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Lesson 35, Topic 1
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# Floyd’s Algorithm

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Floyd-Warshall Algorithm is an algorithm for finding the shortest path between all the pairs of vertices in a weighted graph. This algorithm works for both the directed and undirected weighted graphs. But, it does not work for the graphs with negative cycles (where the sum of the edges in a cycle is negative).

A weighted graph is a graph in which each edge has a numerical value associated with it.

Floyd-Warhshall algorithm is also called as Floyd’s algorithm, Roy-Floyd algorithm, Roy-Warshall algorithm, or WFI algorithm.

This algorithm follows the dynamic programming approach to find the shortest paths.

Example:

```Input:
graph[][] = { {0,   5,  INF, 10},
{INF,  0,  3,  INF},
{INF, INF, 0,   1},
{INF, INF, INF, 0} }
which represents the following graph
10
(0)------->(3)
|         /|\
5 |          |
|          | 1
\|/         |
(1)------->(2)
3
Note that the value of graph[i][j] is 0 if i is equal to j
And graph[i][j] is INF (infinite) if there is no edge from vertex i to j.

Output:
Shortest distance matrix
0      5      8      9
INF      0      3      4
INF    INF      0      1
INF    INF    INF      0 ```

## Floyd-Warshall Algorithm

```n = no of vertices
A = matrix of dimension n*n
for k = 1 to n
for i = 1 to n
for j = 1 to n
Ak[i, j] = min (Ak-1[i, j], Ak-1[i, k] + Ak-1[k, j])
return A```

Implementation of Floyd’s Algorithm

``````// C Program for Floyd Warshall Algorithm
#include<stdio.h>

// Number of vertices in the graph
#define V 4

/* Define Infinite as a large enough value. This value will be used
for vertices not connected to each other */
#define INF 99999

// A function to print the solution matrix
void printSolution(int dist[][V]);

// Solves the all-pairs shortest path problem using Floyd Warshall algorithm
void floydWarshall (int graph[][V])
{
/* dist[][] will be the output matrix that will finally have the shortest
distances between every pair of vertices */
int dist[V][V], i, j, k;

/* Initialize the solution matrix same as input graph matrix. Or
we can say the initial values of shortest distances are based
on shortest paths considering no intermediate vertex. */
for (i = 0; i < V; i++)
for (j = 0; j < V; j++)
dist[i][j] = graph[i][j];

/* Add all vertices one by one to the set of intermediate vertices.
---> Before start of an iteration, we have shortest distances between all
pairs of vertices such that the shortest distances consider only the
vertices in set {0, 1, 2, .. k-1} as intermediate vertices.
----> After the end of an iteration, vertex no. k is added to the set of
intermediate vertices and the set becomes {0, 1, 2, .. k} */
for (k = 0; k < V; k++)
{
// Pick all vertices as source one by one
for (i = 0; i < V; i++)
{
// Pick all vertices as destination for the
// above picked source
for (j = 0; j < V; j++)
{
// If vertex k is on the shortest path from
// i to j, then update the value of dist[i][j]
if (dist[i][k] + dist[k][j] < dist[i][j])
dist[i][j] = dist[i][k] + dist[k][j];
}
}
}

// Print the shortest distance matrix
printSolution(dist);
}

/* A utility function to print solution */
void printSolution(int dist[][V])
{
printf ("The following matrix shows the shortest distances"
" between every pair of vertices \n");
for (int i = 0; i < V; i++)
{
for (int j = 0; j < V; j++)
{
if (dist[i][j] == INF)
printf("%7s", "INF");
else
printf ("%7d", dist[i][j]);
}
printf("\n");
}
}

// driver program to test above function
int main()
{
/* Let us create the following weighted graph
10
(0)------->(3)
|         /|\
5 |          |
|          | 1
\|/         |
(1)------->(2)
3           */
int graph[V][V] = { {0,   5,  INF, 10},
{INF, 0,   3, INF},
{INF, INF, 0,   1},
{INF, INF, INF, 0}
};

// Print the solution
floydWarshall(graph);
return 0;
} ``````

Time Complexity

``````Following matrix shows the shortest distances between every pair of vertices
0      5      8      9
INF      0      3      4
INF    INF      0      1
INF    INF    INF      0``````

## Floyd Warshall Algorithm Complexity

### Time Complexity

There are three loops. Each loop has constant complexities. So, the time complexity of the Floyd-Warshall algorithm is `O(n3)`.

### Space Complexity

The space complexity of the Floyd-Warshall algorithm is `O(n2)`.

## Floyd Warshall Algorithm Applications

• To find the shortest path is a directed graph
• To find the transitive closure of directed graphs
• To find the Inversion of real matrices
• For testing whether an undirected graph is bipartite