Floyd-Warshall Shortest Path Algorithm
Floyd-Warshall shortest path for all node pairs
Below are the key points of Floyd-Warshall algorithm
- Floyd-Warshall algorithm finds the shortest path between all pairs of vertices (in terms of distance / cost ) in a directed weighted graph containing positive and negative edge weights.
- Floyd-Warshall algorithm works on graphs without any negative weight cycles.
- Floyd-Warshall algorithm is based on a recursive function Shortest_Path (i, j, k), where i is the source, j is the destination and [ 1…k ] are the intermediate nodes that can be used for going from i to j. Thus, Shortest_Path ( i, j, k ) = Minimum ( Shortest_Path ( i, j, k-1 ) , Shortest_Path ( i, k, k-1 ) + Shortest_Path ( k, j, k-1 ) ) Where, The shortest path does not visit an intermediate node k Shortest_Path ( i, j, k-1 ) : The shortest path from i to j does not take an intermediate node k but there are still ( k-1 ) nodes to choose from. The shortest path visits an intermediate node k Shortest_Path ( i, k, k-1 ) : The shortest path from i to k with intermediate nodes ( k-1 ) to choose from. Shortest_Path ( k, j, k-1 ) : The shortest path from k to j with intermediate nodes ( k-1 ) to choose from. Using this recursive function as a base, the Floyd-Warshall finds the shortest path from node i to node j using all the available intermediate nodes [ 1 … k ].
Algorithm : Floyd-Warshall 1. Create a two-dimensional array of size n x n for storing the length of the shortest path from every node to every other node. n is the number of nodes in the graph. Initialize all the cells of this array to ∞. 2. For every node i in the graph, initialize the distance of the node to itself as 0. Distance [ i ] [ i ] = 0. 3. For every edge (u, v) in the graph, initialize the distance array as the weight of the edge. Distance [ u ] [ v ] = Weight of (u, v) 4. For every node k in [ 1..n ] do For every node i in [ 1..n ] do For every node j in [ 1..n ] do If ( Distance [ i ] [ j ] > Distance [ i ] [ k ] + Distance [ k ] [ j ] ) : Distance [ i ] [ j ] = Distance [ i ] [ k ] + Distance [ k ] [ j ]
Floyd-Warshall path construction example
Consider the shortest path in terms of weight / distance between node Source ( 1 ) - Destination ( 3 ) : [ 1 2 3 ] Case 1 : The shortest path from a node to itself is the node itself. i.e next [ i ] [ i ] = i. Example : next [ 1 ] [ 1 ] = 1 i.e Next node from 1 to reach 1 is 1. Example : next [ 3 ] [ 3 ] = 3 i.e Next node from 3 to reach 3 is 3. Case 2 : If the shortest path from node i to node j does not visit any intermediate node, next [ i ] [ j ] = j. Example : next [ 1 ] [ 2 ] = 2 i.e Next node from 1 to reach 2 is 2. Example : next [ 2 ] [ 3 ] = 3 i.e Next node from 2 to reach 3 is 3. Case 3 : If the shortest path from node i to node j visits an intermediate node k, **next [ i ] [ j ] = next [ i ] [ k ]** Example : next [ 1 ] [ 3 ] = next [ 1 ] [ 2 ] = 2 i.e Next node from 1 to 3 is 2.
To construct the path from Source 1 - Destination 3 the algorithm follows the below steps
- Insert source node 1 in the path [ 1 ]
- Find the next node in the path from 1 - 3 i.e next [ 1 ] [ 3 ] = 2. Insert node 2 in the path [ 1 2 ]. 2 becomes the source node.
- Find the next node in the path from 2 - 3 i.e next [ 2 ] [ 3 ] = 3. Insert node 3 in the path [ 1 2 3]. 3 becomes the source which is now same as the destination.
Algorithm : Floyd-Warshall path construction (Source src, Destination dst) 1. If next [ src ] [ dst ] == -1 then no path exist. return empty path [ ]. 2. Insert path = [ src ] 3. While ( src != dst ) 4. source = next [ src ] [ dst ] 5. Append src to path
Graph type : Designed for directed weighted graphs containing positive and negative edge weights and no negative weight cycles.
Data structure used for storing graph : Edge list for storing the edges and their corresponding weights.
Time complexity of Flyod-Warshall’s algorithm : O (N^3). N is the number of vertices / node in the graph.
C++ program for Floyd-Warshall’s all pairs shortest path algorithm.
#include<iostream>
#include<map>
#include<vector>
using namespace std;
typedef pair<int,int> PII;
/*
1
/|\
/ | \
9/ |7 \3
/ 3 | \
0----4----2
\ | 1 /
2\ |1 /-2
\ | /
\|/
3
*/
class Graph {
private:
map<PII, int> edge_weight;
int nodes;
vector<vector<int>> distance; // stores the shortest distance between [src][dst]
vector<vector<int>> next; // stores the node next to src for a path between [src][dst]
public:
Graph() {}
Graph (int n) {
nodes = n;
distance.resize(n);
next.resize(n);
for (int i=0; i<n; i++) {
distance[i].resize(n, 999999999); // 999999999 indicates infinite distance
next[i].resize(n, -1);
}
}
void AddEdgeWeight (int src, int dst, int weight, bool isbidirectional = true) {
edge_weight.insert(pair<PII,int>(make_pair(src, dst), weight));
if (isbidirectional)
edge_weight.insert(pair<PII,int>(make_pair(dst, src), weight));
}
void Floyd_Warshall() {
for (int i=0; i<nodes; i++) {
distance[i][i] = 0;
next[i][i] = i;
}
for (auto& it: edge_weight) {
PII edge = it.first;
int weight = it.second;
int u = edge.first;
int v = edge.second;
distance[u][v] = weight;
next[u][v] = v;
}
for (int k=0; k<nodes; k++) {
for (int i=0; i<nodes; i++) {
for (int j=0; j<nodes; j++) {
if (distance[i][j] > distance[i][k] + distance[k][j]) {
distance[i][j] = distance[i][k] + distance[k][j];
next[i][j] = next[i][k];
}
}
}
}
cout << "Shortest distance between nodes" << endl;
for (int u=0; u<nodes; u++) {
for (int v=u+1; v<nodes; v++) {
cout << "Distance ( " << u << " - " << v << " ) : " << distance[u][v];
PathConstruction(u, v);
}
}
}
// Construct path from source node to destination node
void PathConstruction (int src, int dst) {
cout << " # Path between " << src << " and " << dst << " : ";
if (next[src][dst] == -1) {
cout << "No path exists" << endl;
} else {
vector<int> path;
path.push_back(src);
while (src != dst) {
src = next[src][dst];
path.push_back(src);
}
for (auto& it : path)
cout << it << " ";
cout << endl;
}
}
};
int main()
{
Graph g(5);
// Edges from node 0
g.AddEdgeWeight(0, 1, 9);
g.AddEdgeWeight(0, 3, 2);
g.AddEdgeWeight(0, 4, 3);
// Edges from node 1
g.AddEdgeWeight(1, 2, 3);
g.AddEdgeWeight(1, 4, 7);
// Edges from node 2
// Edge from 2 -> 3 is unidirectional. If it was bidirectional, it would introduce negative weight cycle
// causing the Floyd-Warshall algorithm to fail.
g.AddEdgeWeight(2, 3, -2, false);
g.AddEdgeWeight(2, 4, 1);
// Edges from node 3
g.AddEdgeWeight(3, 4, 1);
g.Floyd_Warshall();
return 0;
}
Output
Shortest distance between nodes
Distance ( 0 - 1 ) : 7 # Path between 0 and 1 : 0 4 2 1
Distance ( 0 - 2 ) : 4 # Path between 0 and 2 : 0 4 2
Distance ( 0 - 3 ) : 2 # Path between 0 and 3 : 0 3
Distance ( 0 - 4 ) : 3 # Path between 0 and 4 : 0 4
Distance ( 1 - 2 ) : 3 # Path between 1 and 2 : 1 2
Distance ( 1 - 3 ) : 1 # Path between 1 and 3 : 1 2 3
Distance ( 1 - 4 ) : 2 # Path between 1 and 4 : 1 2 3 4
Distance ( 2 - 3 ) : -2 # Path between 2 and 3 : 2 3
Distance ( 2 - 4 ) : -1 # Path between 2 and 4 : 2 3 4
Distance ( 3 - 4 ) : 1 # Path between 3 and 4 : 3 4