Minimum number of edges between two vertices of a Graph
You are given a undirected graph G(V, E) with N vertices and M edges. We need to find the minimum number of edges between a given pair of vertices (u, v).
Examples:
Input : For given graph G. Find minimum number of edges between (1, 5).
Output : 2 Explanation: (1, 2) and (2, 5) are the only edges resulting into shortest path between 1 and 5.
The idea is to perform BFS from one of given input vertex(u). At the time of BFS maintain an array of distance[n] and initialize it to zero for all vertices. Now, suppose during BFS, vertex x is popped from queue and we are pushing all adjacent non-visited vertices(i) back into queue at the same time we should update distance[i] = distance[x] + 1;.
Finally, distance[v] gives the minimum number of edges between u and v.
Algorithm:
int minEdgeBFS(int u, int v, int n) { // visited[n] for keeping track of visited // node in BFS bool visited[n] = {0}; // Initialize distances as 0 int distance[n] = {0}; // queue to do BFS. queue Q; distance[u] = 0; Q.push(u); visited[u] = true; while (!Q.empty()) { int x = Q.front(); Q.pop(); for (int i=0; i<edges[x].size(); i++) { if (visited[edges[x][i]]) continue; // update distance for i distance[edges[x][i]] = distance[x] + 1; Q.push(edges[x][i]); visited[edges[x][i]] = 1; } } return distance[v]; }
C++
// C++ program to find minimum edge // between given two vertex of Graph #include<bits/stdc++.h> using namespace std; // function for finding minimum no. of edge // using BFS int minEdgeBFS(vector < int > edges[], int u, int v, int n) { // visited[n] for keeping track of visited // node in BFS vector< bool > visited(n, 0); // Initialize distances as 0 vector< int > distance(n, 0); // queue to do BFS. queue < int > Q; distance[u] = 0; Q.push(u); visited[u] = true ; while (!Q.empty()) { int x = Q.front(); Q.pop(); for ( int i=0; i<edges[x].size(); i++) { if (visited[edges[x][i]]) continue ; // update distance for i distance[edges[x][i]] = distance[x] + 1; Q.push(edges[x][i]); visited[edges[x][i]] = 1; } } return distance[v]; } // function for addition of edge void addEdge(vector < int > edges[], int u, int v) { edges[u].push_back(v); edges[v].push_back(u); } // Driver function int main() { // To store adjacency list of graph int n = 9; vector < int > edges[9]; addEdge(edges, 0, 1); addEdge(edges, 0, 7); addEdge(edges, 1, 7); addEdge(edges, 1, 2); addEdge(edges, 2, 3); addEdge(edges, 2, 5); addEdge(edges, 2, 8); addEdge(edges, 3, 4); addEdge(edges, 3, 5); addEdge(edges, 4, 5); addEdge(edges, 5, 6); addEdge(edges, 6, 7); addEdge(edges, 7, 8); int u = 0; int v = 5; cout << minEdgeBFS(edges, u, v, n); return 0; } |
Java
// Java program to find minimum edge // between given two vertex of Graph import java.util.LinkedList; import java.util.Queue; import java.util.Vector; class Test { // Method for finding minimum no. of edge // using BFS static int minEdgeBFS(Vector <Integer> edges[], int u, int v, int n) { // visited[n] for keeping track of visited // node in BFS Vector<Boolean> visited = new Vector<Boolean>(n); for ( int i = 0 ; i < n; i++) { visited.addElement( false ); } // Initialize distances as 0 Vector<Integer> distance = new Vector<Integer>(n); for ( int i = 0 ; i < n; i++) { distance.addElement( 0 ); } // queue to do BFS. Queue<Integer> Q = new LinkedList<>(); distance.setElementAt( 0 , u); Q.add(u); visited.setElementAt( true , u); while (!Q.isEmpty()) { int x = Q.peek(); Q.poll(); for ( int i= 0 ; i<edges[x].size(); i++) { if (visited.elementAt(edges[x].get(i))) continue ; // update distance for i distance.setElementAt(distance.get(x) + 1 ,edges[x].get(i)); Q.add(edges[x].get(i)); visited.setElementAt( true ,edges[x].get(i)); } } return distance.get(v); } // method for addition of edge static void addEdge(Vector <Integer> edges[], int u, int v) { edges[u].add(v); edges[v].add(u); } // Driver method public static void main(String args[]) { // To store adjacency list of graph int n = 9 ; Vector <Integer> edges[] = new Vector[ 9 ]; for ( int i = 0 ; i < edges.length; i++) { edges[i] = new Vector<>(); } addEdge(edges, 0 , 1 ); addEdge(edges, 0 , 7 ); addEdge(edges, 1 , 7 ); addEdge(edges, 1 , 2 ); addEdge(edges, 2 , 3 ); addEdge(edges, 2 , 5 ); addEdge(edges, 2 , 8 ); addEdge(edges, 3 , 4 ); addEdge(edges, 3 , 5 ); addEdge(edges, 4 , 5 ); addEdge(edges, 5 , 6 ); addEdge(edges, 6 , 7 ); addEdge(edges, 7 , 8 ); int u = 0 ; int v = 5 ; System.out.println(minEdgeBFS(edges, u, v, n)); } } // This code is contributed by Gaurav Miglani |
Python3
# Python3 program to find minimum edge # between given two vertex of Graph import queue # function for finding minimum # no. of edge using BFS def minEdgeBFS(edges, u, v, n): # visited[n] for keeping track # of visited node in BFS visited = [ 0 ] * n # Initialize distances as 0 distance = [ 0 ] * n # queue to do BFS. Q = queue.Queue() distance[u] = 0 Q.put(u) visited[u] = True while ( not Q.empty()): x = Q.get() for i in range ( len (edges[x])): if (visited[edges[x][i]]): continue # update distance for i distance[edges[x][i]] = distance[x] + 1 Q.put(edges[x][i]) visited[edges[x][i]] = 1 return distance[v] # function for addition of edge def addEdge(edges, u, v): edges[u].append(v) edges[v].append(u) # Driver Code if __name__ = = '__main__' : # To store adjacency list of graph n = 9 edges = [[] for i in range (n)] addEdge(edges, 0 , 1 ) addEdge(edges, 0 , 7 ) addEdge(edges, 1 , 7 ) addEdge(edges, 1 , 2 ) addEdge(edges, 2 , 3 ) addEdge(edges, 2 , 5 ) addEdge(edges, 2 , 8 ) addEdge(edges, 3 , 4 ) addEdge(edges, 3 , 5 ) addEdge(edges, 4 , 5 ) addEdge(edges, 5 , 6 ) addEdge(edges, 6 , 7 ) addEdge(edges, 7 , 8 ) u = 0 v = 5 print (minEdgeBFS(edges, u, v, n)) # This code is contributed by PranchalK |
C#
// C# program to find minimum edge // between given two vertex of Graph using System; using System.Collections; using System.Collections.Generic; class GFG{ // Method for finding minimum no. of edge // using BFS static int minEdgeBFS(ArrayList []edges, int u, int v, int n) { // visited[n] for keeping track of visited // node in BFS ArrayList visited = new ArrayList(); for ( int i = 0; i < n; i++) { visited.Add( false ); } // Initialize distances as 0 ArrayList distance = new ArrayList(); for ( int i = 0; i < n; i++) { distance.Add(0); } // queue to do BFS. Queue Q = new Queue(); distance[u] = 0; Q.Enqueue(u); visited[u] = true ; while (Q.Count != 0) { int x = ( int )Q.Dequeue(); for ( int i = 0; i < edges[x].Count; i++) { if (( bool )visited[( int )edges[x][i]]) continue ; // Update distance for i distance[( int )edges[x][i]] = ( int )distance[x] + 1; Q.Enqueue(( int )edges[x][i]); visited[( int )edges[x][i]] = true ; } } return ( int )distance[v]; } // Method for addition of edge static void addEdge(ArrayList []edges, int u, int v) { edges[u].Add(v); edges[v].Add(u); } // Driver code public static void Main( string []args) { // To store adjacency list of graph int n = 9; ArrayList []edges = new ArrayList[9]; for ( int i = 0; i < 9; i++) { edges[i] = new ArrayList(); } addEdge(edges, 0, 1); addEdge(edges, 0, 7); addEdge(edges, 1, 7); addEdge(edges, 1, 2); addEdge(edges, 2, 3); addEdge(edges, 2, 5); addEdge(edges, 2, 8); addEdge(edges, 3, 4); addEdge(edges, 3, 5); addEdge(edges, 4, 5); addEdge(edges, 5, 6); addEdge(edges, 6, 7); addEdge(edges, 7, 8); int u = 0; int v = 5; Console.Write(minEdgeBFS(edges, u, v, n)); } } // This code is contributed by rutvik_56 |
Javascript
<script> // JavaScript program to find minimum edge // between given two vertex of Graph // Method for finding minimum no. of edge // using BFS function minEdgeBFS(edges,u,v,n) { // visited[n] for keeping track of visited // node in BFS let visited = []; for (let i = 0; i < n; i++) { visited.push( false ); } // Initialize distances as 0 let distance = []; for (let i = 0; i < n; i++) { distance.push(0); } // queue to do BFS. let Q = []; distance[u] = 0; Q.push(u); visited[u] = true ; while (Q.length!=0) { let x = Q.shift(); for (let i=0; i<edges[x].length; i++) { if (visited[edges[x][i]]) continue ; // update distance for i distance[edges[x][i]] = distance[x] + 1; Q.push(edges[x][i]); visited[edges[x][i]]= true ; } } return distance[v]; } // method for addition of edge function addEdge(edges,u,v) { edges[u].push(v); edges[v].push(u); } // Driver method // To store adjacency list of graph let n = 9; let edges = new Array(9); for (let i = 0; i < edges.length; i++) { edges[i] = []; } addEdge(edges, 0, 1); addEdge(edges, 0, 7); addEdge(edges, 1, 7); addEdge(edges, 1, 2); addEdge(edges, 2, 3); addEdge(edges, 2, 5); addEdge(edges, 2, 8); addEdge(edges, 3, 4); addEdge(edges, 3, 5); addEdge(edges, 4, 5); addEdge(edges, 5, 6); addEdge(edges, 6, 7); addEdge(edges, 7, 8); let u = 0; let v = 5; document.write(minEdgeBFS(edges, u, v, n)); // This code is contributed by rag2127 </script> |
Output:
3
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