# Floyd-Warshall Shortest Path for all node pairs

### Floyd-Warshall shortest path for all node pairs

The gist of Floyd-Warshall all pairs shortest path 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

1. Insert source node 1 in the path **[ 1 ]**
2. 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.
3. 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++ : Implementation of Floyd-Warshall’s all pairs shortest path algorithm in C++11

``````#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

// Edges from node 1

// 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.

// Edges from node 3

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
``````