{243}
|
||||
\indexentry{border|hyperpage}{244}
|
||||
\indexentry{lexicographical order|hyperpage}{244}
|
||||
\indexentry{trie|hyperpage}{244}
|
||||
\indexentry{hashing|hyperpage}{245}
|
||||
\indexentry{string hashing|hyperpage}{245}
|
||||
\indexentry{hash value|hyperpage}{245}
|
||||
\indexentry{polynomial hashing|hyperpage}{245}
|
||||
\indexentry{collision|hyperpage}{246}
|
||||
\indexentry{birthday paradox|hyperpage}{247}
|
||||
\indexentry{Z-algorithm|hyperpage}{247}
|
||||
\indexentry{Z-array|hyperpage}{247}
|
||||
\indexentry{square root algorithm|hyperpage}{251}
|
||||
\indexentry{Mo's algorithm|hyperpage}{255}
|
||||
\indexentry{segment tree|hyperpage}{257}
|
||||
\indexentry{lazy propagation|hyperpage}{258}
|
||||
\indexentry{lazy segment tree|hyperpage}{258}
|
||||
\indexentry{dynamic segment tree|hyperpage}{261}
|
||||
\indexentry{sparse segment tree|hyperpage}{261}
|
||||
\indexentry{persistent segment tree|hyperpage}{262}
|
||||
\indexentry{two-dimensional segment tree|hyperpage}{264}
|
||||
\indexentry{geometry|hyperpage}{265}
|
||||
\indexentry{Heron's formula|hyperpage}{265}
|
||||
\indexentry{complex number|hyperpage}{266}
|
||||
\indexentry{point|hyperpage}{266}
|
||||
\indexentry{vector|hyperpage}{266}
|
||||
\indexentry{complex@\texttt{complex}|hyperpage}{266}
|
||||
\indexentry{cross product|hyperpage}{268}
|
||||
\indexentry{line segment intersection|hyperpage}{269}
|
||||
\indexentry{shoelace formula|hyperpage}{271}
|
||||
\indexentry{Pick's theorem|hyperpage}{272}
|
||||
\indexentry{distance function|hyperpage}{272}
|
||||
\indexentry{Euclidean distance|hyperpage}{272}
|
||||
\indexentry{Manhattan distance|hyperpage}{272}
|
||||
\indexentry{sweep line|hyperpage}{275}
|
||||
\indexentry{intersection point|hyperpage}{276}
|
||||
\indexentry{closest pair|hyperpage}{277}
|
||||
\indexentry{Andrew's algorithm|hyperpage}{279}
|
||||
|
|
@ -1,6 +0,0 @@
|
|||
This is makeindex, version 2.15 [TeX Live 2020] (kpathsea + Thai support).
|
||||
Scanning input file output/book.idx....done (345 entries accepted, 0 rejected).
|
||||
Sorting entries......done (3076 comparisons).
|
||||
Generating output file output/book.ind....done (401 lines written, 0 warnings).
|
||||
Output written in output/book.ind.
|
||||
Transcript written in output/book.ilg.
|
||||
401
output/book.ind
401
output/book.ind
|
|
@ -1,401 +0,0 @@
|
|||
\begin{theindex}
|
||||
|
||||
\item 2SAT problem, \hyperpage{160}
|
||||
\item 2SUM problem, \hyperpage{78}
|
||||
\item 3SAT problem, \hyperpage{162}
|
||||
\item 3SUM problem, \hyperpage{79}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item adjacency list, \hyperpage{113}
|
||||
\item adjacency matrix, \hyperpage{114}
|
||||
\item alphabet, \hyperpage{243}
|
||||
\item amortized analysis, \hyperpage{77}
|
||||
\item ancestor, \hyperpage{163}
|
||||
\item and operation, \hyperpage{96}
|
||||
\item Andrew's algorithm, \hyperpage{279}
|
||||
\item antichain, \hyperpage{193}
|
||||
\item arithmetic progression, \hyperpage{10}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item backtracking, \hyperpage{50}
|
||||
\item Bellman–Ford algorithm, \hyperpage{123}
|
||||
\item binary code, \hyperpage{62}
|
||||
\item binary indexed tree, \hyperpage{86}
|
||||
\item binary search, \hyperpage{31}
|
||||
\item binary tree, \hyperpage{139}
|
||||
\item Binet's formula, \hyperpage{14}
|
||||
\item binomial coefficient, \hyperpage{208}
|
||||
\item binomial distribution, \hyperpage{230}
|
||||
\item bipartite graph, \hyperpage{112}, \hyperpage{122}
|
||||
\item birthday paradox, \hyperpage{247}
|
||||
\item bit representation, \hyperpage{95}
|
||||
\item bit shift, \hyperpage{97}
|
||||
\item bitset, \hyperpage{41}
|
||||
\item border, \hyperpage{244}
|
||||
\item breadth-first search, \hyperpage{119}
|
||||
\item bubble sort, \hyperpage{25}
|
||||
\item Burnside's lemma, \hyperpage{214}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Catalan number, \hyperpage{210}
|
||||
\item Cayley's formula, \hyperpage{215}
|
||||
\item child, \hyperpage{133}
|
||||
\item Chinese remainder theorem, \hyperpage{205}
|
||||
\item closest pair, \hyperpage{277}
|
||||
\item codeword, \hyperpage{62}
|
||||
\item cofactor, \hyperpage{219}
|
||||
\item collision, \hyperpage{246}
|
||||
\item coloring, \hyperpage{112}, \hyperpage{233}
|
||||
\item combinatorics, \hyperpage{207}
|
||||
\item comparison function, \hyperpage{31}
|
||||
\item comparison operator, \hyperpage{30}
|
||||
\item complement, \hyperpage{12}
|
||||
\item complete graph, \hyperpage{111}
|
||||
\item \texttt{complex}, \hyperpage{266}
|
||||
\item complex number, \hyperpage{266}
|
||||
\item complexity classes, \hyperpage{20}
|
||||
\item component, \hyperpage{110}
|
||||
\item component graph, \hyperpage{157}
|
||||
\item conditional probability, \hyperpage{227}
|
||||
\item conjuction, \hyperpage{13}
|
||||
\item connected graph, \hyperpage{110}, \hyperpage{121}
|
||||
\item constant factor, \hyperpage{21}
|
||||
\item constant-time algorithm, \hyperpage{20}
|
||||
\item coprime, \hyperpage{201}
|
||||
\item counting sort, \hyperpage{28}
|
||||
\item cross product, \hyperpage{268}
|
||||
\item cubic algorithm, \hyperpage{20}
|
||||
\item cut, \hyperpage{182}
|
||||
\item cycle, \hyperpage{109}, \hyperpage{121}, \hyperpage{149},
|
||||
\hyperpage{155}
|
||||
\item cycle detection, \hyperpage{155}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item data compression, \hyperpage{62}
|
||||
\item data structure, \hyperpage{35}
|
||||
\item De Bruijn sequence, \hyperpage{178}
|
||||
\item degree, \hyperpage{111}
|
||||
\item depth-first search, \hyperpage{117}
|
||||
\item deque, \hyperpage{42}
|
||||
\item derangement, \hyperpage{213}
|
||||
\item determinant, \hyperpage{219}
|
||||
\item diameter, \hyperpage{135}
|
||||
\item difference, \hyperpage{12}
|
||||
\item difference array, \hyperpage{93}
|
||||
\item Dijkstra's algorithm, \hyperpage{126}, \hyperpage{153}
|
||||
\item Dilworth's theorem, \hyperpage{193}
|
||||
\item Diophantine equation, \hyperpage{204}
|
||||
\item Dirac's theorem, \hyperpage{177}
|
||||
\item directed graph, \hyperpage{110}
|
||||
\item disjunction, \hyperpage{13}
|
||||
\item distance function, \hyperpage{272}
|
||||
\item distribution, \hyperpage{229}
|
||||
\item divisibility, \hyperpage{197}
|
||||
\item divisor, \hyperpage{197}
|
||||
\item dynamic array, \hyperpage{35}
|
||||
\item dynamic programming, \hyperpage{65}
|
||||
\item dynamic segment tree, \hyperpage{261}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item edge, \hyperpage{109}
|
||||
\item edge list, \hyperpage{115}
|
||||
\item edit distance, \hyperpage{74}
|
||||
\item Edmonds–Karp algorithm, \hyperpage{184}
|
||||
\item equivalence, \hyperpage{13}
|
||||
\item Euclid's algorithm, \hyperpage{200}
|
||||
\item Euclid's formula, \hyperpage{206}
|
||||
\item Euclidean distance, \hyperpage{272}
|
||||
\item Euler tour technique, \hyperpage{168}
|
||||
\item Euler's theorem, \hyperpage{202}
|
||||
\item Euler's totient function, \hyperpage{201}
|
||||
\item Eulerian circuit, \hyperpage{174}
|
||||
\item Eulerian path, \hyperpage{173}
|
||||
\item expected value, \hyperpage{229}
|
||||
\item extended Euclid's algorithm, \hyperpage{204}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item factor, \hyperpage{197}
|
||||
\item factorial, \hyperpage{14}
|
||||
\item Faulhaber's formula, \hyperpage{10}
|
||||
\item Fenwick tree, \hyperpage{86}
|
||||
\item Fermat's theorem, \hyperpage{202}
|
||||
\item Fibonacci number, \hyperpage{14}, \hyperpage{206},
|
||||
\hyperpage{220}
|
||||
\item floating point number, \hyperpage{7}
|
||||
\item flow, \hyperpage{181}
|
||||
\item Floyd's algorithm, \hyperpage{156}
|
||||
\item Floyd–Warshall algorithm, \hyperpage{129}
|
||||
\item Ford–Fulkerson algorithm, \hyperpage{182}
|
||||
\item Freivalds' algoritm, \hyperpage{232}
|
||||
\item functional graph, \hyperpage{154}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item geometric distribution, \hyperpage{230}
|
||||
\item geometric progression, \hyperpage{11}
|
||||
\item geometry, \hyperpage{265}
|
||||
\item Goldbach's conjecture, \hyperpage{199}
|
||||
\item graph, \hyperpage{109}
|
||||
\item greatest common divisor, \hyperpage{200}
|
||||
\item greedy algorithm, \hyperpage{57}
|
||||
\item Grundy number, \hyperpage{238}
|
||||
\item Grundy's game, \hyperpage{241}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Hall's theorem, \hyperpage{189}
|
||||
\item Hamiltonian circuit, \hyperpage{177}
|
||||
\item Hamiltonian path, \hyperpage{177}
|
||||
\item Hamming distance, \hyperpage{100}
|
||||
\item harmonic sum, \hyperpage{11}, \hyperpage{200}
|
||||
\item hash value, \hyperpage{245}
|
||||
\item hashing, \hyperpage{245}
|
||||
\item heap, \hyperpage{43}
|
||||
\item Heron's formula, \hyperpage{265}
|
||||
\item heuristic, \hyperpage{179}
|
||||
\item Hierholzer's algorithm, \hyperpage{175}
|
||||
\item Huffman coding, \hyperpage{63}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item identity matrix, \hyperpage{218}
|
||||
\item implication, \hyperpage{13}
|
||||
\item in-order, \hyperpage{139}
|
||||
\item inclusion-exclusion, \hyperpage{212}
|
||||
\item indegree, \hyperpage{111}
|
||||
\item independence, \hyperpage{228}
|
||||
\item independent set, \hyperpage{190}
|
||||
\item index compression, \hyperpage{93}
|
||||
\item input and output, \hyperpage{4}
|
||||
\item integer, \hyperpage{6}
|
||||
\item intersection, \hyperpage{12}
|
||||
\item intersection point, \hyperpage{276}
|
||||
\item inverse matrix, \hyperpage{220}
|
||||
\item inversion, \hyperpage{26}
|
||||
\item iterator, \hyperpage{39}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Kadane's algorithm, \hyperpage{23}
|
||||
\item Kirchhoff's theorem, \hyperpage{223}
|
||||
\item knapsack, \hyperpage{72}
|
||||
\item knight's tour, \hyperpage{179}
|
||||
\item Kosaraju's algorithm, \hyperpage{158}
|
||||
\item Kruskal's algorithm, \hyperpage{142}
|
||||
\item Kőnig's theorem, \hyperpage{189}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Lagrange's theorem, \hyperpage{205}
|
||||
\item Laplacean matrix, \hyperpage{224}
|
||||
\item Las Vegas algorithm, \hyperpage{231}
|
||||
\item lazy propagation, \hyperpage{258}
|
||||
\item lazy segment tree, \hyperpage{258}
|
||||
\item leaf, \hyperpage{133}
|
||||
\item least common multiple, \hyperpage{200}
|
||||
\item Legendre's conjecture, \hyperpage{199}
|
||||
\item Levenshtein distance, \hyperpage{74}
|
||||
\item lexicographical order, \hyperpage{244}
|
||||
\item line segment intersection, \hyperpage{269}
|
||||
\item linear algorithm, \hyperpage{20}
|
||||
\item linear recurrence, \hyperpage{220}
|
||||
\item logarithm, \hyperpage{14}
|
||||
\item logarithmic algorithm, \hyperpage{20}
|
||||
\item logic, \hyperpage{13}
|
||||
\item longest increasing subsequence, \hyperpage{70}
|
||||
\item losing state, \hyperpage{235}
|
||||
\item lowest common ancestor, \hyperpage{167}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item macro, \hyperpage{9}
|
||||
\item Manhattan distance, \hyperpage{272}
|
||||
\item map, \hyperpage{38}
|
||||
\item Markov chain, \hyperpage{230}
|
||||
\item matching, \hyperpage{187}
|
||||
\item matrix, \hyperpage{217}
|
||||
\item matrix multiplication, \hyperpage{218}, \hyperpage{232}
|
||||
\item matrix power, \hyperpage{219}
|
||||
\item maximum flow, \hyperpage{181}
|
||||
\item maximum independent set, \hyperpage{190}
|
||||
\item maximum matching, \hyperpage{187}
|
||||
\item maximum query, \hyperpage{83}
|
||||
\item maximum spanning tree, \hyperpage{142}
|
||||
\item maximum subarray sum, \hyperpage{21}
|
||||
\item meet in the middle, \hyperpage{54}
|
||||
\item memoization, \hyperpage{67}
|
||||
\item merge sort, \hyperpage{27}
|
||||
\item mex function, \hyperpage{238}
|
||||
\item minimum cut, \hyperpage{182}, \hyperpage{185}
|
||||
\item minimum node cover, \hyperpage{189}
|
||||
\item minimum query, \hyperpage{83}
|
||||
\item minimum spanning tree, \hyperpage{141}
|
||||
\item misère game, \hyperpage{238}
|
||||
\item Mo's algorithm, \hyperpage{255}
|
||||
\item modular arithmetic, \hyperpage{6}, \hyperpage{201}
|
||||
\item modular inverse, \hyperpage{202}
|
||||
\item Monte Carlo algorithm, \hyperpage{231}
|
||||
\item multinomial coefficient, \hyperpage{210}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item natural logarithm, \hyperpage{15}
|
||||
\item nearest smaller elements, \hyperpage{79}
|
||||
\item negation, \hyperpage{13}
|
||||
\item negative cycle, \hyperpage{125}
|
||||
\item neighbor, \hyperpage{111}
|
||||
\item \texttt{next\_permutation}, \hyperpage{49}
|
||||
\item nim game, \hyperpage{237}
|
||||
\item nim sum, \hyperpage{237}
|
||||
\item node, \hyperpage{109}
|
||||
\item node cover, \hyperpage{189}
|
||||
\item not operation, \hyperpage{97}
|
||||
\item NP-hard problem, \hyperpage{20}
|
||||
\item number theory, \hyperpage{197}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item or operation, \hyperpage{96}
|
||||
\item order statistic, \hyperpage{232}
|
||||
\item Ore's theorem, \hyperpage{177}
|
||||
\item outdegree, \hyperpage{111}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item \texttt{pair}, \hyperpage{30}
|
||||
\item parent, \hyperpage{133}
|
||||
\item parenthesis expression, \hyperpage{211}
|
||||
\item Pascal's triangle, \hyperpage{209}
|
||||
\item path, \hyperpage{109}
|
||||
\item path cover, \hyperpage{190}
|
||||
\item pattern matching, \hyperpage{243}
|
||||
\item perfect matching, \hyperpage{189}
|
||||
\item perfect number, \hyperpage{198}
|
||||
\item period, \hyperpage{243}
|
||||
\item permutation, \hyperpage{49}
|
||||
\item persistent segment tree, \hyperpage{262}
|
||||
\item Pick's theorem, \hyperpage{272}
|
||||
\item point, \hyperpage{266}
|
||||
\item polynomial algorithm, \hyperpage{20}
|
||||
\item polynomial hashing, \hyperpage{245}
|
||||
\item post-order, \hyperpage{139}
|
||||
\item pre-order, \hyperpage{139}
|
||||
\item predicate, \hyperpage{13}
|
||||
\item prefix, \hyperpage{243}
|
||||
\item prefix sum array, \hyperpage{84}
|
||||
\item Prim's algorithm, \hyperpage{147}
|
||||
\item prime, \hyperpage{197}
|
||||
\item prime decomposition, \hyperpage{197}
|
||||
\item priority queue, \hyperpage{43}
|
||||
\item probability, \hyperpage{225}
|
||||
\item programming language, \hyperpage{3}
|
||||
\item Prüfer code, \hyperpage{216}
|
||||
\item Pythagorean triple, \hyperpage{206}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item quadratic algorithm, \hyperpage{20}
|
||||
\item quantifier, \hyperpage{13}
|
||||
\item queen problem, \hyperpage{50}
|
||||
\item queue, \hyperpage{43}
|
||||
\item quickselect, \hyperpage{232}
|
||||
\item quicksort, \hyperpage{232}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item random variable, \hyperpage{228}
|
||||
\item \texttt{random\_shuffle}, \hyperpage{39}
|
||||
\item randomized algorithm, \hyperpage{231}
|
||||
\item range query, \hyperpage{83}
|
||||
\item regular graph, \hyperpage{111}
|
||||
\item remainder, \hyperpage{6}
|
||||
\item \texttt{reverse}, \hyperpage{39}
|
||||
\item root, \hyperpage{133}
|
||||
\item rooted tree, \hyperpage{133}
|
||||
\item rotation, \hyperpage{243}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item scaling algorithm, \hyperpage{185}
|
||||
\item segment tree, \hyperpage{89}, \hyperpage{257}
|
||||
\item set, \hyperpage{12}, \hyperpage{37}
|
||||
\item set theory, \hyperpage{12}
|
||||
\item shoelace formula, \hyperpage{271}
|
||||
\item shortest path, \hyperpage{123}
|
||||
\item sieve of Eratosthenes, \hyperpage{200}
|
||||
\item simple graph, \hyperpage{112}
|
||||
\item sliding window, \hyperpage{81}
|
||||
\item sliding window minimum, \hyperpage{81}
|
||||
\item \texttt{sort}, \hyperpage{29}, \hyperpage{39}
|
||||
\item sorting, \hyperpage{25}
|
||||
\item spanning tree, \hyperpage{141}, \hyperpage{223}
|
||||
\item sparse segment tree, \hyperpage{261}
|
||||
\item sparse table, \hyperpage{85}
|
||||
\item SPFA algorithm, \hyperpage{126}
|
||||
\item Sprague–Grundy theorem, \hyperpage{238}
|
||||
\item square matrix, \hyperpage{217}
|
||||
\item square root algorithm, \hyperpage{251}
|
||||
\item stack, \hyperpage{42}
|
||||
\item string, \hyperpage{36}, \hyperpage{243}
|
||||
\item string hashing, \hyperpage{245}
|
||||
\item strongly connected component, \hyperpage{157}
|
||||
\item strongly connected graph, \hyperpage{157}
|
||||
\item subsequence, \hyperpage{243}
|
||||
\item subset, \hyperpage{12}, \hyperpage{47}
|
||||
\item substring, \hyperpage{243}
|
||||
\item subtree, \hyperpage{133}
|
||||
\item successor graph, \hyperpage{154}
|
||||
\item suffix, \hyperpage{243}
|
||||
\item sum query, \hyperpage{83}
|
||||
\item sweep line, \hyperpage{275}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item time complexity, \hyperpage{17}
|
||||
\item topological sorting, \hyperpage{149}
|
||||
\item transpose, \hyperpage{217}
|
||||
\item tree, \hyperpage{110}, \hyperpage{133}
|
||||
\item tree query, \hyperpage{163}
|
||||
\item tree traversal array, \hyperpage{164}
|
||||
\item trie, \hyperpage{244}
|
||||
\item \texttt{tuple}, \hyperpage{30}
|
||||
\item \texttt{typedef}, \hyperpage{8}
|
||||
\item twin prime, \hyperpage{199}
|
||||
\item two pointers method, \hyperpage{77}
|
||||
\item two-dimensional segment tree, \hyperpage{264}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item uniform distribution, \hyperpage{230}
|
||||
\item union, \hyperpage{12}
|
||||
\item union-find structure, \hyperpage{145}
|
||||
\item universal set, \hyperpage{12}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item vector, \hyperpage{35}, \hyperpage{217}, \hyperpage{266}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Warnsdorf's rule, \hyperpage{179}
|
||||
\item weighted graph, \hyperpage{111}
|
||||
\item Wilson's theorem, \hyperpage{206}
|
||||
\item winning state, \hyperpage{235}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item xor operation, \hyperpage{97}
|
||||
|
||||
\indexspace
|
||||
|
||||
\item Z-algorithm, \hyperpage{247}
|
||||
\item Z-array, \hyperpage{247}
|
||||
\item Zeckendorf's theorem, \hyperpage{206}
|
||||
|
||||
\end{theindex}
|
||||
2245
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output/book.out
|
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@ -1,154 +0,0 @@
|
|||
\BOOKMARK [0][-]{chapter*.2}{Preface}{}% 1
|
||||
\BOOKMARK [-1][-]{part.1}{I Basic techniques}{}% 2
|
||||
\BOOKMARK [0][-]{chapter.1}{Introduction}{part.1}% 3
|
||||
\BOOKMARK [1][-]{section.1.1}{Programming languages}{chapter.1}% 4
|
||||
\BOOKMARK [1][-]{section.1.2}{Input and output}{chapter.1}% 5
|
||||
\BOOKMARK [1][-]{section.1.3}{Working with numbers}{chapter.1}% 6
|
||||
\BOOKMARK [1][-]{section.1.4}{Shortening code}{chapter.1}% 7
|
||||
\BOOKMARK [1][-]{section.1.5}{Mathematics}{chapter.1}% 8
|
||||
\BOOKMARK [1][-]{section.1.6}{Contests and resources}{chapter.1}% 9
|
||||
\BOOKMARK [0][-]{chapter.2}{Time complexity}{part.1}% 10
|
||||
\BOOKMARK [1][-]{section.2.1}{Calculation rules}{chapter.2}% 11
|
||||
\BOOKMARK [1][-]{section.2.2}{Complexity classes}{chapter.2}% 12
|
||||
\BOOKMARK [1][-]{section.2.3}{Estimating efficiency}{chapter.2}% 13
|
||||
\BOOKMARK [1][-]{section.2.4}{Maximum subarray sum}{chapter.2}% 14
|
||||
\BOOKMARK [0][-]{chapter.3}{Sorting}{part.1}% 15
|
||||
\BOOKMARK [1][-]{section.3.1}{Sorting theory}{chapter.3}% 16
|
||||
\BOOKMARK [1][-]{section.3.2}{Sorting in C++}{chapter.3}% 17
|
||||
\BOOKMARK [1][-]{section.3.3}{Binary search}{chapter.3}% 18
|
||||
\BOOKMARK [0][-]{chapter.4}{Data structures}{part.1}% 19
|
||||
\BOOKMARK [1][-]{section.4.1}{Dynamic arrays}{chapter.4}% 20
|
||||
\BOOKMARK [1][-]{section.4.2}{Set structures}{chapter.4}% 21
|
||||
\BOOKMARK [1][-]{section.4.3}{Map structures}{chapter.4}% 22
|
||||
\BOOKMARK [1][-]{section.4.4}{Iterators and ranges}{chapter.4}% 23
|
||||
\BOOKMARK [1][-]{section.4.5}{Other structures}{chapter.4}% 24
|
||||
\BOOKMARK [1][-]{section.4.6}{Comparison to sorting}{chapter.4}% 25
|
||||
\BOOKMARK [0][-]{chapter.5}{Complete search}{part.1}% 26
|
||||
\BOOKMARK [1][-]{section.5.1}{Generating subsets}{chapter.5}% 27
|
||||
\BOOKMARK [1][-]{section.5.2}{Generating permutations}{chapter.5}% 28
|
||||
\BOOKMARK [1][-]{section.5.3}{Backtracking}{chapter.5}% 29
|
||||
\BOOKMARK [1][-]{section.5.4}{Pruning the search}{chapter.5}% 30
|
||||
\BOOKMARK [1][-]{section.5.5}{Meet in the middle}{chapter.5}% 31
|
||||
\BOOKMARK [0][-]{chapter.6}{Greedy algorithms}{part.1}% 32
|
||||
\BOOKMARK [1][-]{section.6.1}{Coin problem}{chapter.6}% 33
|
||||
\BOOKMARK [1][-]{section.6.2}{Scheduling}{chapter.6}% 34
|
||||
\BOOKMARK [1][-]{section.6.3}{Tasks and deadlines}{chapter.6}% 35
|
||||
\BOOKMARK [1][-]{section.6.4}{Minimizing sums}{chapter.6}% 36
|
||||
\BOOKMARK [1][-]{section.6.5}{Data compression}{chapter.6}% 37
|
||||
\BOOKMARK [0][-]{chapter.7}{Dynamic programming}{part.1}% 38
|
||||
\BOOKMARK [1][-]{section.7.1}{Coin problem}{chapter.7}% 39
|
||||
\BOOKMARK [1][-]{section.7.2}{Longest increasing subsequence}{chapter.7}% 40
|
||||
\BOOKMARK [1][-]{section.7.3}{Paths in a grid}{chapter.7}% 41
|
||||
\BOOKMARK [1][-]{section.7.4}{Knapsack problems}{chapter.7}% 42
|
||||
\BOOKMARK [1][-]{section.7.5}{Edit distance}{chapter.7}% 43
|
||||
\BOOKMARK [1][-]{section.7.6}{Counting tilings}{chapter.7}% 44
|
||||
\BOOKMARK [0][-]{chapter.8}{Amortized analysis}{part.1}% 45
|
||||
\BOOKMARK [1][-]{section.8.1}{Two pointers method}{chapter.8}% 46
|
||||
\BOOKMARK [1][-]{section.8.2}{Nearest smaller elements}{chapter.8}% 47
|
||||
\BOOKMARK [1][-]{section.8.3}{Sliding window minimum}{chapter.8}% 48
|
||||
\BOOKMARK [0][-]{chapter.9}{Range queries}{part.1}% 49
|
||||
\BOOKMARK [1][-]{section.9.1}{Static array queries}{chapter.9}% 50
|
||||
\BOOKMARK [1][-]{section.9.2}{Binary indexed tree}{chapter.9}% 51
|
||||
\BOOKMARK [1][-]{section.9.3}{Segment tree}{chapter.9}% 52
|
||||
\BOOKMARK [1][-]{section.9.4}{Additional techniques}{chapter.9}% 53
|
||||
\BOOKMARK [0][-]{chapter.10}{Bit manipulation}{part.1}% 54
|
||||
\BOOKMARK [1][-]{section.10.1}{Bit representation}{chapter.10}% 55
|
||||
\BOOKMARK [1][-]{section.10.2}{Bit operations}{chapter.10}% 56
|
||||
\BOOKMARK [1][-]{section.10.3}{Representing sets}{chapter.10}% 57
|
||||
\BOOKMARK [1][-]{section.10.4}{Bit optimizations}{chapter.10}% 58
|
||||
\BOOKMARK [1][-]{section.10.5}{Dynamic programming}{chapter.10}% 59
|
||||
\BOOKMARK [-1][-]{part.2}{II Graph algorithms}{}% 60
|
||||
\BOOKMARK [0][-]{chapter.11}{Basics of graphs}{part.2}% 61
|
||||
\BOOKMARK [1][-]{section.11.1}{Graph terminology}{chapter.11}% 62
|
||||
\BOOKMARK [1][-]{section.11.2}{Graph representation}{chapter.11}% 63
|
||||
\BOOKMARK [0][-]{chapter.12}{Graph traversal}{part.2}% 64
|
||||
\BOOKMARK [1][-]{section.12.1}{Depth-first search}{chapter.12}% 65
|
||||
\BOOKMARK [1][-]{section.12.2}{Breadth-first search}{chapter.12}% 66
|
||||
\BOOKMARK [1][-]{section.12.3}{Applications}{chapter.12}% 67
|
||||
\BOOKMARK [0][-]{chapter.13}{Shortest paths}{part.2}% 68
|
||||
\BOOKMARK [1][-]{section.13.1}{Bellman\205Ford algorithm}{chapter.13}% 69
|
||||
\BOOKMARK [1][-]{section.13.2}{Dijkstra's algorithm}{chapter.13}% 70
|
||||
\BOOKMARK [1][-]{section.13.3}{Floyd\205Warshall algorithm}{chapter.13}% 71
|
||||
\BOOKMARK [0][-]{chapter.14}{Tree algorithms}{part.2}% 72
|
||||
\BOOKMARK [1][-]{section.14.1}{Tree traversal}{chapter.14}% 73
|
||||
\BOOKMARK [1][-]{section.14.2}{Diameter}{chapter.14}% 74
|
||||
\BOOKMARK [1][-]{section.14.3}{All longest paths}{chapter.14}% 75
|
||||
\BOOKMARK [1][-]{section.14.4}{Binary trees}{chapter.14}% 76
|
||||
\BOOKMARK [0][-]{chapter.15}{Spanning trees}{part.2}% 77
|
||||
\BOOKMARK [1][-]{section.15.1}{Kruskal's algorithm}{chapter.15}% 78
|
||||
\BOOKMARK [1][-]{section.15.2}{Union-find structure}{chapter.15}% 79
|
||||
\BOOKMARK [1][-]{section.15.3}{Prim's algorithm}{chapter.15}% 80
|
||||
\BOOKMARK [0][-]{chapter.16}{Directed graphs}{part.2}% 81
|
||||
\BOOKMARK [1][-]{section.16.1}{Topological sorting}{chapter.16}% 82
|
||||
\BOOKMARK [1][-]{section.16.2}{Dynamic programming}{chapter.16}% 83
|
||||
\BOOKMARK [1][-]{section.16.3}{Successor paths}{chapter.16}% 84
|
||||
\BOOKMARK [1][-]{section.16.4}{Cycle detection}{chapter.16}% 85
|
||||
\BOOKMARK [0][-]{chapter.17}{Strong connectivity}{part.2}% 86
|
||||
\BOOKMARK [1][-]{section.17.1}{Kosaraju's algorithm}{chapter.17}% 87
|
||||
\BOOKMARK [1][-]{section.17.2}{2SAT problem}{chapter.17}% 88
|
||||
\BOOKMARK [0][-]{chapter.18}{Tree queries}{part.2}% 89
|
||||
\BOOKMARK [1][-]{section.18.1}{Finding ancestors}{chapter.18}% 90
|
||||
\BOOKMARK [1][-]{section.18.2}{Subtrees and paths}{chapter.18}% 91
|
||||
\BOOKMARK [1][-]{section.18.3}{Lowest common ancestor}{chapter.18}% 92
|
||||
\BOOKMARK [1][-]{section.18.4}{Offline algorithms}{chapter.18}% 93
|
||||
\BOOKMARK [0][-]{chapter.19}{Paths and circuits}{part.2}% 94
|
||||
\BOOKMARK [1][-]{section.19.1}{Eulerian paths}{chapter.19}% 95
|
||||
\BOOKMARK [1][-]{section.19.2}{Hamiltonian paths}{chapter.19}% 96
|
||||
\BOOKMARK [1][-]{section.19.3}{De Bruijn sequences}{chapter.19}% 97
|
||||
\BOOKMARK [1][-]{section.19.4}{Knight's tours}{chapter.19}% 98
|
||||
\BOOKMARK [0][-]{chapter.20}{Flows and cuts}{part.2}% 99
|
||||
\BOOKMARK [1][-]{section.20.1}{Ford\205Fulkerson algorithm}{chapter.20}% 100
|
||||
\BOOKMARK [1][-]{section.20.2}{Disjoint paths}{chapter.20}% 101
|
||||
\BOOKMARK [1][-]{section.20.3}{Maximum matchings}{chapter.20}% 102
|
||||
\BOOKMARK [1][-]{section.20.4}{Path covers}{chapter.20}% 103
|
||||
\BOOKMARK [-1][-]{part.3}{III Advanced topics}{}% 104
|
||||
\BOOKMARK [0][-]{chapter.21}{Number theory}{part.3}% 105
|
||||
\BOOKMARK [1][-]{section.21.1}{Primes and factors}{chapter.21}% 106
|
||||
\BOOKMARK [1][-]{section.21.2}{Modular arithmetic}{chapter.21}% 107
|
||||
\BOOKMARK [1][-]{section.21.3}{Solving equations}{chapter.21}% 108
|
||||
\BOOKMARK [1][-]{section.21.4}{Other results}{chapter.21}% 109
|
||||
\BOOKMARK [0][-]{chapter.22}{Combinatorics}{part.3}% 110
|
||||
\BOOKMARK [1][-]{section.22.1}{Binomial coefficients}{chapter.22}% 111
|
||||
\BOOKMARK [1][-]{section.22.2}{Catalan numbers}{chapter.22}% 112
|
||||
\BOOKMARK [1][-]{section.22.3}{Inclusion-exclusion}{chapter.22}% 113
|
||||
\BOOKMARK [1][-]{section.22.4}{Burnside's lemma}{chapter.22}% 114
|
||||
\BOOKMARK [1][-]{section.22.5}{Cayley's formula}{chapter.22}% 115
|
||||
\BOOKMARK [0][-]{chapter.23}{Matrices}{part.3}% 116
|
||||
\BOOKMARK [1][-]{section.23.1}{Operations}{chapter.23}% 117
|
||||
\BOOKMARK [1][-]{section.23.2}{Linear recurrences}{chapter.23}% 118
|
||||
\BOOKMARK [1][-]{section.23.3}{Graphs and matrices}{chapter.23}% 119
|
||||
\BOOKMARK [0][-]{chapter.24}{Probability}{part.3}% 120
|
||||
\BOOKMARK [1][-]{section.24.1}{Calculation}{chapter.24}% 121
|
||||
\BOOKMARK [1][-]{section.24.2}{Events}{chapter.24}% 122
|
||||
\BOOKMARK [1][-]{section.24.3}{Random variables}{chapter.24}% 123
|
||||
\BOOKMARK [1][-]{section.24.4}{Markov chains}{chapter.24}% 124
|
||||
\BOOKMARK [1][-]{section.24.5}{Randomized algorithms}{chapter.24}% 125
|
||||
\BOOKMARK [0][-]{chapter.25}{Game theory}{part.3}% 126
|
||||
\BOOKMARK [1][-]{section.25.1}{Game states}{chapter.25}% 127
|
||||
\BOOKMARK [1][-]{section.25.2}{Nim game}{chapter.25}% 128
|
||||
\BOOKMARK [1][-]{section.25.3}{Sprague\205Grundy theorem}{chapter.25}% 129
|
||||
\BOOKMARK [0][-]{chapter.26}{String algorithms}{part.3}% 130
|
||||
\BOOKMARK [1][-]{section.26.1}{String terminology}{chapter.26}% 131
|
||||
\BOOKMARK [1][-]{section.26.2}{Trie structure}{chapter.26}% 132
|
||||
\BOOKMARK [1][-]{section.26.3}{String hashing}{chapter.26}% 133
|
||||
\BOOKMARK [1][-]{section.26.4}{Z-algorithm}{chapter.26}% 134
|
||||
\BOOKMARK [0][-]{chapter.27}{Square root algorithms}{part.3}% 135
|
||||
\BOOKMARK [1][-]{section.27.1}{Combining algorithms}{chapter.27}% 136
|
||||
\BOOKMARK [1][-]{section.27.2}{Integer partitions}{chapter.27}% 137
|
||||
\BOOKMARK [1][-]{section.27.3}{Mo's algorithm}{chapter.27}% 138
|
||||
\BOOKMARK [0][-]{chapter.28}{Segment trees revisited}{part.3}% 139
|
||||
\BOOKMARK [1][-]{section.28.1}{Lazy propagation}{chapter.28}% 140
|
||||
\BOOKMARK [1][-]{section.28.2}{Dynamic trees}{chapter.28}% 141
|
||||
\BOOKMARK [1][-]{section.28.3}{Data structures}{chapter.28}% 142
|
||||
\BOOKMARK [1][-]{section.28.4}{Two-dimensionality}{chapter.28}% 143
|
||||
\BOOKMARK [0][-]{chapter.29}{Geometry}{part.3}% 144
|
||||
\BOOKMARK [1][-]{section.29.1}{Complex numbers}{chapter.29}% 145
|
||||
\BOOKMARK [1][-]{section.29.2}{Points and lines}{chapter.29}% 146
|
||||
\BOOKMARK [1][-]{section.29.3}{Polygon area}{chapter.29}% 147
|
||||
\BOOKMARK [1][-]{section.29.4}{Distance functions}{chapter.29}% 148
|
||||
\BOOKMARK [0][-]{chapter.30}{Sweep line algorithms}{part.3}% 149
|
||||
\BOOKMARK [1][-]{section.30.1}{Intersection points}{chapter.30}% 150
|
||||
\BOOKMARK [1][-]{section.30.2}{Closest pair problem}{chapter.30}% 151
|
||||
\BOOKMARK [1][-]{section.30.3}{Convex hull problem}{chapter.30}% 152
|
||||
\BOOKMARK [0][-]{section*.3}{Bibliography}{part.3}% 153
|
||||
\BOOKMARK [0][-]{chapter*.5}{Index}{part.3}% 154
|
||||
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|
|||
\babel@toc {english}{}
|
||||
\contentsline {chapter}{Preface}{ix}{chapter*.2}%
|
||||
\contentsline {part}{I\hspace {1em}Basic techniques}{1}{part.1}%
|
||||
\contentsline {chapter}{\numberline {1}Introduction}{3}{chapter.1}%
|
||||
\contentsline {section}{\numberline {1.1}Programming languages}{3}{section.1.1}%
|
||||
\contentsline {subsubsection}{C++ code template}{4}{section.1.1}%
|
||||
\contentsline {section}{\numberline {1.2}Input and output}{4}{section.1.2}%
|
||||
\contentsline {section}{\numberline {1.3}Working with numbers}{6}{section.1.3}%
|
||||
\contentsline {subsubsection}{Integers}{6}{section.1.3}%
|
||||
\contentsline {subsubsection}{Modular arithmetic}{6}{lstnumber.-14.3}%
|
||||
\contentsline {subsubsection}{Floating point numbers}{7}{lstnumber.-16.2}%
|
||||
\contentsline {section}{\numberline {1.4}Shortening code}{8}{section.1.4}%
|
||||
\contentsline {subsubsection}{Type names}{8}{section.1.4}%
|
||||
\contentsline {subsubsection}{Macros}{9}{lstnumber.-23.2}%
|
||||
\contentsline {section}{\numberline {1.5}Mathematics}{10}{section.1.5}%
|
||||
\contentsline {subsubsection}{Sum formulas}{10}{section.1.5}%
|
||||
\contentsline {subsubsection}{Set theory}{12}{section.1.5}%
|
||||
\contentsline {subsubsection}{Logic}{13}{section.1.5}%
|
||||
\contentsline {subsubsection}{Functions}{13}{section.1.5}%
|
||||
\contentsline {subsubsection}{Logarithms}{14}{section.1.5}%
|
||||
\contentsline {section}{\numberline {1.6}Contests and resources}{15}{section.1.6}%
|
||||
\contentsline {subsubsection}{IOI}{15}{section.1.6}%
|
||||
\contentsline {subsubsection}{ICPC}{15}{section.1.6}%
|
||||
\contentsline {subsubsection}{Online contests}{16}{section.1.6}%
|
||||
\contentsline {subsubsection}{Books}{16}{section.1.6}%
|
||||
\contentsline {chapter}{\numberline {2}Time complexity}{17}{chapter.2}%
|
||||
\contentsline {section}{\numberline {2.1}Calculation rules}{17}{section.2.1}%
|
||||
\contentsline {section}{\numberline {2.2}Complexity classes}{20}{section.2.2}%
|
||||
\contentsline {section}{\numberline {2.3}Estimating efficiency}{21}{section.2.3}%
|
||||
\contentsline {section}{\numberline {2.4}Maximum subarray sum}{21}{section.2.4}%
|
||||
\contentsline {subsubsection}{Algorithm 1}{22}{section.2.4}%
|
||||
\contentsline {subsubsection}{Algorithm 2}{22}{lstnumber.-46.11}%
|
||||
\contentsline {subsubsection}{Algorithm 3}{23}{lstnumber.-47.9}%
|
||||
\contentsline {subsubsection}{Efficiency comparison}{23}{lstnumber.-48.6}%
|
||||
\contentsline {chapter}{\numberline {3}Sorting}{25}{chapter.3}%
|
||||
\contentsline {section}{\numberline {3.1}Sorting theory}{25}{section.3.1}%
|
||||
\contentsline {subsubsection}{$O(n^2)$ algorithms}{25}{section.3.1}%
|
||||
\contentsline {subsubsection}{Inversions}{26}{lstnumber.-49.7}%
|
||||
\contentsline {subsubsection}{$O(n \qopname \relax o{log}n)$ algorithms}{27}{lstnumber.-49.7}%
|
||||
\contentsline {subsubsection}{Sorting lower bound}{28}{Item.7}%
|
||||
\contentsline {subsubsection}{Counting sort}{28}{Item.7}%
|
||||
\contentsline {section}{\numberline {3.2}Sorting in C++}{29}{section.3.2}%
|
||||
\contentsline {subsubsection}{Comparison operators}{30}{lstnumber.-53.2}%
|
||||
\contentsline {subsubsection}{User-defined structs}{30}{lstnumber.-55.5}%
|
||||
\contentsline {subsubsection}{Comparison functions}{31}{lstnumber.-56.7}%
|
||||
\contentsline {section}{\numberline {3.3}Binary search}{31}{section.3.3}%
|
||||
\contentsline {subsubsection}{Method 1}{32}{lstnumber.-59.5}%
|
||||
\contentsline {subsubsection}{Method 2}{32}{lstnumber.-60.9}%
|
||||
\contentsline {subsubsection}{C++ functions}{33}{lstnumber.-61.7}%
|
||||
\contentsline {subsubsection}{Finding the smallest solution}{33}{lstnumber.-64.2}%
|
||||
\contentsline {subsubsection}{Finding the maximum value}{34}{lstnumber.-65.5}%
|
||||
\contentsline {chapter}{\numberline {4}Data structures}{35}{chapter.4}%
|
||||
\contentsline {section}{\numberline {4.1}Dynamic arrays}{35}{section.4.1}%
|
||||
\contentsline {section}{\numberline {4.2}Set structures}{37}{section.4.2}%
|
||||
\contentsline {section}{\numberline {4.3}Map structures}{38}{section.4.3}%
|
||||
\contentsline {section}{\numberline {4.4}Iterators and ranges}{39}{section.4.4}%
|
||||
\contentsline {subsubsection}{Working with ranges}{39}{section.4.4}%
|
||||
\contentsline {subsubsection}{Set iterators}{40}{lstnumber.-87.3}%
|
||||
\contentsline {section}{\numberline {4.5}Other structures}{41}{section.4.5}%
|
||||
\contentsline {subsubsection}{Bitset}{41}{section.4.5}%
|
||||
\contentsline {subsubsection}{Deque}{42}{lstnumber.-98.5}%
|
||||
\contentsline {subsubsection}{Stack}{42}{lstnumber.-99.6}%
|
||||
\contentsline {subsubsection}{Queue}{43}{lstnumber.-100.7}%
|
||||
\contentsline {subsubsection}{Priority queue}{43}{lstnumber.-101.7}%
|
||||
\contentsline {subsubsection}{Policy-based data structures}{44}{lstnumber.-103.1}%
|
||||
\contentsline {section}{\numberline {4.6}Comparison to sorting}{44}{section.4.6}%
|
||||
\contentsline {subsubsection}{Algorithm 1}{45}{section.4.6}%
|
||||
\contentsline {subsubsection}{Algorithm 2}{45}{section.4.6}%
|
||||
\contentsline {subsubsection}{Algorithm 3}{45}{section.4.6}%
|
||||
\contentsline {subsubsection}{Efficiency comparison}{45}{section.4.6}%
|
||||
\contentsline {chapter}{\numberline {5}Complete search}{47}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}Generating subsets}{47}{section.5.1}%
|
||||
\contentsline {subsubsection}{Method 1}{47}{section.5.1}%
|
||||
\contentsline {subsubsection}{Method 2}{48}{lstnumber.-110.10}%
|
||||
\contentsline {section}{\numberline {5.2}Generating permutations}{49}{section.5.2}%
|
||||
\contentsline {subsubsection}{Method 1}{49}{section.5.2}%
|
||||
\contentsline {subsubsection}{Method 2}{49}{lstnumber.-113.14}%
|
||||
\contentsline {section}{\numberline {5.3}Backtracking}{50}{section.5.3}%
|
||||
\contentsline {section}{\numberline {5.4}Pruning the search}{51}{section.5.4}%
|
||||
\contentsline {subsubsection}{Basic algorithm}{52}{section.5.4}%
|
||||
\contentsline {subsubsection}{Optimization 1}{52}{section.5.4}%
|
||||
\contentsline {subsubsection}{Optimization 2}{53}{section.5.4}%
|
||||
\contentsline {subsubsection}{Optimization 3}{53}{section.5.4}%
|
||||
\contentsline {subsubsection}{Optimization 4}{53}{section.5.4}%
|
||||
\contentsline {section}{\numberline {5.5}Meet in the middle}{54}{section.5.5}%
|
||||
\contentsline {chapter}{\numberline {6}Greedy algorithms}{57}{chapter.6}%
|
||||
\contentsline {section}{\numberline {6.1}Coin problem}{57}{section.6.1}%
|
||||
\contentsline {subsubsection}{Greedy algorithm}{57}{section.6.1}%
|
||||
\contentsline {subsubsection}{General case}{58}{section.6.1}%
|
||||
\contentsline {section}{\numberline {6.2}Scheduling}{58}{section.6.2}%
|
||||
\contentsline {section}{\numberline {6.3}Tasks and deadlines}{60}{section.6.3}%
|
||||
\contentsline {section}{\numberline {6.4}Minimizing sums}{61}{section.6.4}%
|
||||
\contentsline {subsubsection}{Case $c=1$}{61}{section.6.4}%
|
||||
\contentsline {subsubsection}{Case $c=2$}{61}{section.6.4}%
|
||||
\contentsline {section}{\numberline {6.5}Data compression}{62}{section.6.5}%
|
||||
\contentsline {subsubsection}{Huffman coding}{63}{section.6.5}%
|
||||
\contentsline {chapter}{\numberline {7}Dynamic programming}{65}{chapter.7}%
|
||||
\contentsline {section}{\numberline {7.1}Coin problem}{65}{section.7.1}%
|
||||
\contentsline {subsubsection}{Recursive formulation}{66}{section.7.1}%
|
||||
\contentsline {subsubsection}{Using memoization}{67}{lstnumber.-116.9}%
|
||||
\contentsline {subsubsection}{Constructing a solution}{68}{lstnumber.-119.9}%
|
||||
\contentsline {subsubsection}{Counting the number of solutions}{69}{lstnumber.-122.4}%
|
||||
\contentsline {section}{\numberline {7.2}Longest increasing subsequence}{70}{section.7.2}%
|
||||
\contentsline {section}{\numberline {7.3}Paths in a grid}{71}{section.7.3}%
|
||||
\contentsline {section}{\numberline {7.4}Knapsack problems}{72}{section.7.4}%
|
||||
\contentsline {section}{\numberline {7.5}Edit distance}{74}{section.7.5}%
|
||||
\contentsline {section}{\numberline {7.6}Counting tilings}{75}{section.7.6}%
|
||||
\contentsline {chapter}{\numberline {8}Amortized analysis}{77}{chapter.8}%
|
||||
\contentsline {section}{\numberline {8.1}Two pointers method}{77}{section.8.1}%
|
||||
\contentsline {subsubsection}{Subarray sum}{77}{section.8.1}%
|
||||
\contentsline {subsubsection}{2SUM problem}{78}{section.8.1}%
|
||||
\contentsline {section}{\numberline {8.2}Nearest smaller elements}{79}{section.8.2}%
|
||||
\contentsline {section}{\numberline {8.3}Sliding window minimum}{81}{section.8.3}%
|
||||
\contentsline {chapter}{\numberline {9}Range queries}{83}{chapter.9}%
|
||||
\contentsline {section}{\numberline {9.1}Static array queries}{84}{section.9.1}%
|
||||
\contentsline {subsubsection}{Sum queries}{84}{section.9.1}%
|
||||
\contentsline {subsubsection}{Minimum queries}{85}{section.9.1}%
|
||||
\contentsline {section}{\numberline {9.2}Binary indexed tree}{86}{section.9.2}%
|
||||
\contentsline {subsubsection}{Structure}{86}{section.9.2}%
|
||||
\contentsline {subsubsection}{Implementation}{88}{section.9.2}%
|
||||
\contentsline {section}{\numberline {9.3}Segment tree}{89}{section.9.3}%
|
||||
\contentsline {subsubsection}{Structure}{89}{section.9.3}%
|
||||
\contentsline {subsubsection}{Implementation}{90}{section.9.3}%
|
||||
\contentsline {subsubsection}{Other queries}{92}{lstnumber.-135.7}%
|
||||
\contentsline {section}{\numberline {9.4}Additional techniques}{93}{section.9.4}%
|
||||
\contentsline {subsubsection}{Index compression}{93}{section.9.4}%
|
||||
\contentsline {subsubsection}{Range updates}{93}{section.9.4}%
|
||||
\contentsline {chapter}{\numberline {10}Bit manipulation}{95}{chapter.10}%
|
||||
\contentsline {section}{\numberline {10.1}Bit representation}{95}{section.10.1}%
|
||||
\contentsline {section}{\numberline {10.2}Bit operations}{96}{section.10.2}%
|
||||
\contentsline {subsubsection}{And operation}{96}{section.10.2}%
|
||||
\contentsline {subsubsection}{Or operation}{96}{section.10.2}%
|
||||
\contentsline {subsubsection}{Xor operation}{97}{section.10.2}%
|
||||
\contentsline {subsubsection}{Not operation}{97}{section.10.2}%
|
||||
\contentsline {subsubsection}{Bit shifts}{97}{section.10.2}%
|
||||
\contentsline {subsubsection}{Applications}{97}{section.10.2}%
|
||||
\contentsline {section}{\numberline {10.3}Representing sets}{98}{section.10.3}%
|
||||
\contentsline {subsubsection}{Set implementation}{98}{section.10.3}%
|
||||
\contentsline {subsubsection}{Set operations}{99}{lstnumber.-141.4}%
|
||||
\contentsline {subsubsection}{Iterating through subsets}{99}{lstnumber.-142.4}%
|
||||
\contentsline {section}{\numberline {10.4}Bit optimizations}{100}{section.10.4}%
|
||||
\contentsline {subsubsection}{Hamming distances}{100}{section.10.4}%
|
||||
\contentsline {subsubsection}{Counting subgrids}{101}{lstnumber.-147.3}%
|
||||
\contentsline {section}{\numberline {10.5}Dynamic programming}{102}{section.10.5}%
|
||||
\contentsline {subsubsection}{Optimal selection}{102}{section.10.5}%
|
||||
\contentsline {subsubsection}{From permutations to subsets}{103}{lstnumber.-152.11}%
|
||||
\contentsline {subsubsection}{Counting subsets}{105}{lstnumber.-155.18}%
|
||||
\contentsline {part}{II\hspace {1em}Graph algorithms}{107}{part.2}%
|
||||
\contentsline {chapter}{\numberline {11}Basics of graphs}{109}{chapter.11}%
|
||||
\contentsline {section}{\numberline {11.1}Graph terminology}{109}{section.11.1}%
|
||||
\contentsline {subsubsection}{Connectivity}{110}{section.11.1}%
|
||||
\contentsline {subsubsection}{Edge directions}{110}{section.11.1}%
|
||||
\contentsline {subsubsection}{Edge weights}{111}{section.11.1}%
|
||||
\contentsline {subsubsection}{Neighbors and degrees}{111}{section.11.1}%
|
||||
\contentsline {subsubsection}{Colorings}{112}{section.11.1}%
|
||||
\contentsline {subsubsection}{Simplicity}{112}{section.11.1}%
|
||||
\contentsline {section}{\numberline {11.2}Graph representation}{113}{section.11.2}%
|
||||
\contentsline {subsubsection}{Adjacency list representation}{113}{section.11.2}%
|
||||
\contentsline {subsubsection}{Adjacency matrix representation}{114}{lstnumber.-163.3}%
|
||||
\contentsline {subsubsection}{Edge list representation}{115}{lstnumber.-164.1}%
|
||||
\contentsline {chapter}{\numberline {12}Graph traversal}{117}{chapter.12}%
|
||||
\contentsline {section}{\numberline {12.1}Depth-first search}{117}{section.12.1}%
|
||||
\contentsline {section}{\numberline {12.2}Breadth-first search}{119}{section.12.2}%
|
||||
\contentsline {section}{\numberline {12.3}Applications}{121}{section.12.3}%
|
||||
\contentsline {subsubsection}{Connectivity check}{121}{section.12.3}%
|
||||
\contentsline {subsubsection}{Finding cycles}{121}{section.12.3}%
|
||||
\contentsline {subsubsection}{Bipartiteness check}{122}{section.12.3}%
|
||||
\contentsline {chapter}{\numberline {13}Shortest paths}{123}{chapter.13}%
|
||||
\contentsline {section}{\numberline {13.1}Bellman–Ford algorithm}{123}{section.13.1}%
|
||||
\contentsline {subsubsection}{Example}{123}{section.13.1}%
|
||||
\contentsline {subsubsection}{Implementation}{125}{section.13.1}%
|
||||
\contentsline {subsubsection}{Negative cycles}{125}{lstnumber.-174.9}%
|
||||
\contentsline {subsubsection}{SPFA algorithm}{126}{lstnumber.-174.9}%
|
||||
\contentsline {section}{\numberline {13.2}Dijkstra's algorithm}{126}{section.13.2}%
|
||||
\contentsline {subsubsection}{Example}{126}{section.13.2}%
|
||||
\contentsline {subsubsection}{Negative edges}{128}{section.13.2}%
|
||||
\contentsline {subsubsection}{Implementation}{128}{section.13.2}%
|
||||
\contentsline {section}{\numberline {13.3}Floyd–Warshall algorithm}{129}{section.13.3}%
|
||||
\contentsline {subsubsection}{Example}{129}{section.13.3}%
|
||||
\contentsline {subsubsection}{Implementation}{131}{section.13.3}%
|
||||
\contentsline {chapter}{\numberline {14}Tree algorithms}{133}{chapter.14}%
|
||||
\contentsline {section}{\numberline {14.1}Tree traversal}{134}{section.14.1}%
|
||||
\contentsline {subsubsection}{Dynamic programming}{134}{lstnumber.-179.1}%
|
||||
\contentsline {section}{\numberline {14.2}Diameter}{135}{section.14.2}%
|
||||
\contentsline {subsubsection}{Algorithm 1}{135}{section.14.2}%
|
||||
\contentsline {subsubsection}{Algorithm 2}{136}{section.14.2}%
|
||||
\contentsline {section}{\numberline {14.3}All longest paths}{137}{section.14.3}%
|
||||
\contentsline {section}{\numberline {14.4}Binary trees}{139}{section.14.4}%
|
||||
\contentsline {chapter}{\numberline {15}Spanning trees}{141}{chapter.15}%
|
||||
\contentsline {section}{\numberline {15.1}Kruskal's algorithm}{142}{section.15.1}%
|
||||
\contentsline {subsubsection}{Example}{142}{section.15.1}%
|
||||
\contentsline {subsubsection}{Why does this work?}{144}{section.15.1}%
|
||||
\contentsline {subsubsection}{Implementation}{145}{section.15.1}%
|
||||
\contentsline {section}{\numberline {15.2}Union-find structure}{145}{section.15.2}%
|
||||
\contentsline {subsubsection}{Structure}{145}{section.15.2}%
|
||||
\contentsline {subsubsection}{Implementation}{146}{section.15.2}%
|
||||
\contentsline {section}{\numberline {15.3}Prim's algorithm}{147}{section.15.3}%
|
||||
\contentsline {subsubsection}{Example}{147}{section.15.3}%
|
||||
\contentsline {subsubsection}{Implementation}{148}{section.15.3}%
|
||||
\contentsline {chapter}{\numberline {16}Directed graphs}{149}{chapter.16}%
|
||||
\contentsline {section}{\numberline {16.1}Topological sorting}{149}{section.16.1}%
|
||||
\contentsline {subsubsection}{Algorithm}{150}{section.16.1}%
|
||||
\contentsline {subsubsection}{Example 1}{150}{section.16.1}%
|
||||
\contentsline {subsubsection}{Example 2}{151}{section.16.1}%
|
||||
\contentsline {section}{\numberline {16.2}Dynamic programming}{151}{section.16.2}%
|
||||
\contentsline {subsubsection}{Counting the number of paths}{152}{section.16.2}%
|
||||
\contentsline {subsubsection}{Extending Dijkstra's algorithm}{153}{section.16.2}%
|
||||
\contentsline {subsubsection}{Representing problems as graphs}{153}{section.16.2}%
|
||||
\contentsline {section}{\numberline {16.3}Successor paths}{154}{section.16.3}%
|
||||
\contentsline {section}{\numberline {16.4}Cycle detection}{155}{section.16.4}%
|
||||
\contentsline {subsubsection}{Floyd's algorithm}{156}{section.16.4}%
|
||||
\contentsline {chapter}{\numberline {17}Strong connectivity}{157}{chapter.17}%
|
||||
\contentsline {section}{\numberline {17.1}Kosaraju's algorithm}{158}{section.17.1}%
|
||||
\contentsline {subsubsection}{Search 1}{158}{section.17.1}%
|
||||
\contentsline {subsubsection}{Search 2}{159}{section.17.1}%
|
||||
\contentsline {section}{\numberline {17.2}2SAT problem}{160}{section.17.2}%
|
||||
\contentsline {chapter}{\numberline {18}Tree queries}{163}{chapter.18}%
|
||||
\contentsline {section}{\numberline {18.1}Finding ancestors}{163}{section.18.1}%
|
||||
\contentsline {section}{\numberline {18.2}Subtrees and paths}{164}{section.18.2}%
|
||||
\contentsline {subsubsection}{Subtree queries}{165}{section.18.2}%
|
||||
\contentsline {subsubsection}{Path queries}{166}{section.18.2}%
|
||||
\contentsline {section}{\numberline {18.3}Lowest common ancestor}{167}{section.18.3}%
|
||||
\contentsline {subsubsection}{Method 1}{167}{section.18.3}%
|
||||
\contentsline {subsubsection}{Method 2}{168}{section.18.3}%
|
||||
\contentsline {subsubsection}{Distances of nodes}{169}{section.18.3}%
|
||||
\contentsline {section}{\numberline {18.4}Offline algorithms}{170}{section.18.4}%
|
||||
\contentsline {subsubsection}{Merging data structures}{170}{section.18.4}%
|
||||
\contentsline {subsubsection}{Lowest common ancestors}{171}{lstnumber.-189.1}%
|
||||
\contentsline {chapter}{\numberline {19}Paths and circuits}{173}{chapter.19}%
|
||||
\contentsline {section}{\numberline {19.1}Eulerian paths}{173}{section.19.1}%
|
||||
\contentsline {subsubsection}{Existence}{174}{section.19.1}%
|
||||
\contentsline {subsubsection}{Hierholzer's algorithm}{175}{section.19.1}%
|
||||
\contentsline {subsubsection}{Example}{176}{section.19.1}%
|
||||
\contentsline {section}{\numberline {19.2}Hamiltonian paths}{177}{section.19.2}%
|
||||
\contentsline {subsubsection}{Existence}{177}{section.19.2}%
|
||||
\contentsline {subsubsection}{Construction}{178}{section.19.2}%
|
||||
\contentsline {section}{\numberline {19.3}De Bruijn sequences}{178}{section.19.3}%
|
||||
\contentsline {section}{\numberline {19.4}Knight's tours}{179}{section.19.4}%
|
||||
\contentsline {subsubsection}{Warnsdorf's rule}{179}{section.19.4}%
|
||||
\contentsline {chapter}{\numberline {20}Flows and cuts}{181}{chapter.20}%
|
||||
\contentsline {subsubsection}{Maximum flow}{181}{chapter.20}%
|
||||
\contentsline {subsubsection}{Minimum cut}{182}{chapter.20}%
|
||||
\contentsline {section}{\numberline {20.1}Ford–Fulkerson algorithm}{182}{section.20.1}%
|
||||
\contentsline {subsubsection}{Algorithm description}{183}{section.20.1}%
|
||||
\contentsline {subsubsection}{Finding paths}{184}{section.20.1}%
|
||||
\contentsline {subsubsection}{Minimum cuts}{185}{section.20.1}%
|
||||
\contentsline {section}{\numberline {20.2}Disjoint paths}{186}{section.20.2}%
|
||||
\contentsline {subsubsection}{Edge-disjoint paths}{186}{section.20.2}%
|
||||
\contentsline {subsubsection}{Node-disjoint paths}{186}{section.20.2}%
|
||||
\contentsline {section}{\numberline {20.3}Maximum matchings}{187}{section.20.3}%
|
||||
\contentsline {subsubsection}{Finding maximum matchings}{188}{section.20.3}%
|
||||
\contentsline {subsubsection}{Hall's theorem}{189}{section.20.3}%
|
||||
\contentsline {subsubsection}{Kőnig's theorem}{189}{section.20.3}%
|
||||
\contentsline {section}{\numberline {20.4}Path covers}{190}{section.20.4}%
|
||||
\contentsline {subsubsection}{Node-disjoint path cover}{191}{section.20.4}%
|
||||
\contentsline {subsubsection}{General path cover}{192}{section.20.4}%
|
||||
\contentsline {subsubsection}{Dilworth's theorem}{193}{section.20.4}%
|
||||
\contentsline {part}{III\hspace {1em}Advanced topics}{195}{part.3}%
|
||||
\contentsline {chapter}{\numberline {21}Number theory}{197}{chapter.21}%
|
||||
\contentsline {section}{\numberline {21.1}Primes and factors}{197}{section.21.1}%
|
||||
\contentsline {subsubsection}{Number of primes}{198}{section.21.1}%
|
||||
\contentsline {subsubsection}{Density of primes}{198}{section.21.1}%
|
||||
\contentsline {subsubsection}{Conjectures}{199}{section.21.1}%
|
||||
\contentsline {subsubsection}{Basic algorithms}{199}{section.21.1}%
|
||||
\contentsline {subsubsection}{Sieve of Eratosthenes}{200}{lstnumber.-191.11}%
|
||||
\contentsline {subsubsection}{Euclid's algorithm}{200}{lstnumber.-192.6}%
|
||||
\contentsline {subsubsection}{Euler's totient function}{201}{lstnumber.-193.4}%
|
||||
\contentsline {section}{\numberline {21.2}Modular arithmetic}{201}{section.21.2}%
|
||||
\contentsline {subsubsection}{Modular exponentiation}{202}{section.21.2}%
|
||||
\contentsline {subsubsection}{Fermat's theorem and Euler's theorem}{202}{lstnumber.-194.7}%
|
||||
\contentsline {subsubsection}{Modular inverse}{202}{lstnumber.-194.7}%
|
||||
\contentsline {subsubsection}{Computer arithmetic}{203}{lstnumber.-194.7}%
|
||||
\contentsline {section}{\numberline {21.3}Solving equations}{204}{section.21.3}%
|
||||
\contentsline {subsubsection}{Chinese remainder theorem}{205}{section.21.3}%
|
||||
\contentsline {section}{\numberline {21.4}Other results}{205}{section.21.4}%
|
||||
\contentsline {subsubsection}{Lagrange's theorem}{205}{section.21.4}%
|
||||
\contentsline {subsubsection}{Zeckendorf's theorem}{206}{section.21.4}%
|
||||
\contentsline {subsubsection}{Pythagorean triples}{206}{section.21.4}%
|
||||
\contentsline {subsubsection}{Wilson's theorem}{206}{section.21.4}%
|
||||
\contentsline {chapter}{\numberline {22}Combinatorics}{207}{chapter.22}%
|
||||
\contentsline {section}{\numberline {22.1}Binomial coefficients}{208}{section.22.1}%
|
||||
\contentsline {subsubsection}{Formula 1}{208}{section.22.1}%
|
||||
\contentsline {subsubsection}{Formula 2}{208}{section.22.1}%
|
||||
\contentsline {subsubsection}{Properties}{208}{section.22.1}%
|
||||
\contentsline {subsubsection}{Boxes and balls}{209}{section.22.1}%
|
||||
\contentsline {subsubsection}{Multinomial coefficients}{210}{section.22.1}%
|
||||
\contentsline {section}{\numberline {22.2}Catalan numbers}{210}{section.22.2}%
|
||||
\contentsline {subsubsection}{Parenthesis expressions}{211}{section.22.2}%
|
||||
\contentsline {subsubsection}{Formula 1}{211}{section.22.2}%
|
||||
\contentsline {subsubsection}{Formula 2}{211}{section.22.2}%
|
||||
\contentsline {subsubsection}{Counting trees}{212}{section.22.2}%
|
||||
\contentsline {section}{\numberline {22.3}Inclusion-exclusion}{212}{section.22.3}%
|
||||
\contentsline {subsubsection}{Derangements}{213}{section.22.3}%
|
||||
\contentsline {section}{\numberline {22.4}Burnside's lemma}{214}{section.22.4}%
|
||||
\contentsline {section}{\numberline {22.5}Cayley's formula}{215}{section.22.5}%
|
||||
\contentsline {subsubsection}{Prüfer code}{216}{section.22.5}%
|
||||
\contentsline {chapter}{\numberline {23}Matrices}{217}{chapter.23}%
|
||||
\contentsline {section}{\numberline {23.1}Operations}{217}{section.23.1}%
|
||||
\contentsline {subsubsection}{Matrix multiplication}{218}{section.23.1}%
|
||||
\contentsline {subsubsection}{Matrix power}{219}{section.23.1}%
|
||||
\contentsline {subsubsection}{Determinant}{219}{section.23.1}%
|
||||
\contentsline {section}{\numberline {23.2}Linear recurrences}{220}{section.23.2}%
|
||||
\contentsline {subsubsection}{Fibonacci numbers}{220}{section.23.2}%
|
||||
\contentsline {subsubsection}{General case}{221}{section.23.2}%
|
||||
\contentsline {section}{\numberline {23.3}Graphs and matrices}{222}{section.23.3}%
|
||||
\contentsline {subsubsection}{Counting paths}{222}{section.23.3}%
|
||||
\contentsline {subsubsection}{Shortest paths}{222}{section.23.3}%
|
||||
\contentsline {subsubsection}{Kirchhoff's theorem}{223}{section.23.3}%
|
||||
\contentsline {chapter}{\numberline {24}Probability}{225}{chapter.24}%
|
||||
\contentsline {section}{\numberline {24.1}Calculation}{225}{section.24.1}%
|
||||
\contentsline {section}{\numberline {24.2}Events}{226}{section.24.2}%
|
||||
\contentsline {subsubsection}{Complement}{227}{section.24.2}%
|
||||
\contentsline {subsubsection}{Union}{227}{section.24.2}%
|
||||
\contentsline {subsubsection}{Conditional probability}{227}{section.24.2}%
|
||||
\contentsline {subsubsection}{Intersection}{228}{section.24.2}%
|
||||
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\indexentry{tree|hyperpage}{1}
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\indexentry{leaf|hyperpage}{1}
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\indexentry{root|hyperpage}{1}
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\indexentry{rooted tree|hyperpage}{1}
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\indexentry{child|hyperpage}{1}
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\indexentry{parent|hyperpage}{1}
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\indexentry{subtree|hyperpage}{1}
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\indexentry{diameter|hyperpage}{3}
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\indexentry{binary tree|hyperpage}{7}
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\indexentry{pre-order|hyperpage}{7}
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\indexentry{in-order|hyperpage}{7}
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\indexentry{post-order|hyperpage}{7}
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|
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This is makeindex, version 2.15 [TeX Live 2020] (kpathsea + Thai support).
|
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Scanning input file output/test.idx....done (12 entries accepted, 0 rejected).
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|
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Generating output file output/test.ind....done (40 lines written, 0 warnings).
|
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Output written in output/test.ind.
|
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Transcript written in output/test.ilg.
|
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|
|
@ -1,40 +0,0 @@
|
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\begin{theindex}
|
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|
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\item binary tree, \hyperpage{7}
|
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\indexspace
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\item child, \hyperpage{1}
|
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|
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\indexspace
|
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|
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\item diameter, \hyperpage{3}
|
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|
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\indexspace
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\item in-order, \hyperpage{7}
|
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\indexspace
|
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\item leaf, \hyperpage{1}
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\indexspace
|
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|
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\item parent, \hyperpage{1}
|
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\item post-order, \hyperpage{7}
|
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\item pre-order, \hyperpage{7}
|
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|
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\indexspace
|
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|
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\item root, \hyperpage{1}
|
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\item rooted tree, \hyperpage{1}
|
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|
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\indexspace
|
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|
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\item subtree, \hyperpage{1}
|
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|
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\indexspace
|
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|
||||
\item tree, \hyperpage{1}
|
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|
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\end{theindex}
|
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1451
output/test.log
1451
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|
|
@ -1,5 +0,0 @@
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\BOOKMARK [0][-]{chapter.1}{Tree algorithms}{}% 1
|
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\BOOKMARK [1][-]{section.1.1}{Tree traversal}{chapter.1}% 2
|
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\BOOKMARK [1][-]{section.1.2}{Diameter}{chapter.1}% 3
|
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\BOOKMARK [1][-]{section.1.3}{All longest paths}{chapter.1}% 4
|
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\BOOKMARK [1][-]{section.1.4}{Binary trees}{chapter.1}% 5
|
||||
BIN
output/test.pdf
BIN
output/test.pdf
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|
|
@ -1,12 +0,0 @@
|
|||
\babel@toc {english}{}
|
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\contentsline {part}{I\hspace {1em}Graph algorithms}{1}{part.1}%
|
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\contentsline {chapter}{\numberline {1}Tree algorithms}{3}{chapter.1}%
|
||||
\contentsline {section}{\numberline {1.1}Tree traversal}{4}{section.1.1}%
|
||||
\contentsline {subsubsection}{Dynamic programming}{4}{lstnumber.-2.1}%
|
||||
\contentsline {section}{\numberline {1.2}Diameter}{5}{section.1.2}%
|
||||
\contentsline {subsubsection}{Algorithm 1}{5}{section.1.2}%
|
||||
\contentsline {subsubsection}{Algorithm 2}{6}{section.1.2}%
|
||||
\contentsline {section}{\numberline {1.3}All longest paths}{7}{section.1.3}%
|
||||
\contentsline {section}{\numberline {1.4}Binary trees}{9}{section.1.4}%
|
||||
\contentsline {chapter}{Bibliography}{11}{section*.2}%
|
||||
\contentsline {chapter}{Index}{17}{chapter*.4}%
|
||||
34
preface.tex
34
preface.tex
|
|
@ -2,32 +2,10 @@
|
|||
\markboth{\MakeUppercase{Preface}}{}
|
||||
\addcontentsline{toc}{chapter}{Preface}
|
||||
|
||||
The purpose of this book is to give you
|
||||
a thorough introduction to competitive programming.
|
||||
It is assumed that you already
|
||||
know the basics of programming, but no previous
|
||||
background in competitive programming is needed.
|
||||
This script is based on the Competitive Programmer's Handbook
|
||||
by Antti Laaksonen.
|
||||
|
||||
The book is especially intended for
|
||||
students who want to learn algorithms and
|
||||
possibly participate in
|
||||
the International Olympiad in Informatics (IOI) or
|
||||
in the International Collegiate Programming Contest (ICPC).
|
||||
Of course, the book is also suitable for
|
||||
anybody else interested in competitive programming.
|
||||
|
||||
It takes a long time to become a good competitive
|
||||
programmer, but it is also an opportunity to learn a lot.
|
||||
You can be sure that you will get
|
||||
a good general understanding of algorithms
|
||||
if you spend time reading the book,
|
||||
solving problems and taking part in contests.
|
||||
|
||||
The book is under continuous development.
|
||||
You can always send feedback on the book to
|
||||
\texttt{ahslaaks@cs.helsinki.fi}.
|
||||
|
||||
\begin{flushright}
|
||||
Helsinki, August 2019 \\
|
||||
Antti Laaksonen
|
||||
\end{flushright}
|
||||
It contains the topics relevant for the graph day of the
|
||||
SOI Camp 2021.
|
||||
Most of the code was modified slightly, and also some minor
|
||||
adjustments were made on the text.
|
||||
|
|
|
|||
100
test.tex
100
test.tex
|
|
@ -1,100 +0,0 @@
|
|||
\documentclass[twoside,12pt,a4paper,english]{book}
|
||||
|
||||
%\includeonly{chapter04,list}
|
||||
|
||||
\usepackage[english]{babel}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{listings}
|
||||
\usepackage[table]{xcolor}
|
||||
\usepackage{tikz}
|
||||
\usepackage{multicol}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{array}
|
||||
\usepackage{microtype}
|
||||
|
||||
\usepackage{fouriernc}
|
||||
\usepackage[T1]{fontenc}
|
||||
|
||||
\usepackage{graphicx}
|
||||
\usepackage{framed}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{amsmath}
|
||||
|
||||
\usepackage{pifont}
|
||||
\usepackage{ifthen}
|
||||
\usepackage{makeidx}
|
||||
\usepackage{enumitem}
|
||||
|
||||
\usepackage{titlesec}
|
||||
|
||||
\usepackage{skak}
|
||||
\usepackage[scaled=0.95]{inconsolata}
|
||||
|
||||
|
||||
\usetikzlibrary{patterns,snakes}
|
||||
\pagestyle{plain}
|
||||
|
||||
\definecolor{keywords}{HTML}{44548A}
|
||||
\definecolor{strings}{HTML}{00999A}
|
||||
\definecolor{comments}{HTML}{990000}
|
||||
|
||||
\lstset{language=C++,frame=single,basicstyle=\ttfamily \small,showstringspaces=false,columns=flexible}
|
||||
\lstset{
|
||||
literate={ö}{{\"o}}1
|
||||
{ä}{{\"a}}1
|
||||
{ü}{{\"u}}1
|
||||
}
|
||||
\lstset{xleftmargin=20pt,xrightmargin=5pt}
|
||||
\lstset{aboveskip=12pt,belowskip=8pt}
|
||||
|
||||
\lstset{
|
||||
commentstyle=\color{comments},
|
||||
keywordstyle=\color{keywords},
|
||||
stringstyle=\color{strings}
|
||||
}
|
||||
|
||||
\date{Draft \today}
|
||||
|
||||
\usepackage[a4paper,vmargin=30mm,hmargin=33mm,footskip=15mm]{geometry}
|
||||
|
||||
\title{\Huge Competitive Programmer's Handbook}
|
||||
\author{\Large Antti Laaksonen}
|
||||
|
||||
\makeindex
|
||||
\usepackage[totoc]{idxlayout}
|
||||
|
||||
\titleformat{\subsubsection}
|
||||
{\normalfont\large\bfseries\sffamily}{\thesubsection}{1em}{}
|
||||
|
||||
\begin{document}
|
||||
|
||||
%\selectlanguage{finnish}
|
||||
|
||||
%\setcounter{page}{1}
|
||||
%\pagenumbering{roman}
|
||||
|
||||
%\frontmatter
|
||||
%\maketitle
|
||||
%\setcounter{tocdepth}{1}
|
||||
%\tableofcontents
|
||||
|
||||
%\include{preface}
|
||||
|
||||
\mainmatter
|
||||
\pagenumbering{arabic}
|
||||
\setcounter{page}{1}
|
||||
|
||||
\newcommand{\key}[1] {\textbf{#1}}
|
||||
|
||||
%\part{Graph algorithms}
|
||||
\include{chapter14}
|
||||
|
||||
%\cleardoublepage
|
||||
%\phantomsection
|
||||
%\addcontentsline{toc}{chapter}{Bibliography}
|
||||
%\include{list}
|
||||
|
||||
%\cleardoublepage
|
||||
%\printindex
|
||||
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue