In this tutorial, you will learn how the floyd-warshall algorithm works. Additionally, you will discover working instances of the Floyd-warshall algorithm in C, C++, Java, and Python.

The Floyd-Warshall algorithm, additionally differently known as Floyd’s algorithm, the Roy-Floyd algorithm, the Roy-Warshall algorithm, or the WFI algorithm, is an algorithm for productively and at the same time tracking down the shortest paths (i.e., graph geodesics) between each pair of vertices in a weighted and potentially directed graph.

Floyd–Warshall’s Algorithm is an algorithm for tracking down the most brief way between every one of the sets of vertices in a weighted graph. This algorithm works for both the directed and undirected weighted graphs. In any case, it doesn’t work for the graphs with negative cycles (where the sum of the edges in a cycle is negative).

A weighted graph is a graph where each edge has a numerical value related with it.

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

This algorithm follows the dynamic programming way to deal with track down the most brief ways.

Related: What is an Algorithm?

## How Floyd-Warshall Algorithm Works?

Let the given graph be:

Follow the steps underneath to track down the shortest path between every one of the sets of vertices.

1. Create a matrix A0 of measurement n*n where n is the number of vertices. The row and the column are indexed as I and j respectively. I and j are the vertices of the graph.

Every cell A[i][j] is filled with the distance from the ith vertex to the jth vertex. In the event that there is no way from ith vertex to jth vertex, the cell is left as infinity.

2. Presently, create a matrix A1 using matrix A0. The elements in the first column and the first row are left as they are. The remaining cells are filled in an accompanying manner.

Let k be the middle vertex in the most brief way from source to objective. In this progression, k is the first vertex. A[i][j] is filled with (A[i][k] + A[k][j]) if (A[i][j] > A[i][k] + A[k][j]).

That is, if the direct distance from the source to the objective is greater than the way through the vertex k, at that point the cell is loaded up with A[i][k] + A[k][j].

In this progression, k is vertex 1. We calculate the distance from the source vertex to the objective vertex through this vertex k.

For instance: For A1[2, 4], the direct distance from vertex 2 to 4 will be 4 and the sum of the distance from vertex 2 to 4 through vertex (ie. from vertex 2 to 1 and from vertex 1 to 4) is 7. Since 4 < 7, A0[2, 4] is loaded up with 4.

3. Additionally, A2 is created using A1. The elements in the second column and the second row are left as they are.

In this progression, k is the second vertex (for example vertex 2). The remaining steps are equivalent to in step 2.

4. Additionally, A3 and A4 is likewise created.

5. A4 gives the shortest path between each pair of vertices.

## 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```

## Python, Java, and C/C++ Examples

Python

```# Floyd Warshall Algorithm in python

# The number of vertices
nV = 4

INF = 999

# Algorithm implementation
def floyd_warshall(G):
distance = list(map(lambda i: list(map(lambda j: j, i)), G))

for k in range(nV):
for i in range(nV):
for j in range(nV):
distance[i][j] = min(distance[i][j], distance[i][k] + distance[k][j])
print_solution(distance)

# Printing the solution
def print_solution(distance):
for i in range(nV):
for j in range(nV):
if(distance[i][j] == INF):
print("INF", end=" ")
else:
print(distance[i][j], end="  ")
print(" ")

G = [[0, 3, INF, 5],
[2, 0, INF, 4],
[INF, 1, 0, INF],
[INF, INF, 2, 0]]
floyd_warshall(G)```

Java

```// Floyd Warshall Algorithm in Java

class FloydWarshall {
final static int INF = 9999, nV = 4;

// Implementing floyd warshall algorithm
void floydWarshall(int graph[][]) {
int matrix[][] = new int[nV][nV];
int i, j, k;

for (i = 0; i < nV; i++)
for (j = 0; j < nV; j++)
matrix[i][j] = graph[i][j];

for (k = 0; k < nV; k++) {
for (i = 0; i < nV; i++) {
for (j = 0; j < nV; j++) {
if (matrix[i][k] + matrix[k][j] < matrix[i][j])
matrix[i][j] = matrix[i][k] + matrix[k][j];
}
}
}
printMatrix(matrix);
}

void printMatrix(int matrix[][]) {
for (int i = 0; i < nV; ++i) {
for (int j = 0; j < nV; ++j) {
if (matrix[i][j] == INF)
System.out.print("INF ");
else
System.out.print(matrix[i][j] + "  ");
}
System.out.println();
}
}

public static void main(String[] args) {
int graph[][] = { { 0, 3, INF, 5 }, { 2, 0, INF, 4 }, { INF, 1, 0, INF }, { INF, INF, 2, 0 } };
FloydWarshall a = new FloydWarshall();
a.floydWarshall(graph);
}
}```

C

```// Floyd-Warshall Algorithm in C

#include <stdio.h>

// defining the number of vertices
#define nV 4

#define INF 999

void printMatrix(int matrix[][nV]);

// Implementing floyd warshall algorithm
void floydWarshall(int graph[][nV]) {
int matrix[nV][nV], i, j, k;

for (i = 0; i < nV; i++)
for (j = 0; j < nV; j++)
matrix[i][j] = graph[i][j];

for (k = 0; k < nV; k++) {
for (i = 0; i < nV; i++) {
for (j = 0; j < nV; j++) {
if (matrix[i][k] + matrix[k][j] < matrix[i][j])
matrix[i][j] = matrix[i][k] + matrix[k][j];
}
}
}
printMatrix(matrix);
}

void printMatrix(int matrix[][nV]) {
for (int i = 0; i < nV; i++) {
for (int j = 0; j < nV; j++) {
if (matrix[i][j] == INF)
printf("%4s", "INF");
else
printf("%4d", matrix[i][j]);
}
printf("\n");
}
}

int main() {
int graph[nV][nV] = {{0, 3, INF, 5},
{2, 0, INF, 4},
{INF, 1, 0, INF},
{INF, INF, 2, 0}};
floydWarshall(graph);
}```

C++

```// Floyd-Warshall Algorithm in C++

#include <iostream>
using namespace std;

// defining the number of vertices
#define nV 4

#define INF 999

void printMatrix(int matrix[][nV]);

// Implementing floyd warshall algorithm
void floydWarshall(int graph[][nV]) {
int matrix[nV][nV], i, j, k;

for (i = 0; i < nV; i++)
for (j = 0; j < nV; j++)
matrix[i][j] = graph[i][j];

for (k = 0; k < nV; k++) {
for (i = 0; i < nV; i++) {
for (j = 0; j < nV; j++) {
if (matrix[i][k] + matrix[k][j] < matrix[i][j])
matrix[i][j] = matrix[i][k] + matrix[k][j];
}
}
}
printMatrix(matrix);
}

void printMatrix(int matrix[][nV]) {
for (int i = 0; i < nV; i++) {
for (int j = 0; j < nV; j++) {
if (matrix[i][j] == INF)
printf("%4s", "INF");
else
printf("%4d", matrix[i][j]);
}
printf("\n");
}
}

int main() {
int graph[nV][nV] = {{0, 3, INF, 5},
{2, 0, INF, 4},
{INF, 1, 0, INF},
{INF, INF, 2, 0}};
floydWarshall(graph);
}```

## Floyd Warshall Algorithm Complexity

Time Complexity

There are three loops. Each loop has constant complexities. Thus, 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 track down the shortest path is a directed graph
• To track down the transitive closure of directed graphs
• To discover the Inversion of genuine matrices
• For testing whether an undirected graph is bipartite

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