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@ -2,17 +2,18 @@
Many programming problems can be solved by Many programming problems can be solved by
interpreting the problem as a graph problem interpreting the problem as a graph problem
and using a suitable graph algorithm. and using an appropriate graph algorithm.
A typical example of a graph is a network A typical example of a graph is a network
of roads and cities in a country. of roads and cities in a country.
Sometimes, though, the graph is hidden Sometimes, though, the graph is hidden
in the problem and it can be difficult to detect it. in the problem and it can be difficult to detect it.
This part of the book discusses techniques and algorithms This part of the book discusses graph algorithms,
involving graphs especially focusing on topics that
that are important in competitive programming. are important in competitive programming.
We will first go through graph terminology In this chapter, we go through terminology
and different ways to store graphs in algorithms. related to graphs,
and study different ways to represent graphs in algorithms.
\section{Terminology} \section{Terminology}
@ -26,10 +27,10 @@ In this book,
the variable $n$ denotes the number of nodes the variable $n$ denotes the number of nodes
in a graph, and the variable $m$ denotes in a graph, and the variable $m$ denotes
the number of edges. the number of edges.
In addition, the nodes are numbered The nodes are numbered
using integers $1,2,\ldots,n$. using integers $1,2,\ldots,n$.
For example, the following graph contains 5 nodes and 7 edges: For example, the following graph consists of 5 nodes and 7 edges:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -51,12 +52,12 @@ For example, the following graph contains 5 nodes and 7 edges:
\index{path} \index{path}
A \key{path} is a route from node $a$ to node $b$ A \key{path} leads from node $a$ to node $b$
that goes through the edges in the graph. through edges of the graph.
The \key{length} of a path is the number of The \key{length} of a path is the number of
edges in the path. edges in it.
For example, in the above graph, paths For example, in the above graph, there
from node 1 to node 5 are: are several paths from node 1 to node 5:
\begin{itemize} \begin{itemize}
\item $1 \rightarrow 2 \rightarrow 5$ (length 2) \item $1 \rightarrow 2 \rightarrow 5$ (length 2)
@ -71,7 +72,7 @@ from node 1 to node 5 are:
\index{connected graph} \index{connected graph}
A graph is \key{connected}, if there is path A graph is \key{connected} if there is path
between any two nodes. between any two nodes.
For example, the following graph is connected: For example, the following graph is connected:
\begin{center} \begin{center}
@ -88,9 +89,9 @@ For example, the following graph is connected:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
The following graph is not connected The following graph is not connected,
because it is not possible to get to other because it is not possible to get
nodes from node 4. from node 4 to any other node:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1,3) {$1$}; \node[draw, circle] (1) at (1,3) {$1$};
@ -103,10 +104,10 @@ nodes from node 4.
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
\index{compomnent} \index{component}
The connected parts of a graph are The connected parts of a graph are
its \key{components}. called its \key{components}.
For example, the following graph For example, the following graph
contains three components: contains three components:
$\{1,\,2,\,3\}$, $\{1,\,2,\,3\}$,
@ -138,9 +139,9 @@ $\{8\}$.
\index{tree} \index{tree}
A \key{tree} is a connected graph A \key{tree} is a connected graph
that contains $n$ nodes and $n-1$ edges. that consists of $n$ nodes and $n-1$ edges.
In a tree, there is a unique path There is a unique path
between any two nodes. between any two nodes in a tree.
For example, the following graph is a tree: For example, the following graph is a tree:
\begin{center} \begin{center}
@ -165,8 +166,8 @@ For example, the following graph is a tree:
\index{directed graph} \index{directed graph}
A graph is \key{directed} A graph is \key{directed}
if the edges can be travelled only if the edges can be traversed
in one direction. in one direction only.
For example, the following graph is directed: For example, the following graph is directed:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -185,10 +186,9 @@ For example, the following graph is directed:
\end{center} \end{center}
The above graph contains a path from The above graph contains a path from
node $3$ to $5$ using edges node $3$ to node $5$ through the edges
$3 \rightarrow 1 \rightarrow 2 \rightarrow 5$. $3 \rightarrow 1 \rightarrow 2 \rightarrow 5$,
However, the graph doesn't contain but there is no path from node $5$ to node $3$.
a path from node $5$ to $3$.
\index{cycle} \index{cycle}
\index{acyclic graph} \index{acyclic graph}
@ -198,7 +198,7 @@ last node is the same.
For example, the above graph contains For example, the above graph contains
a cycle a cycle
$1 \rightarrow 2 \rightarrow 4 \rightarrow 1$. $1 \rightarrow 2 \rightarrow 4 \rightarrow 1$.
If a graph doesn't contain any cycles, If a graph does not contain any cycles,
it is called \key{acyclic}. it is called \key{acyclic}.
\subsubsection{Edge weights} \subsubsection{Edge weights}
@ -225,14 +225,14 @@ For example, the following graph is weighted:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Now the length of a path is the sum of The length of a path in a weighted graph
edge weights. is the sum of edge weights on the path.
For example, in the above graph For example, in the above graph,
the length of path the length of the path
$1 \rightarrow 2 \rightarrow 5$ $1 \rightarrow 2 \rightarrow 5$ is $12$
is $12$, and the length of path and the length of the path
$1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ is $11$. $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ is $11$.
The latter is the shortest path from node $1$ to node $5$. The latter path is the \key{shortest} path from node $1$ to node $5$.
\subsubsection{Neighbors and degrees} \subsubsection{Neighbors and degrees}
@ -240,7 +240,7 @@ The latter is the shortest path from node $1$ to node $5$.
\index{degree} \index{degree}
Two nodes are \key{neighbors} or \key{adjacent} Two nodes are \key{neighbors} or \key{adjacent}
if there is a edge between them. if there is an edge between them.
The \key{degree} of a node The \key{degree} of a node
is the number of its neighbors. is the number of its neighbors.
For example, in the following graph, For example, in the following graph,
@ -265,11 +265,11 @@ so its degree is 3.
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
The sum of degrees in a graph is always $2m$ The sum of degrees in a graph is always $2m$,
where $m$ is the number of edges. where $m$ is the number of edges,
The reason for this is that each edge because each edge
increases the degree of two nodes by one. increases the degree of two nodes by one.
Thus, the sum of degrees is always even. For this reason, the sum of degrees is always even.
\index{regular graph} \index{regular graph}
\index{complete graph} \index{complete graph}
@ -285,11 +285,13 @@ between the nodes.
\index{outdegree} \index{outdegree}
In a directed graph, the \key{indegree} In a directed graph, the \key{indegree}
and \key{outdegree} of a node is of a node is the number of edges
the number of edges that end and begin that end at the node,
at the node, respectively. and the \key{outdegree} of a node
is the number of edges that start at the node.
For example, in the following graph, For example, in the following graph,
node 2 has indegree 2 and outdegree 1. the indegree of node 2 is 2
and the outdegree of the node is 1.
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -320,8 +322,8 @@ no adjacent nodes have the same color.
A graph is \key{bipartite} if A graph is \key{bipartite} if
it is possible to color it using two colors. it is possible to color it using two colors.
It turns out that a graph is bipartite It turns out that a graph is bipartite
exactly when it doesn't contain a cycle exactly when it does not contain a cycle
with odd number of edges. with an odd number of edges.
For example, the graph For example, the graph
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -339,7 +341,7 @@ For example, the graph
\path[draw,thick,-] (5) -- (6); \path[draw,thick,-] (5) -- (6);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
is bipartite because we can color it as follows: is bipartite, because it can be colored as follows:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
\node[draw, circle, fill=blue!40] (1) at (1,3) {$2$}; \node[draw, circle, fill=blue!40] (1) at (1,3) {$2$};
@ -356,16 +358,34 @@ is bipartite because we can color it as follows:
\path[draw,thick,-] (5) -- (6); \path[draw,thick,-] (5) -- (6);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
However, the following graph is not bipartite:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1,3) {$2$};
\node[draw, circle] (2) at (4,3) {$3$};
\node[draw, circle] (3) at (1,1) {$5$};
\node[draw, circle] (4) at (4,1) {$6$};
\node[draw, circle] (5) at (-2,1) {$4$};
\node[draw, circle] (6) at (-2,3) {$1$};
\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (3) -- (6);
\path[draw,thick,-] (5) -- (6);
\path[draw,thick,-] (1) -- (6);
\end{tikzpicture}
\end{center}
\subsubsection{Simplicity} \subsubsection{Simplicity}
\index{simple graph} \index{simple graph}
A graph is \key{simple} A graph is \key{simple}
if no edge begins and ends at the same node, if no edge starts and ends at the same node,
and there are no multiple and there are no multiple
edges between two nodes. edges between two nodes.
Often we will assume that the graph is simple. Often we assume that graphs are simple.
For example, the graph For example, the graph
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -389,41 +409,39 @@ For example, the graph
\path[draw,thick,-] (5) edge [loop left] (5); \path[draw,thick,-] (5) edge [loop left] (5);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
is \emph{not} simple because there is an edge that begins is \emph{not} simple, because there is an edge that starts
and ends at node 4, and there are two edges and ends at node 4, and there are two edges
between nodes 2 and 3. between nodes 2 and 3.
\section{Graph representation} \section{Graph representation}
There are several ways how to represent graphs in memory There are several ways to represent graphs
in an algorithm. in algorithms.
The choice of a data structure The choice of a data structure
depends on the size of the graph and depends on the size of the graph and
how the algorithm manipulates it. the way the algorithm processes it.
Next we will go through three representations. Next we will go through three possible representations.
\subsubsection{Adjacency list representation} \subsubsection{Adjacency list representation}
\index{adjacency list} \index{adjacency list}
A usual way to represent a graph is In the adjacency list representation,
to create an \key{adjacency list} for each node. each node $x$ in the graph is assigned an \key{adjacency list}
An adjacency list contains contains all nodes that consists of nodes
that can be reached from the node using a single edge. to which there is an edge from $x$.
The adjacency list representation is the most popular Adjacency lists are the most popular
way to store a graph, and most algorithms can be way to represent a graph, and most algorithms can be
efficiently implemented using it. efficiently implemented using them.
A good way to store the adjacency lists is to allocate an array A convenient way to store the adjacency lists is to declare
whose each element is a vector: an array of vectors as follows:
\begin{lstlisting} \begin{lstlisting}
vector<int> v[N]; vector<int> v[N];
\end{lstlisting} \end{lstlisting}
The adjacency list for node $s$ is in position The constant $N$ is chosen so that there
$\texttt{v}[s]$ in the array. is space for all adjacency lists.
The constant $N$ is so chosen that all
adjacency lists can be stored.
For example, the graph For example, the graph
\begin{center} \begin{center}
@ -450,18 +468,18 @@ v[4].push_back(1);
\end{lstlisting} \end{lstlisting}
If the graph is undirected, it can be stored in a similar way, If the graph is undirected, it can be stored in a similar way,
but each edge each is store in both directions. but each edge is stored in both directions.
For an weighted graph, the structure can be extended For a weighted graph, the structure can be extended
as follows: as follows:
\begin{lstlisting} \begin{lstlisting}
vector<pair<int,int>> v[N]; vector<pair<int,int>> v[N];
\end{lstlisting} \end{lstlisting}
Now each adjacency list contains pairs whose first If there is an edge from node $a$ to node $b$
element is the target node, with weight $w$, the adjacency list of node $a$
and the second element is the edge weight. contains the pair $(b,w)$.
For example, the graph For example, the graph
\begin{center} \begin{center}
@ -487,11 +505,11 @@ v[3].push_back({4,5});
v[4].push_back({1,2}); v[4].push_back({1,2});
\end{lstlisting} \end{lstlisting}
The benefit in the adjacency list representation is that The benefit in using adjacency lists is that
we can efficiently find the nodes that can be we can efficiently find the nodes to which
reached from a certain node. we can move from a certain node through an edge.
For example, the following loop goes trough all nodes For example, the following loop goes through all nodes
that can be reached from node $s$: to which we can move from node $s$:
\begin{lstlisting} \begin{lstlisting}
for (auto u : v[s]) { for (auto u : v[s]) {
@ -504,17 +522,14 @@ for (auto u : v[s]) {
\index{adjacency matrix} \index{adjacency matrix}
An \key{adjacency matrix} is a two-dimensional array An \key{adjacency matrix} is a two-dimensional array
that indicates for each possible edge if it is that indicates which edges exist in the graph.
included in the graph. We can efficiently check from an adjacency matrix
Using an adjacency matrix, we can efficiently check
if there is an edge between two nodes. if there is an edge between two nodes.
On the other hand, the matrix takes a lot of memory The matrix can be stored as an array
if the graph is large.
We can store the matrix as an array
\begin{lstlisting} \begin{lstlisting}
int v[N][N]; int v[N][N];
\end{lstlisting} \end{lstlisting}
where the value $\texttt{v}[a][b]$ indicates where each value $\texttt{v}[a][b]$ indicates
whether the graph contains an edge from whether the graph contains an edge from
node $a$ to node $b$. node $a$ to node $b$.
If the edge is included in the graph, If the edge is included in the graph,
@ -619,22 +634,29 @@ corresponds to the following matrix:
\end{center} \end{center}
\end{samepage} \end{samepage}
The drawback in the adjacency matrix representation
is that there are $n^2$ elements in the matrix
and usually most of them are zero.
For this reason, the representation cannot be used
if the graph is large.
\subsubsection{Edge list representation} \subsubsection{Edge list representation}
\index{edge list} \index{edge list}
An \key{edge list} contains all edges of a graph. An \key{edge list} contains all edges of a graph
This is a convenient way to represent a graph, in some order.
if the algorithm will go trough all edges of the graph, This is a convenient way to represent a graph
and it is not needed to find edges that begin if the algorithm processes all edges of the graph,
and it is not needed to find edges that start
at a given node. at a given node.
The edge list can be stored in a vector The edge list can be stored in a vector
\begin{lstlisting} \begin{lstlisting}
vector<pair<int,int>> v; vector<pair<int,int>> v;
\end{lstlisting} \end{lstlisting}
where each element contains the starting where each pair $(a,b)$ denotes that
and ending node of an edge. there is an edge from node $a$ to node $b$.
Thus, the graph Thus, the graph
\begin{center} \begin{center}
@ -661,14 +683,14 @@ v.push_back({4,1});
\end{lstlisting} \end{lstlisting}
\noindent \noindent
If the graph is weighted, we can extend the If the graph is weighted, the structure can
structure as follows: be extended as follows:
\begin{lstlisting} \begin{lstlisting}
vector<pair<pair<int,int>,int>> v; vector<tuple<int,int,int>> v;
\end{lstlisting} \end{lstlisting}
Now the list contains pairs whose first element Each element in this list is of the
contains the starting and ending node of an edge, form $(a,b,w)$, which means that there
and the second element corresponds to the edge weight. is an edge from node $a$ to node $b$ with weight $w$.
For example, the graph For example, the graph
\begin{center} \begin{center}
@ -688,10 +710,10 @@ For example, the graph
\begin{samepage} \begin{samepage}
can be represented as follows: can be represented as follows:
\begin{lstlisting} \begin{lstlisting}
v.push_back({{1,2},5}); v.push_back(make_tuple(1,2,5));
v.push_back({{2,3},7}); v.push_back(make_tuple(2,3,7));
v.push_back({{2,4},6}); v.push_back(make_tuple(2,4,6));
v.push_back({{3,4},5}); v.push_back(make_tuple(3,4,5));
v.push_back({{4,1},2}); v.push_back(make_tuple(4,1,2));
\end{lstlisting} \end{lstlisting}
\end{samepage} \end{samepage}