### Given a directed graph, design an algorithm to find out whether there is a route between two nodes.

Given a Directed Graph and two vertices in it, check whether there is a path from the first given vertex to second. For example, in the following graph, there is a path from vertex 1 to 3. As another example, there is no path from 3 to 0.

We can either use Breadth First Search (BFS) or Depth First Search (DFS) to find path between two vertices. Take the first vertex as source in BFS (or DFS), follow the standard BFS (or DFS). If we see the second vertex in our traversal, then return true. Else return false.

Following is C++ code that uses BFS for finding reachability of second vertex from first vertex.

`// Program to check if there is exist a path between two vertices of a graph.` `#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` ` ` `bool` `isReachable(` `int` `s, ` `int` `d); ` `// returns true if there is a path from s to d` `};` `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.` `}` `// A BFS based function to check whether d is reachable from s.` `bool` `Graph::isReachable(` `int` `s, ` `int` `d)` `{` ` ` `// Base case` ` ` `if` `(s == d)` ` ` `return` `true` `;` ` ` `// 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);` ` ` `// it 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();` ` ` `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 this adjacent node is the destination node, then return true` ` ` `if` `(*i == d)` ` ` `return` `true` `;` ` ` `// Else, continue to do BFS` ` ` `if` `(!visited[*i])` ` ` `{` ` ` `visited[*i] = ` `true` `;` ` ` `queue.push_back(*i);` ` ` `}` ` ` `}` ` ` `}` ` ` `return` `false` `;` `}` `// 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);` ` ` `int` `u = 1, v = 3;` ` ` `if` `(g.isReachable(u, v))` ` ` `cout<< ` `"\n There is a path from "` `<< u << ` `" to "` `<< v;` ` ` `else` ` ` `cout<< ` `"\n There is no path from "` `<< u << ` `" to "` `<< v;` ` ` `u = 3, v = 1;` ` ` `if` `(g.isReachable(u, v))` ` ` `cout<< ` `"\n There is a path from "` `<< u << ` `" to "` `<< v;` ` ` `else` ` ` `cout<< ` `"\n There is no path from "` `<< u << ` `" to "` `<< v;` ` ` `return` `0;` `}` |

Output:

There is a path from 1 to 3 There is no path from 3 to 1

As an exercise, try an extended version of the problem where the complete path between two vertices is also needed.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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