cphb/chapter11.tex

\chapter{Basics of graphs}

Many programming problems can be solved by
modeling the problem as a graph problem
and using an appropriate graph algorithm.
A typical example of a graph is a network
of roads and cities in a country.
Sometimes, though, the graph is hidden
in the problem and it may be difficult to detect it.

This part of the book discusses graph algorithms,
especially focusing on topics that
are important in competitive programming.
In this chapter, we go through concepts
related to graphs,
and study different ways to represent graphs in algorithms.

\section{Graph terminology}

\index{graph}
\index{node}
\index{edge}

A \key{graph} consists of \key{nodes}
and \key{edges}. In this book,
the variable $n$ denotes the number of nodes
in a graph, and the variable $m$ denotes
the number of edges.
The nodes are numbered
using integers $1,2,\ldots,n$.

For example, the following graph consists of 5 nodes and 7 edges:

\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};

\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
\path[draw,thick,-] (4) -- (5);
\end{tikzpicture}
\end{center}

\index{path}

A \key{path} leads from node $a$ to node $b$
through edges of the graph.
The \key{length} of a path is the number of
edges in it.
For example, the above graph contains
a path $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$
of length 3
from node 1 to node 5:

\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};

\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
\path[draw,thick,-] (4) -- (5);

\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}

\index{cycle}

A path is a \key{cycle} if the first and last
node is the same.
For example, the above graph contains
a cycle $1 \rightarrow 3 \rightarrow 4 \rightarrow 1$.
A path is \key{simple} if each node appears
at most once in the path.


% 
% \begin{itemize}
% \item $1 \rightarrow 2 \rightarrow 5$ (length 2)
% \item $1 \rightarrow 4 \rightarrow 5$ (length 2)
% \item $1 \rightarrow 2 \rightarrow 4 \rightarrow 5$ (length 3)
% \item $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ (length 3)
% \item $1 \rightarrow 4 \rightarrow 2 \rightarrow 5$ (length 3)
% \item $1 \rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 5$ (length 4)
% \end{itemize}

\subsubsection{Connectivity}

\index{connected graph}

A graph is \key{connected} if there is a path
between any two nodes.
For example, the following graph is connected:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$4$};
\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (2) -- (3);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\end{tikzpicture}
\end{center}

The following graph is not connected,
because it is not possible to get
from node 4 to any other node:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$4$};
\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (2) -- (3);
\end{tikzpicture}
\end{center}

\index{component}

The connected parts of a graph are
called its \key{components}.
For example, the following graph
contains three components:
$\{1,\,2,\,3\}$,
$\{4,\,5,\,6,\,7\}$ and
$\{8\}$.
\begin{center}
\begin{tikzpicture}[scale=0.8]
\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] (6) at (6,1) {$6$};
\node[draw, circle] (7) at (9,1) {$7$};
\node[draw, circle] (4) at (6,3) {$4$};
\node[draw, circle] (5) at (9,3) {$5$};

\node[draw, circle] (8) at (11,2) {$8$};

\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (2) -- (3);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (4) -- (5);
\path[draw,thick,-] (5) -- (7);
\path[draw,thick,-] (6) -- (7);
\path[draw,thick,-] (6) -- (4);
\end{tikzpicture}
\end{center}

\index{tree}

A \key{tree} is a connected graph
that consists of $n$ nodes and $n-1$ edges.
There is a unique path
between any two nodes of a tree.
For example, the following graph is a tree:

\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};

\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
%\path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (2) -- (5);
\path[draw,thick,-] (2) -- (4);
%\path[draw,thick,-] (4) -- (5);
\end{tikzpicture}
\end{center}

\subsubsection{Edge directions}

\index{directed graph}

A graph is \key{directed}
if the edges can be traversed
in one direction only.
For example, the following graph is directed:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};
\path[draw,thick,->,>=latex] (1) -- (2);
\path[draw,thick,->,>=latex] (2) -- (4);
\path[draw,thick,->,>=latex] (2) -- (5);
\path[draw,thick,->,>=latex] (4) -- (5);
\path[draw,thick,->,>=latex] (4) -- (1);
\path[draw,thick,->,>=latex] (3) -- (1);
\end{tikzpicture}
\end{center}

The above graph contains
a path $3 \rightarrow 1 \rightarrow 2 \rightarrow 5$
from node $3$ to node $5$,
but there is no path from node $5$ to node $3$.

\subsubsection{Edge weights}

\index{weighted graph}

In a \key{weighted} graph, each edge is assigned
a \key{weight}.
The weights are often interpreted as edge lengths.
For example, the following graph is weighted:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};
\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:1] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:7] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:7] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:3] {} (5);
\end{tikzpicture}
\end{center}

The length of a path in a weighted graph
is the sum of the edge weights on the path.
For example, in the above graph,
the length of the path
$1 \rightarrow 2 \rightarrow 5$ is $12$,
and the length of the path
$1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ is $11$.
The latter path is the \key{shortest} path from node $1$ to node $5$.

\subsubsection{Neighbors and degrees}

\index{neighbor}
\index{degree}

Two nodes are \key{neighbors} or \key{adjacent}
if there is an edge between them.
The \key{degree} of a node
is the number of its neighbors.
For example, in the following graph,
the neighbors of node 2 are 1, 4 and 5,
so its degree is 3.

\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};

\path[draw,thick,-] (1) -- (2);
\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
%\path[draw,thick,-] (4) -- (5);
\end{tikzpicture}
\end{center}

The sum of degrees in a graph is always $2m$,
where $m$ is the number of edges,
because each edge
increases the degree of exactly two nodes by one.
For this reason, the sum of degrees is always even.

\index{regular graph}
\index{complete graph}

A graph is \key{regular} if the
degree of every node is a constant $d$.
A graph is \key{complete} if the
degree of every node is $n-1$, i.e.,
the graph contains all possible edges
between the nodes.

\index{indegree}
\index{outdegree}

In a directed graph, the \key{indegree}
of a node is the number of edges
that end at the node,
and the \key{outdegree} of a node
is the number of edges that start at the node.
For example, in the following graph,
the indegree of node 2 is 2,
and the outdegree of node 2 is 1.

\begin{center}
\begin{tikzpicture}[scale=0.9]
\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) {$5$};

\path[draw,thick,->,>=latex] (1) -- (2);
\path[draw,thick,->,>=latex] (1) -- (3);
\path[draw,thick,->,>=latex] (1) -- (4);
\path[draw,thick,->,>=latex] (3) -- (4);
\path[draw,thick,->,>=latex] (2) -- (4);
\path[draw,thick,<-,>=latex] (2) -- (5);
\end{tikzpicture}
\end{center}

\subsubsection{Colorings}

\index{coloring}
\index{bipartite graph}

In a \key{coloring} of a graph,
each node is assigned a color so that
no adjacent nodes have the same color.

A graph is \key{bipartite} if
it is possible to color it using two colors.
It turns out that a graph is bipartite
exactly when it does not contain a cycle
with an odd number of edges.
For example, the graph
\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);
\end{tikzpicture}
\end{center}
is bipartite, because it can be colored as follows:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle, fill=blue!40] (1) at (1,3) {$2$};
\node[draw, circle, fill=red!40] (2) at (4,3) {$3$};
\node[draw, circle, fill=red!40] (3) at (1,1) {$5$};
\node[draw, circle, fill=blue!40] (4) at (4,1) {$6$};
\node[draw, circle, fill=red!40] (5) at (-2,1) {$4$};
\node[draw, circle, fill=blue!40] (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);
\end{tikzpicture}
\end{center}
However, the graph
\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}
is not bipartite, because it is not possible to color
the following cycle of three nodes using two colors:
\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);

\path[draw=red,thick,-,line width=2pt] (1) -- (3);
\path[draw=red,thick,-,line width=2pt] (3) -- (6);
\path[draw=red,thick,-,line width=2pt] (6) -- (1);
\end{tikzpicture}
\end{center}

\subsubsection{Simplicity}

\index{simple graph}

A graph is \key{simple}
if no edge starts and ends at the same node,
and there are no multiple
edges between two nodes.
Often we assume that graphs are simple.
For example, the following graph is \emph{not} simple:
\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) edge [bend right=20] (2);
\path[draw,thick,-] (2) edge [bend right=20] (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);

\tikzset{every loop/.style={in=135,out=190}}
\path[draw,thick,-] (5) edge [loop left] (5);
\end{tikzpicture}
\end{center}

\section{Graph representation}

There are several ways to represent graphs
in algorithms.
The choice of a data structure
depends on the size of the graph and
the way the algorithm processes it.
Next we will go through three common representations.

\subsubsection{Adjacency list representation}

\index{adjacency list}

In the adjacency list representation,
each node $x$ in the graph is assigned an \key{adjacency list}
that consists of nodes
to which there is an edge from $x$.
Adjacency lists are the most popular
way to represent graphs, and most algorithms can be
efficiently implemented using them.

A convenient way to store the adjacency lists is to declare
an array of vectors as follows:
\begin{lstlisting}
vector<int> adj[N];
\end{lstlisting}

The constant $N$ is chosen so that all
adjacency lists can be stored.
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) {$4$};

\path[draw,thick,->,>=latex] (1) -- (2);
\path[draw,thick,->,>=latex] (2) -- (3);
\path[draw,thick,->,>=latex] (2) -- (4);
\path[draw,thick,->,>=latex] (3) -- (4);
\path[draw,thick,->,>=latex] (4) -- (1);
\end{tikzpicture}
\end{center}
can be stored as follows:
\begin{lstlisting}
adj[1].push_back(2);
adj[2].push_back(3);
adj[2].push_back(4);
adj[3].push_back(4);
adj[4].push_back(1);
\end{lstlisting}

If the graph is undirected, it can be stored in a similar way,
but each edge is added in both directions.

For a weighted graph, the structure can be extended
as follows:

\begin{lstlisting}
vector<pair<int,int>> adj[N];
\end{lstlisting}

In this case, the adjacency list of node $a$
contains the pair $(b,w)$
always when there is an edge from node $a$ to node $b$
with weight $w$. 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) {$4$};

\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
\end{tikzpicture}
\end{center}
can be stored as follows:
\begin{lstlisting}
adj[1].push_back({2,5});
adj[2].push_back({3,7});
adj[2].push_back({4,6});
adj[3].push_back({4,5});
adj[4].push_back({1,2});
\end{lstlisting}

The benefit of using adjacency lists is that
we can efficiently find the nodes to which
we can move from a given node through an edge.
For example, the following loop goes through all nodes
to which we can move from node $s$:

\begin{lstlisting}
for (auto u : adj[s]) {
    // process node u
}
\end{lstlisting}

\subsubsection{Adjacency matrix representation}

\index{adjacency matrix}

An \key{adjacency matrix} is a two-dimensional array
that indicates which edges the graph contains.
We can efficiently check from an adjacency matrix
if there is an edge between two nodes.
The matrix can be stored as an array
\begin{lstlisting}
int adj[N][N];
\end{lstlisting}
where each value $\texttt{adj}[a][b]$ indicates
whether the graph contains an edge from
node $a$ to node $b$.
If the edge is included in the graph,
then $\texttt{adj}[a][b]=1$,
and otherwise $\texttt{adj}[a][b]=0$.
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) {$4$};

\path[draw,thick,->,>=latex] (1) -- (2);
\path[draw,thick,->,>=latex] (2) -- (3);
\path[draw,thick,->,>=latex] (2) -- (4);
\path[draw,thick,->,>=latex] (3) -- (4);
\path[draw,thick,->,>=latex] (4) -- (1);
\end{tikzpicture}
\end{center}
can be represented as follows:
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw (0,0) grid (4,4);
\node at (0.5,0.5) {1};
\node at (1.5,0.5) {0};
\node at (2.5,0.5) {0};
\node at (3.5,0.5) {0};
\node at (0.5,1.5) {0};
\node at (1.5,1.5) {0};
\node at (2.5,1.5) {0};
\node at (3.5,1.5) {1};
\node at (0.5,2.5) {0};
\node at (1.5,2.5) {0};
\node at (2.5,2.5) {1};
\node at (3.5,2.5) {1};
\node at (0.5,3.5) {0};
\node at (1.5,3.5) {1};
\node at (2.5,3.5) {0};
\node at (3.5,3.5) {0};
\node at (-0.5,0.5) {4};
\node at (-0.5,1.5) {3};
\node at (-0.5,2.5) {2};
\node at (-0.5,3.5) {1};
\node at (0.5,4.5) {1};
\node at (1.5,4.5) {2};
\node at (2.5,4.5) {3};
\node at (3.5,4.5) {4};
\end{tikzpicture}
\end{center}

If the graph is weighted, the adjacency matrix
representation can be extended so that
the matrix contains the weight of the edge
if the edge exists.
Using this representation, 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) {$4$};

\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
\end{tikzpicture}
\end{center}
\begin{samepage}
corresponds to the following matrix:
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw (0,0) grid (4,4);
\node at (0.5,0.5) {2};
\node at (1.5,0.5) {0};
\node at (2.5,0.5) {0};
\node at (3.5,0.5) {0};
\node at (0.5,1.5) {0};
\node at (1.5,1.5) {0};
\node at (2.5,1.5) {0};
\node at (3.5,1.5) {5};
\node at (0.5,2.5) {0};
\node at (1.5,2.5) {0};
\node at (2.5,2.5) {7};
\node at (3.5,2.5) {6};
\node at (0.5,3.5) {0};
\node at (1.5,3.5) {5};
\node at (2.5,3.5) {0};
\node at (3.5,3.5) {0};
\node at (-0.5,0.5) {4};
\node at (-0.5,1.5) {3};
\node at (-0.5,2.5) {2};
\node at (-0.5,3.5) {1};
\node at (0.5,4.5) {1};
\node at (1.5,4.5) {2};
\node at (2.5,4.5) {3};
\node at (3.5,4.5) {4};
\end{tikzpicture}
\end{center}
\end{samepage}

The drawback of the adjacency matrix representation
is that the matrix contains $n^2$ elements,
and usually most of them are zero.
For this reason, the representation cannot be used
if the graph is large.

\subsubsection{Edge list representation}

\index{edge list}

An \key{edge list} contains all edges of a graph
in some order.
This is a convenient way to represent a graph
if the algorithm processes all edges of the graph
and it is not needed to find edges that start
at a given node.

The edge list can be stored in a vector
\begin{lstlisting}
vector<pair<int,int>> edges;
\end{lstlisting}
where each pair $(a,b)$ denotes that
there is an edge from node $a$ to node $b$.
Thus, 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) {$4$};

\path[draw,thick,->,>=latex] (1) -- (2);
\path[draw,thick,->,>=latex] (2) -- (3);
\path[draw,thick,->,>=latex] (2) -- (4);
\path[draw,thick,->,>=latex] (3) -- (4);
\path[draw,thick,->,>=latex] (4) -- (1);
\end{tikzpicture}
\end{center}
can be represented as follows:
\begin{lstlisting}
edges.push_back({1,2});
edges.push_back({2,3});
edges.push_back({2,4});
edges.push_back({3,4});
edges.push_back({4,1});
\end{lstlisting}

\noindent
If the graph is weighted, the structure can
be extended as follows:
\begin{lstlisting}
vector<tuple<int,int,int>> edges;
\end{lstlisting}
Each element in this list is of the
form $(a,b,w)$, which means that there
is an edge from node $a$ to node $b$ with weight $w$.
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) {$4$};

\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
\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\}}).}:
\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});
\end{lstlisting}
\end{samepage}