Given a graph and two nodes u and v, the task is to print the shortest path between u and v using the Floyd Warshall algorithm.
Examples:
Input: u = 1, v = 3
Output: 1 -> 2 -> 3
Explanation:
Shortest path from 1 to 3 is through vertex 2 with total cost 3.
The first edge is 1 -> 2 with cost 2 and the second edge is 2 -> 3 with cost 1.
Input: u = 0, v = 2
Output: 0 -> 1 -> 2
Explanation:
Shortest path from 0 to 2 is through vertex 1 with total cost = 5
Approach:
- The main idea here is to use a matrix(2D array) that will keep track of the next node to point if the shortest path changes for any pair of nodes. Initially, the shortest path between any two nodes u and v is v (that is the direct edge from u -> v).
- Initialising the Next array
If the path exists between two nodes then Next[u][v] = v
else we set Next[u][v] = -1
- Modification in Floyd Warshall Algorithm
Inside the if condition of Floyd Warshall Algorithm we’ll add a statement Next[i][j] = Next[i][k]
(that means we found the shortest path between i, j through an intermediate node k).
- This is how our if condition would look like
if(dis[i][j] > dis[i][k] + dis[k][j]) { dis[i][j] = dis[i][k] + dis[k][j]; Next[i][j] = Next[i][k]; }
- For constructing path using these nodes we’ll simply start looping through the node u while updating its value to next[u][v] until we reach node v.
path = [u] while u != v: u = Next[u][v] path.append(u)
Below is the implementation of the above approach.
C++
// C++ program to find the shortest // path between any two nodes using // Floyd Warshall Algorithm. #include <bits/stdc++.h> using namespace std; #define MAXN 100 // Infinite value for array const int INF = 1e7; int dis[MAXN][MAXN]; int Next[MAXN][MAXN]; // Initializing the distance and // Next array void initialise( int V, vector<vector< int > >& graph) { for ( int i = 0; i < V; i++) { for ( int j = 0; j < V; j++) { dis[i][j] = graph[i][j]; // No edge between node // i and j if (graph[i][j] == INF) Next[i][j] = -1; else Next[i][j] = j; } } } // Function construct the shotest // path between u and v vector< int > constructPath( int u, int v) { // If there's no path between // node u and v, simply return // an empty array if (Next[u][v] == -1) return {}; // Storing the path in a vector vector< int > path = { u }; while (u != v) { u = Next[u][v]; path.push_back(u); } return path; } // Standard Floyd Warshall Algorithm // with little modification Now if we find // that dis[i][j] > dis[i][k] + dis[k][j] // then we modify next[i][j] = next[i][k] void floydWarshall( int V) { for ( int k = 0; k < V; k++) { for ( int i = 0; i < V; i++) { for ( int j = 0; j < V; j++) { // We cannot travel through // edge that doesn't exist if (dis[i][k] == INF || dis[k][j] == INF) continue ; if (dis[i][j] > dis[i][k] + dis[k][j]) { dis[i][j] = dis[i][k] + dis[k][j]; Next[i][j] = Next[i][k]; } } } } } // Print the shortest path void printPath(vector< int >& path) { int n = path.size(); for ( int i = 0; i < n - 1; i++) cout << path[i] << " -> " ; cout << path[n - 1] << endl; } // Driver code int main() { int V = 4; vector<vector< int > > graph = { { 0, 3, INF, 7 }, { 8, 0, 2, INF }, { 5, INF, 0, 1 }, { 2, INF, INF, 0 } }; // Function to initialise the // distance and Next array initialise(V, graph); // Calling Floyd Warshall Algorithm, // this will update the shortest // distance as well as Next array floydWarshall(V); vector< int > path; // Path from node 1 to 3 cout << "Shortest path from 1 to 3: " ; path = constructPath(1, 3); printPath(path); // Path from node 0 to 2 cout << "Shortest path from 0 to 2: " ; path = constructPath(0, 2); printPath(path); // path from node 3 to 2 cout << "Shortest path from 3 to 2: " ; path = constructPath(3, 2); printPath(path); return 0; } |
Java
// Java program to find the shortest // path between any two nodes using // Floyd Warshall Algorithm. import java.util.*; class GFG{ static final int MAXN = 100 ; // Infinite value for array static int INF = ( int ) 1e7; static int [][]dis = new int [MAXN][MAXN]; static int [][]Next = new int [MAXN][MAXN]; // Initializing the distance and // Next array static void initialise( int V, int [][] graph) { for ( int i = 0 ; i < V; i++) { for ( int j = 0 ; j < V; j++) { dis[i][j] = graph[i][j]; // No edge between node // i and j if (graph[i][j] == INF) Next[i][j] = - 1 ; else Next[i][j] = j; } } } // Function conthe shotest // path between u and v static Vector<Integer> constructPath( int u, int v) { // If there's no path between // node u and v, simply return // an empty array if (Next[u][v] == - 1 ) return null ; // Storing the path in a vector Vector<Integer> path = new Vector<Integer>(); path.add(u); while (u != v) { u = Next[u][v]; path.add(u); } return path; } // Standard Floyd Warshall Algorithm // with little modification Now if we find // that dis[i][j] > dis[i][k] + dis[k][j] // then we modify next[i][j] = next[i][k] static void floydWarshall( int V) { for ( int k = 0 ; k < V; k++) { for ( int i = 0 ; i < V; i++) { for ( int j = 0 ; j < V; j++) { // We cannot travel through // edge that doesn't exist if (dis[i][k] == INF || dis[k][j] == INF) continue ; if (dis[i][j] > dis[i][k] + dis[k][j]) { dis[i][j] = dis[i][k] + dis[k][j]; Next[i][j] = Next[i][k]; } } } } } // Print the shortest path static void printPath(Vector<Integer> path) { int n = path.size(); for ( int i = 0 ; i < n - 1 ; i++) System.out.print(path.get(i) + " -> " ); System.out.print(path.get(n - 1 ) + "\n" ); } // Driver code public static void main(String[] args) { int V = 4 ; int [][] graph = { { 0 , 3 , INF, 7 }, { 8 , 0 , 2 , INF }, { 5 , INF, 0 , 1 }, { 2 , INF, INF, 0 } }; // Function to initialise the // distance and Next array initialise(V, graph); // Calling Floyd Warshall Algorithm, // this will update the shortest // distance as well as Next array floydWarshall(V); Vector<Integer> path; // Path from node 1 to 3 System.out.print( "Shortest path from 1 to 3: " ); path = constructPath( 1 , 3 ); printPath(path); // Path from node 0 to 2 System.out.print( "Shortest path from 0 to 2: " ); path = constructPath( 0 , 2 ); printPath(path); // Path from node 3 to 2 System.out.print( "Shortest path from 3 to 2: " ); path = constructPath( 3 , 2 ); printPath(path); } } // This code is contributed by Amit Katiyar |
C#
// C# program to find the shortest // path between any two nodes using // Floyd Warshall Algorithm. using System; using System.Collections.Generic; class GFG{ static readonly int MAXN = 100; // Infinite value for array static int INF = ( int )1e7; static int [,]dis = new int [MAXN, MAXN]; static int [,]Next = new int [MAXN, MAXN]; // Initializing the distance and // Next array static void initialise( int V, int [,] graph) { for ( int i = 0; i < V; i++) { for ( int j = 0; j < V; j++) { dis[i, j] = graph[i, j]; // No edge between node // i and j if (graph[i, j] == INF) Next[i, j] = -1; else Next[i, j] = j; } } } // Function conthe shotest // path between u and v static List< int > constructPath( int u, int v) { // If there's no path between // node u and v, simply return // an empty array if (Next[u, v] == -1) return null ; // Storing the path in a vector List< int > path = new List< int >(); path.Add(u); while (u != v) { u = Next[u, v]; path.Add(u); } return path; } // Standard Floyd Warshall Algorithm // with little modification Now if we find // that dis[i,j] > dis[i,k] + dis[k,j] // then we modify next[i,j] = next[i,k] static void floydWarshall( int V) { for ( int k = 0; k < V; k++) { for ( int i = 0; i < V; i++) { for ( int j = 0; j < V; j++) { // We cannot travel through // edge that doesn't exist if (dis[i, k] == INF || dis[k, j] == INF) continue ; if (dis[i, j] > dis[i, k] + dis[k, j]) { dis[i, j] = dis[i, k] + dis[k, j]; Next[i, j] = Next[i, k]; } } } } } // Print the shortest path static void printPath(List< int > path) { int n = path.Count; for ( int i = 0; i < n - 1; i++) Console.Write(path[i] + " -> " ); Console.Write(path[n - 1] + "\n" ); } // Driver code public static void Main(String[] args) { int V = 4; int [,] graph = { { 0, 3, INF, 7 }, { 8, 0, 2, INF }, { 5, INF, 0, 1 }, { 2, INF, INF, 0 } }; // Function to initialise the // distance and Next array initialise(V, graph); // Calling Floyd Warshall Algorithm, // this will update the shortest // distance as well as Next array floydWarshall(V); List< int > path; // Path from node 1 to 3 Console.Write( "Shortest path from 1 to 3: " ); path = constructPath(1, 3); printPath(path); // Path from node 0 to 2 Console.Write( "Shortest path from 0 to 2: " ); path = constructPath(0, 2); printPath(path); // Path from node 3 to 2 Console.Write( "Shortest path from 3 to 2: " ); path = constructPath(3, 2); printPath(path); } } // This code is contributed by Amit Katiyar |
Python3
# Python3 program to find the shortest # path between any two nodes using # Floyd Warshall Algorithm. # Initializing the distance and # Next array def initialise(V): global dis, Next for i in range (V): for j in range (V): dis[i][j] = graph[i][j] # No edge between node # i and j if (graph[i][j] = = INF): Next [i][j] = - 1 else : Next [i][j] = j # Function construct the shotest # path between u and v def constructPath(u, v): global graph, Next # If there's no path between # node u and v, simply return # an empty array if ( Next [u][v] = = - 1 ): return {} # Storing the path in a vector path = [u] while (u ! = v): u = Next [u][v] path.append(u) return path # Standard Floyd Warshall Algorithm # with little modification Now if we find # that dis[i][j] > dis[i][k] + dis[k][j] # then we modify next[i][j] = next[i][k] def floydWarshall(V): global dist, Next for k in range (V): for i in range (V): for j in range (V): # We cannot travel through # edge that doesn't exist if (dis[i][k] = = INF or dis[k][j] = = INF): continue if (dis[i][j] > dis[i][k] + dis[k][j]): dis[i][j] = dis[i][k] + dis[k][j] Next [i][j] = Next [i][k] # Prthe shortest path def printPath(path): n = len (path) for i in range (n - 1 ): print (path[i], end = " -> " ) print (path[n - 1 ]) # Driver code if __name__ = = '__main__' : MAXM,INF = 100 , 10 * * 7 dis = [[ - 1 for i in range (MAXM)] for i in range (MAXM)] Next = [[ - 1 for i in range (MAXM)] for i in range (MAXM)] V = 4 graph = [ [ 0 , 3 , INF, 7 ], [ 8 , 0 , 2 , INF ], [ 5 , INF, 0 , 1 ], [ 2 , INF, INF, 0 ] ] # Function to initialise the # distance and Next array initialise(V) # Calling Floyd Warshall Algorithm, # this will update the shortest # distance as well as Next array floydWarshall(V) path = [] # Path from node 1 to 3 print ( "Shortest path from 1 to 3: " , end = "") path = constructPath( 1 , 3 ) printPath(path) # Path from node 0 to 2 print ( "Shortest path from 0 to 2: " , end = "") path = constructPath( 0 , 2 ) printPath(path) # Path from node 3 to 2 print ( "Shortest path from 3 to 2: " , end = "") path = constructPath( 3 , 2 ) printPath(path) # This code is contributed by mohit kumar 29 |
Shortest path from 1 to 3: 1 -> 2 -> 3 Shortest path from 0 to 2: 0 -> 1 -> 2 Shortest path from 3 to 2: 3 -> 0 -> 1 -> 2
Complexity Analysis:
- The time complexity for Floyd Warshall Algorithm is O(V3)
- For finding shortest path time complexity is O(V) per query.
Note: It would be efficient to use the Floyd Warshall Algorithm when your graph contains a couple of hundred vertices and you need to answer multiple queries related to the shortest path.
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