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📜  打印从给定源到目标的所有路径

📅  最后修改于: 2021-05-24 19:40:18             🧑  作者: Mango

给定有向图,源顶点“ s”和目标顶点“ d”,打印从给定“ s”到“ d”的所有路径。
考虑以下有向图。令s为2,d为3。从2到3有4条不同的路径。

allPaths

方法:

  1. 这个想法是对给定的有向图进行深度优先遍历。
  2. 从源开始DFS遍历。
  3. 继续将访问的顶点存储在数组或HashMap中,说“ path []”。
  4. 如果到达目标顶点,则打印path []的内容。
  5. 重要的是将path []中的当前顶点也标记为已访问,以使遍历不会以一个周期进行。

以下是上述想法的实现。

C++14
// C++ program to print all paths
// from a source to destination.
#include 
#include 
using namespace std;
  
// A directed graph using
// adjacency list representation
class Graph {
    int V; // No. of vertices in graph
    list* adj; // Pointer to an array containing adjacency lists
  
    // A recursive function used by printAllPaths()
    void printAllPathsUtil(int, int, bool[], int[], int&);
  
public:
    Graph(int V); // Constructor
    void addEdge(int u, int v);
    void printAllPaths(int s, int d);
};
  
Graph::Graph(int V)
{
    this->V = V;
    adj = new list[V];
}
  
void Graph::addEdge(int u, int v)
{
    adj[u].push_back(v); // Add v to u’s list.
}
  
// Prints all paths from 's' to 'd'
void Graph::printAllPaths(int s, int d)
{
    // Mark all the vertices as not visited
    bool* visited = new bool[V];
  
    // Create an array to store paths
    int* path = new int[V];
    int path_index = 0; // Initialize path[] as empty
  
    // Initialize all vertices as not visited
    for (int i = 0; i < V; i++)
        visited[i] = false;
  
    // Call the recursive helper function to print all paths
    printAllPathsUtil(s, d, visited, path, path_index);
}
  
// A recursive function to print all paths from 'u' to 'd'.
// visited[] keeps track of vertices in current path.
// path[] stores actual vertices and path_index is current
// index in path[]
void Graph::printAllPathsUtil(int u, int d, bool visited[],
                              int path[], int& path_index)
{
    // Mark the current node and store it in path[]
    visited[u] = true;
    path[path_index] = u;
    path_index++;
  
    // If current vertex is same as destination, then print
    // current path[]
    if (u == d) {
        for (int i = 0; i < path_index; i++)
            cout << path[i] << " ";
        cout << endl;
    }
    else // If current vertex is not destination
    {
        // Recur for all the vertices adjacent to current vertex
        list::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (!visited[*i])
                printAllPathsUtil(*i, d, visited, path, path_index);
    }
  
    // Remove current vertex from path[] and mark it as unvisited
    path_index--;
    visited[u] = false;
}
  
// Driver program
int main()
{
    // Create a graph given in the above diagram
    Graph g(4);
    g.addEdge(0, 1);
    g.addEdge(0, 2);
    g.addEdge(0, 3);
    g.addEdge(2, 0);
    g.addEdge(2, 1);
    g.addEdge(1, 3);
  
    int s = 2, d = 3;
    cout << "Following are all different paths from " << s << " to " << d << endl;
    g.printAllPaths(s, d);
  
    return 0;
}


Java
// JAVA program to print all
// paths from a source to
// destination.
import java.util.ArrayList;
import java.util.List;
  
// A directed graph using
// adjacency list representation
public class Graph {
  
    // No. of vertices in graph
    private int v;
  
    // adjacency list
    private ArrayList[] adjList;
  
    // Constructor
    public Graph(int vertices)
    {
  
        // initialise vertex count
        this.v = vertices;
  
        // initialise adjacency list
        initAdjList();
    }
  
    // utility method to initialise
    // adjacency list
    @SuppressWarnings("unchecked")
    private void initAdjList()
    {
        adjList = new ArrayList[v];
  
        for (int i = 0; i < v; i++) {
            adjList[i] = new ArrayList<>();
        }
    }
  
    // add edge from u to v
    public void addEdge(int u, int v)
    {
        // Add v to u's list.
        adjList[u].add(v);
    }
  
    // Prints all paths from
    // 's' to 'd'
    public void printAllPaths(int s, int d)
    {
        boolean[] isVisited = new boolean[v];
        ArrayList pathList = new ArrayList<>();
  
        // add source to path[]
        pathList.add(s);
  
        // Call recursive utility
        printAllPathsUtil(s, d, isVisited, pathList);
    }
  
    // A recursive function to print
    // all paths from 'u' to 'd'.
    // isVisited[] keeps track of
    // vertices in current path.
    // localPathList<> stores actual
    // vertices in the current path
    private void printAllPathsUtil(Integer u, Integer d,
                                   boolean[] isVisited,
                                   List localPathList)
    {
  
        if (u.equals(d)) {
            System.out.println(localPathList);
            // if match found then no need to traverse more till depth
            return;
        }
  
        // Mark the current node
        isVisited[u] = true;
  
        // Recur for all the vertices
        // adjacent to current vertex
        for (Integer i : adjList[u]) {
            if (!isVisited[i]) {
                // store current node
                // in path[]
                localPathList.add(i);
                printAllPathsUtil(i, d, isVisited, localPathList);
  
                // remove current node
                // in path[]
                localPathList.remove(i);
            }
        }
  
        // Mark the current node
        isVisited[u] = false;
    }
  
    // Driver program
    public static void main(String[] args)
    {
        // Create a sample graph
        Graph g = new Graph(4);
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(0, 3);
        g.addEdge(2, 0);
        g.addEdge(2, 1);
        g.addEdge(1, 3);
  
        // arbitrary source
        int s = 2;
  
        // arbitrary destination
        int d = 3;
  
        System.out.println(
            "Following are all different paths from "
            + s + " to " + d);
        g.printAllPaths(s, d);
    }
}
  
// This code is contributed by Himanshu Shekhar.


Python
# Python program to print all paths from a source to destination.
   
from collections import defaultdict
   
# This class represents a directed graph 
# using adjacency list representation
class Graph:
   
    def __init__(self, vertices):
        # No. of vertices
        self.V = vertices 
          
        # 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 recursive function to print all paths from 'u' to 'd'.
    visited[] keeps track of vertices in current path.
    path[] stores actual vertices and path_index is current
    index in path[]'''
    def printAllPathsUtil(self, u, d, visited, path):
  
        # Mark the current node as visited and store in path
        visited[u]= True
        path.append(u)
  
        # If current vertex is same as destination, then print
        # current path[]
        if u == d:
            print path
        else:
            # If current vertex is not destination
            # Recur for all the vertices adjacent to this vertex
            for i in self.graph[u]:
                if visited[i]== False:
                    self.printAllPathsUtil(i, d, visited, path)
                      
        # Remove current vertex from path[] and mark it as unvisited
        path.pop()
        visited[u]= False
   
   
    # Prints all paths from 's' to 'd'
    def printAllPaths(self, s, d):
  
        # Mark all the vertices as not visited
        visited =[False]*(self.V)
  
        # Create an array to store paths
        path = []
  
        # Call the recursive helper function to print all paths
        self.printAllPathsUtil(s, d, visited, path)
   
   
   
# Create a graph given in the above diagram
g = Graph(4)
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(0, 3)
g.addEdge(2, 0)
g.addEdge(2, 1)
g.addEdge(1, 3)
   
s = 2 ; d = 3
print ("Following are all different paths from % d to % d :" %(s, d))
g.printAllPaths(s, d)
# This code is contributed by Neelam Yadav


C#
// C# program to print all
// paths from a source to
// destination.
using System;
using System.Collections.Generic;
  
// A directed graph using
// adjacency list representation
public class Graph {
  
    // No. of vertices in graph
    private int v;
  
    // adjacency list
    private List[] adjList;
  
    // Constructor
    public Graph(int vertices)
    {
  
        // initialise vertex count
        this.v = vertices;
  
        // initialise adjacency list
        initAdjList();
    }
  
    // utility method to initialise
    // adjacency list
    private void initAdjList()
    {
        adjList = new List[v];
  
        for (int i = 0; i < v; i++) {
            adjList[i] = new List();
        }
    }
  
    // add edge from u to v
    public void addEdge(int u, int v)
    {
        // Add v to u's list.
        adjList[u].Add(v);
    }
  
    // Prints all paths from
    // 's' to 'd'
    public void printAllPaths(int s, int d)
    {
        bool[] isVisited = new bool[v];
        List pathList = new List();
  
        // add source to path[]
        pathList.Add(s);
  
        // Call recursive utility
        printAllPathsUtil(s, d, isVisited, pathList);
    }
  
    // A recursive function to print
    // all paths from 'u' to 'd'.
    // isVisited[] keeps track of
    // vertices in current path.
    // localPathList<> stores actual
    // vertices in the current path
    private void printAllPathsUtil(int u, int d,
                                   bool[] isVisited,
                                   List localPathList)
    {
  
        if (u.Equals(d)) {
            Console.WriteLine(string.Join(" ", localPathList));
            // if match found then no need
            // to traverse more till depth
            return;
        }
  
        // Mark the current node
        isVisited[u] = true;
  
        // Recur for all the vertices
        // adjacent to current vertex
        foreach(int i in adjList[u])
        {
            if (!isVisited[i]) {
                // store current node
                // in path[]
                localPathList.Add(i);
                printAllPathsUtil(i, d, isVisited,
                                  localPathList);
  
                // remove current node
                // in path[]
                localPathList.Remove(i);
            }
        }
  
        // Mark the current node
        isVisited[u] = false;
    }
  
    // Driver code
    public static void Main(String[] args)
    {
        // Create a sample graph
        Graph g = new Graph(4);
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(0, 3);
        g.addEdge(2, 0);
        g.addEdge(2, 1);
        g.addEdge(1, 3);
  
        // arbitrary source
        int s = 2;
  
        // arbitrary destination
        int d = 3;
  
        Console.WriteLine("Following are all different"
                          + " paths from " + s + " to " + d);
        g.printAllPaths(s, d);
    }
}
  
// This code contributed by Rajput-Ji


输出:

Following are all different paths from 2 to 3
2 0 1 3
2 0 3
2 1 3 

复杂度分析:

  • 时间复杂度: O(V ^ V)。
    时间复杂度是多项式。每个顶点都有v个顶点,可以从当前顶点访问这些顶点。
  • 辅助空间: O(V ^ V)。
    为了存储路径,需要V ^ V空间。