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@ -1,6 +1,6 @@
\chapter{Graph search} \chapter{Graph traversal}
This chapter introduces two fundamental This chapter discusses two fundamental
graph algorithms: graph algorithms:
depth-first search and breadth-first search. depth-first search and breadth-first search.
Both algorithms are given a starting Both algorithms are given a starting
@ -15,7 +15,7 @@ in which they visit the nodes.
\index{depth-first search} \index{depth-first search}
\key{Depth-first search} (DFS) \key{Depth-first search} (DFS)
is a straightforward graph search technique. is a straightforward graph traversal technique.
The algorithm begins at a starting node, The algorithm begins at a starting node,
and proceeds to all other nodes that are and proceeds to all other nodes that are
reachable from the starting node using reachable from the starting node using
@ -24,14 +24,14 @@ the edges in the graph.
Depth-first search always follows a single Depth-first search always follows a single
path in the graph as long as it finds path in the graph as long as it finds
new nodes. new nodes.
After this, it returns back to previous After this, it returns to previous
nodes and begins to explore other parts of the graph. nodes and begins to explore other parts of the graph.
The algorithm keeps track of visited nodes, The algorithm keeps track of visited nodes,
so that it processes each node only once. so that it processes each node only once.
\subsubsection*{Example} \subsubsection*{Example}
Let's consider how depth-first search processes Let us consider how depth-first search processes
the following graph: the following graph:
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
@ -92,9 +92,9 @@ After this, nodes 3 and 5 will be visited:
\end{center} \end{center}
The neighbors of node 5 are 2 and 3, The neighbors of node 5 are 2 and 3,
but the search has already visited both of them, but the search has already visited both of them,
so it's time to return back. so it is time to return to previous nodes.
Also the neighbors of nodes 3 and 2 Also the neighbors of nodes 3 and 2
have been visited, so we'll next proceed have been visited, so we next move
from node 1 to node 4: from node 1 to node 4:
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
@ -128,7 +128,7 @@ implemented using recursion.
The following function \texttt{dfs} begins The following function \texttt{dfs} begins
a depth-first search at a given node. a depth-first search at a given node.
The function assumes that the graph is The function assumes that the graph is
stored as adjacency lists in array stored as adjacency lists in an array
\begin{lstlisting} \begin{lstlisting}
vector<int> v[N]; vector<int> v[N];
\end{lstlisting} \end{lstlisting}
@ -175,7 +175,7 @@ have been visited.
\subsubsection*{Example} \subsubsection*{Example}
Let's consider how the algorithm processes Let us consider how the algorithm processes
the following graph: the following graph:
\begin{center} \begin{center}
@ -196,7 +196,7 @@ the following graph:
\path[draw,thick,-] (5) -- (6); \path[draw,thick,-] (5) -- (6);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Assume again that the search begins at node 1. Suppose again that the search begins at node 1.
First, we process all nodes that can be reached First, we process all nodes that can be reached
from node 1 using a single edge: from node 1 using a single edge:
\begin{center} \begin{center}
@ -224,7 +224,7 @@ from node 1 using a single edge:
\path[draw=red,thick,->,line width=2pt] (1) -- (4); \path[draw=red,thick,->,line width=2pt] (1) -- (4);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
After this, we procees to nodes 3 and 5: After this, we proceed to nodes 3 and 5:
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
\node[draw, circle,fill=lightgray] (1) at (1,5) {$1$}; \node[draw, circle,fill=lightgray] (1) at (1,5) {$1$};
@ -301,10 +301,10 @@ and $m$ is the number of edges.
\subsubsection*{Implementation} \subsubsection*{Implementation}
Breadth-first search is more difficult Breadth-first search is more difficult
to implement than depth-first search to implement than depth-first search,
because the algorithm visits nodes because the algorithm visits nodes
in different parts in the graph. in different parts of the graph.
A typical implementation is to maintain A typical implementation is based on
a queue of nodes to be processed. a queue of nodes to be processed.
At each step, the next node in the queue At each step, the next node in the queue
will be processed. will be processed.
@ -326,9 +326,9 @@ In addition, the code uses arrays
\begin{lstlisting} \begin{lstlisting}
int z[N], e[N]; int z[N], e[N];
\end{lstlisting} \end{lstlisting}
so that array \texttt{z} indicates so that the array \texttt{z} indicates
which nodes the search already has visited which nodes the search has already visited
and array \texttt{e} will contain the and the array \texttt{e} will contain the
minimum distance to all nodes in the graph. minimum distance to all nodes in the graph.
The search can be implemented as follows: The search can be implemented as follows:
\begin{lstlisting} \begin{lstlisting}
@ -347,12 +347,12 @@ while (!q.empty()) {
\section{Applications} \section{Applications}
Using the graph search algorithms, Using the graph traversal algorithms,
we can check many properties of the graph. we can check many properties of the graph.
Usually, either depth-first search or Usually, either depth-first search or
bredth-first search can be used, bredth-first search can be used,
but in practice, depth-first search but in practice, depth-first search
is a better choice because it is is a better choice, because it is
easier to implement. easier to implement.
In the following applications we will In the following applications we will
assume that the graph is undirected. assume that the graph is undirected.
@ -403,18 +403,18 @@ the following nodes:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Since the search didn't visit all the nodes, Since the search did not visit all the nodes,
we can conclude that the graph is not connected. we can conclude that the graph is not connected.
In a similar way, we can also find all components In a similar way, we can also find all connected components
in a graph by iterating trough the nodes and always of a graph by iterating trough the nodes and always
starting a new depth-first search if the node starting a new depth-first search if the current node
doesn't belong to a component. does not belong to any component yet.
\subsubsection{Finding cycles} \subsubsection{Finding cycles}
\index{cycle} \index{cycle}
A graph contains a cycle if during a graph search, A graph contains a cycle if during a graph traversal,
we find a node whose neighbor (other than the we find a node whose neighbor (other than the
previous node in the current path) has already been previous node in the current path) has already been
visited. visited.
@ -442,7 +442,7 @@ For example, the graph
\end{center} \end{center}
contains a cycle because when we move from contains a cycle because when we move from
node 2 to node 5 it turns out node 2 to node 5 it turns out
that the neighbor node 3 has already been visited. that the neighbor 3 has already been visited.
Thus, the graph contains a cycle that goes through node 3, Thus, the graph contains a cycle that goes through node 3,
for example, $3 \rightarrow 2 \rightarrow 5 \rightarrow 3$. for example, $3 \rightarrow 2 \rightarrow 5 \rightarrow 3$.
@ -452,7 +452,7 @@ in every component.
If a component contains $c$ nodes and no cycle, If a component contains $c$ nodes and no cycle,
it must contain exactly $c-1$ edges. it must contain exactly $c-1$ edges.
If there are $c$ or more edges, the component If there are $c$ or more edges, the component
always contains a cycle. surely contains a cycle.
\subsubsection{Bipartiteness check} \subsubsection{Bipartiteness check}
@ -460,9 +460,9 @@ always contains a cycle.
A graph is bipartite if its nodes can be colored A graph is bipartite if its nodes can be colored
using two colors so that there are no adjacent using two colors so that there are no adjacent
nodes with same color. nodes with the same color.
It is suprisingly easy to check if a graph It is suprisingly easy to check if a graph
is bipartite using graph search algorithms. is bipartite using graph traversal algorithms.
The idea is to color the starting node blue, The idea is to color the starting node blue,
all its neighbors red, all their neighbors blue, and so on. all its neighbors red, all their neighbors blue, and so on.
@ -490,7 +490,7 @@ For example, the graph
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
is not bipartite because a search from node 1 is not bipartite because a search from node 1
produces the following situation: proceeds as follows:
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
\node[draw, circle,fill=red!40] (2) at (5,5) {$2$}; \node[draw, circle,fill=red!40] (2) at (5,5) {$2$};
@ -516,11 +516,11 @@ We notice that the color or both node 2 and node 5
is red, while they are adjacent nodes in the graph. is red, while they are adjacent nodes in the graph.
Thus, the graph is not bipartite. Thus, the graph is not bipartite.
This algorithm always works because when there This algorithm always works, because when there
are only two colors available, are only two colors available,
the color of the starting node in a component the color of the starting node in a component
determines the colors of all other nodes in the component. determines the colors of all other nodes in the component.
It doesn't make any difference whether the It does not make any difference whether the
starting node is red or blue. starting node is red or blue.
Note that in the general case, Note that in the general case,