译自 和
图的广度优先遍历
图的广度优先遍历（或搜索）类似于树的广度优先遍历（参见本文的方法2）。 这里唯一的一点是，与树不同，图可能包含循环，所以我们可能再次来到同一个节点。 为了避免多次处理节点，我们使用一个布尔访问数组。 为了简单起见，假设所有顶点都可以从起始顶点到达。
例如，在下图中，我们从顶点2开始遍历。当我们来到顶点0时，我们寻找它的所有相邻顶点。 顶点2也是顶点0的相邻顶点。如果我们不标记被访问的顶点，则将再次处理2，并且将成为非终止进程。 以下图的广度优先遍历为2，0，3，1。
BFS以下是来自给定来源的简单广度优先遍历的C++和Java实现。
C++版本
// Program to print BFS traversal from a given source vertex. BFS(int s)
// traverses vertices reachable from s.
#include <iostream>
#include <list>
using namespace std;
// This class represents a directed graph using adjacency list representation
class Graph
{
int V; // No. of vertices
list<int> *adj; // Pointer to an array containing adjacency lists
public:
Graph(int V); // Constructor
void addEdge(int v, int w); // function to add an edge to graph
void BFS(int s); // prints BFS traversal from a given source s
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<int>[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w); // Add w to v’s list.
}
void Graph::BFS(int s)
{
// Mark all the vertices as not visited
bool *visited = new bool[V];
for(int i = 0; i < V; i++)
visited[i] = false;
// Create a queue for BFS
list<int> queue;
// Mark the current node as visited and enqueue it
visited[s] = true;
queue.push_back(s);
// 'i' will be used to get all adjacent vertices of a vertex
list<int>::iterator i;
while(!queue.empty())
{
// Dequeue a vertex from queue and print it
s = queue.front();
cout << s << " ";
queue.pop_front();
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it visited
// and enqueue it
for(i = adj[s].begin(); i != adj[s].end(); ++i)
{
if(!visited[*i])
{
visited[*i] = true;
queue.push_back(*i);
}
}
}
}
// Driver program to test methods of graph class
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
cout << "Following is Breadth First Traversal "
<< "(starting from vertex 2) n";
g.BFS(2);
return 0;
}
Java版本
// Java program to print BFS traversal from a given source vertex.
// BFS(int s) traverses vertices reachable from s.
import java.io.*;
import java.util.*;
// This class represents a directed graph using adjacency list
// representation
class Graph
{
private int V; // No. of vertices
private LinkedList<Integer> adj[]; //Adjacency Lists
// Constructor
Graph(int v)
{
V = v;
adj = new LinkedList[v];
for (int i=0; i<v; ++i)
adj[i] = new LinkedList();
}
// Function to add an edge into the graph
void addEdge(int v, int w)
{
adj[v].add(w);
}
// prints BFS traversal from a given source s
void BFS(int s)
{
// Mark all the vertices as not visited(By default
// set as false)
boolean visited[] = new boolean[V];
// Create a queue for BFS
LinkedList<Integer> queue = new LinkedList<Integer>();
// Mark the current node as visited and enqueue it
visited[s]=true;
queue.add(s);
while (queue.size() != 0)
{
// Dequeue a vertex from queue and print it
s = queue.poll();
System.out.print(s+" ");
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
Iterator<Integer> i = adj[s].listIterator();
while (i.hasNext())
{
int n = i.next();
if (!visited[n])
{
visited[n] = true;
queue.add(n);
}
}
}
}
// Driver method to
public static void main(String args[])
{
Graph g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
System.out.println("Following is Breadth First Traversal "+
"(starting from vertex 2)");
g.BFS(2);
}
}
// This code is contributed by Aakash Hasija
Python版本
# Program to print BFS traversal from a given source
# vertex. BFS(int s) traverses vertices reachable
# from s.
from collections import defaultdict
# This class represents a directed graph using adjacency
# list representation
class Graph:
# Constructor
def __init__(self):
# default dictionary to store graph
self.graph = defaultdict(list)
# function to add an edge to graph
def addEdge(self,u,v):
self.graph[u].append(v)
# Function to print a BFS of graph
def BFS(self, s):
# Mark all the vertices as not visited
visited = [False]*(len(self.graph))
# Create a queue for BFS
queue = []
# Mark the source node as visited and enqueue it
queue.append(s)
visited[s] = True
while queue:
# Dequeue a vertex from queue and print it
s = queue.pop(0)
print s,
# Get all adjacent vertices of the dequeued
# vertex s. If a adjacent has not been visited,
# then mark it visited and enqueue it
for i in self.graph[s]:
if visited[i] == False:
queue.append(i)
visited[i] = True
# Driver code
# Create a graph given in the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
print "Following is Breadth First Traversal (starting from vertex 2)"
g.BFS(2)
# This code is contributed by Neelam Yadav
输出
Following is Breadth First Traversal (starting from vertex 2)
2 0 3 1
请注意，上述代码仅遍历从给定源顶点可到达的顶点。 可能无法从给定顶点到达所有顶点（示例断开连接图）。 要打印所有顶点，我们可以修改BFS函数，从所有节点逐个进行遍历（与DFS修改版本一样）。
时间复杂度：O(V + E) 其中V是图的顶点数，E是图的边数。
图的深度优先遍历
图的深度优先遍历（或搜索）也类似于树的深度优先遍历 。 这里唯一的一点是，与树不同，图可能包含循环，所以我们可能再次来到同一个节点。 为了避免多次处理节点，我们使用一个布尔访问数组。
例如，在下图中，我们从顶点2开始遍历。当我们来到顶点0时，我们寻找它的所有相邻顶点。顶点2也是顶点0的相邻顶点。如果我们不标记被访问的顶点，则将再次处理2，并且将成为非终止进程。 以下图的深度优先遍历为2,0,1,3。
DFS对于深度优先穿越的所有应用，请看这篇文章。
以下是简单深度优先遍历的实现。 C++实现使用图的邻接列表表示。 STL的列表容器用于存储相邻节点的列表。
C++版本
// C++ program to print DFS traversal from a given vertex in a given graph
#include<iostream>
#include<list>
using namespace std;
// Graph class represents a directed graph using adjacency list representation
class Graph
{
int V; // No. of vertices
list<int> *adj; // Pointer to an array containing adjacency lists
void DFSUtil(int v, bool visited[]); // A function used by DFS
public:
Graph(int V); // Constructor
void addEdge(int v, int w); // function to add an edge to graph
void DFS(int v); // DFS traversal of the vertices reachable from v
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<int>[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w); // Add w to v’s list.
}
void Graph::DFSUtil(int v, bool visited[])
{
// Mark the current node as visited and print it
visited[v] = true;
cout << v << " ";
// Recur for all the vertices adjacent to this vertex
list<int>::iterator i;
for (i = adj[v].begin(); i != adj[v].end(); ++i)
if (!visited[*i])
DFSUtil(*i, visited);
}
// DFS traversal of the vertices reachable from v.
// It uses recursive DFSUtil()
void Graph::DFS(int v)
{
// Mark all the vertices as not visited
bool *visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Call the recursive helper function to print DFS traversal
DFSUtil(v, visited);
}
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
cout << "Following is Depth First Traversal (starting from vertex 2) n";
g.DFS(2);
return 0;
}
Java版本
// Java program to print DFS traversal from a given given graph
import java.io.*;
import java.util.*;
// This class represents a directed graph using adjacency list
// representation
class Graph
{
private int V; // No. of vertices
// Array of lists for Adjacency List Representation
private LinkedList<Integer> adj[];
// Constructor
Graph(int v)
{
V = v;
adj = new LinkedList[v];
for (int i=0; i<v; ++i)
adj[i] = new LinkedList();
}
//Function to add an edge into the graph
void addEdge(int v, int w)
{
adj[v].add(w); // Add w to v's list.
}
// A function used by DFS
void DFSUtil(int v,boolean visited[])
{
// Mark the current node as visited and print it
visited[v] = true;
System.out.print(v+" ");
// Recur for all the vertices adjacent to this vertex
Iterator<Integer> i = adj[v].listIterator();
while (i.hasNext())
{
int n = i.next();
if (!visited[n])
DFSUtil(n, visited);
}
}
// The function to do DFS traversal. It uses recursive DFSUtil()
void DFS(int v)
{
// Mark all the vertices as not visited(set as
// false by default in java)
boolean visited[] = new boolean[V];
// Call the recursive helper function to print DFS traversal
DFSUtil(v, visited);
}
public static void main(String args[])
{
Graph g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
System.out.println("Following is Depth First Traversal "+
"(starting from vertex 2)");
g.DFS(2);
}
}
// This code is contributed by Aakash Hasija
Python版本
# Python program to print DFS traversal from a
# given given graph
from collections import defaultdict
# This class represents a directed graph using
# adjacency list representation
class Graph:
# Constructor
def __init__(self):
# default dictionary to store graph
self.graph = defaultdict(list)
# function to add an edge to graph
def addEdge(self,u,v):
self.graph[u].append(v)
# A function used by DFS
def DFSUtil(self,v,visited):
# Mark the current node as visited and print it
visited[v]= True
print v,
# Recur for all the vertices adjacent to this vertex
for i in self.graph[v]:
if visited[i] == False:
self.DFSUtil(i, visited)
# The function to do DFS traversal. It uses
# recursive DFSUtil()
def DFS(self,v):
# Mark all the vertices as not visited
visited = [False]*(len(self.graph))
# Call the recursive helper function to print
# DFS traversal
self.DFSUtil(v,visited)
# Driver code
# Create a graph given in the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
print "Following is DFS from (starting from vertex 2)"
g.DFS(2)
# This code is contributed by Neelam Yadav
输出
Following is Depth First Traversal (starting from vertex 2)
2 0 1 3
请注意，上述代码仅遍历从给定源顶点可到达的顶点。可能无法从给定顶点到达所有顶点（示例断开连接图）。 要完成这样的图的DFS遍历，我们必须调用每个顶点的DFSUtil()。 另外，在调用DFSUtil()之前，我们应该检查它是否已被其他调用DFSUtil()打印出来。 下面的实现执行完整的图遍历，即使节点不可达，也是如此。 与上述代码的区别在下面的代码中突出显示。
C++版本
// C++ program to print DFS traversal for a given given graph
#include <iostream>
#include <list>
using namespace std;
class Graph
{
int V; // No. of vertices
list<int> *adj; // Pointer to an array containing adjacency lists
void DFSUtil(int v, bool visited[]); // A function used by DFS
public:
Graph(int V); // Constructor
void addEdge(int v, int w); // function to add an edge to graph
void DFS(); // prints DFS traversal of the complete graph
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<int>[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w); // Add w to v’s list.
}
void Graph::DFSUtil(int v, bool visited[])
{
// Mark the current node as visited and print it
visited[v] = true;
cout << v << " ";
// Recur for all the vertices adjacent to this vertex
list<int>::iterator i;
for(i = adj[v].begin(); i != adj[v].end(); ++i)
if(!visited[*i])
DFSUtil(*i, visited);
}
// The function to do DFS traversal. It uses recursive DFSUtil()
void Graph::DFS()
{
// Mark all the vertices as not visited
bool *visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Call the recursive helper function to print DFS traversal
// starting from all vertices one by one
for (int i = 0; i < V; i++)
if (visited[i] == false)
DFSUtil(i, visited);
}
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
cout << "Following is Depth First Traversal";
g.DFS();
return 0;
}
Java版本
// Java program to print DFS traversal from a given given graph
import java.io.*;
import java.util.*;
// This class represents a directed graph using adjacency list
// representation
class Graph
{
private int V; // No. of vertices
// Array of lists for Adjacency List Representation
private LinkedList<Integer> adj[];
// Constructor
Graph(int v)
{
V = v;
adj = new LinkedList[v];
for (int i=0; i<v; ++i)
adj[i] = new LinkedList();
}
//Function to add an edge into the graph
void addEdge(int v, int w)
{
adj[v].add(w); // Add w to v's list.
}
// A function used by DFS
void DFSUtil(int v,boolean visited[])
{
// Mark the current node as visited and print it
visited[v] = true;
System.out.print(v+" ");
// Recur for all the vertices adjacent to this vertex
Iterator<Integer> i = adj[v].listIterator();
while (i.hasNext())
{
int n = i.next();
if (!visited[n])
DFSUtil(n,visited);
}
}
// The function to do DFS traversal. It uses recursive DFSUtil()
void DFS()
{
// Mark all the vertices as not visited(set as
// false by default in java)
boolean visited[] = new boolean[V];
// Call the recursive helper function to print DFS traversal
// starting from all vertices one by one
for (int i=0; i<V; ++i)
if (visited[i] == false)
DFSUtil(i, visited);
}
public static void main(String args[])
{
Graph g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
System.out.println("Following is Depth First Traversal");
g.DFS();
}
}
// This code is contributed by Aakash Hasija
Python版本
# Python program to print DFS traversal for complete graph
from collections import defaultdict
# This class represents a directed graph using adjacency
# list representation
class Graph:
# Constructor
def __init__(self):
# default dictionary to store graph
self.graph = defaultdict(list)
# function to add an edge to graph
def addEdge(self,u,v):
self.graph[u].append(v)
# A function used by DFS
def DFSUtil(self, v, visited):
# Mark the current node as visited and print it
visited[v]= True
print v,
# Recur for all the vertices adjacent to
# this vertex
for i in self.graph[v]:
if visited[i] == False:
self.DFSUtil(i, visited)
# The function to do DFS traversal. It uses
# recursive DFSUtil()
def DFS(self):
V = len(self.graph) #total vertices
# Mark all the vertices as not visited
visited =[False]*(V)
# Call the recursive helper function to print
# DFS traversal starting from all vertices one
# by one
for i in range(V):
if visited[i] == False:
self.DFSUtil(i, visited)
# Driver code
# Create a graph given in the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
print "Following is Depth First Traversal"
g.DFS()
# This code is contributed by Neelam Yadav
输出
Following is Depth First Traversal
0 1 2 3