FloydWarshall algorithm
The story behind this post
Recently I’ve received +10
karma on StackOverflow. I was curious for what question or answer and clicked to check this. It appeared
to be a sevenyearold answer about a FloydWarshall algorithm. I was surprised both of my bad English back those days and the very small value the answer had. So I’ve revised it and here it is – the brandnew version!
The definitions
Let us have a graph, described by matrix D
, where D[i][j]
is the length of the edge (i > j)
(from graph’s vertex with index i
to the vertex with index j
).
Matrix D
has the size of N * N
, where N
is a total number of vertices in a graph because we can reach the maximum of paths by connecting each graph’s vertex to each other.
Also, we’ll need matrix R
, where we will store shortest paths (R[i][j]
contains the index of a next vertex in the shortest path, starting at vertex i
and ending at vertex j
).
Matrix R
has the same size as D
.
The FloydWarshall algorithm performs these steps:

initialize the matrix of all the paths between any two pairs of vertices in a graph with the edge’s end vertex (this is important since this value will be used for path reconstruction)

for each pair of connected vertices (read: for each edge
(u > v)
),u
andv
, find the vertex, which forms shortest path between them: if the vertexk
defines two valid edges(u > k)
and(k > v)
(if they are present in the graph), which are together shorter than path(u > v)
, then assume the shortest path betweenu
andv
lies throughk
; set the shortest pivot point in matrixR
for edge(u > v)
to be the corresponding pivot point for edge(u > k)
But how do we read the matrix D
?
Inputs
Let us have a graph:
We first create a twodimensional array of size 4
(since there are exactly 4
vertices in our graph).
We initialize its main diagonal (the items, whose indices are equal, for ex. G[0, 0]
, G[1, 1]
, etc.) with zeros, because
the shortest path from vertex to itself has the length 0
and the other elements with a very large number (to indicate there is no edge or an infinitely long edge between them). The defined elements, corresponding to graph’s edges, we fill with edges’ lengths:
int N = 4;
int[,] D = new int[N, N];
for (int i = 0; i < N; i++) {
for (int t = 0; t < N; t++) {
if (i == t) {
D[i, t] = 0;
} else {
D[i, t] = 9999;
}
}
}
D[0, 2] = 1;
D[1, 0] = 7;
D[1, 2] = 6;
D[2, 3] = 5;
D[3, 1] = 2;
The algorithm itself
Now that we are on a same page with definitions, algorithm can be implemented like this:
// Initialise the routes matrix R
for (int i = 0; i < N; i++) {
for (int t = 0; t < N; t++) {
R[i][t] = t;
}
}
// FloydWarshall algorithm:
for (int k = 0; k < N; k++) {
for (int u = 0; u < N; u++) {
for (int v = 0; v < N; v++) {
if (D[u, v] > D[u, k] + D[k, v]) {
D[u, v] = D[u, k] + D[k, v];
R[u, v] = R[u, k];
}
}
}
}
After the algorithm run, the matrix R
will be filled with vertices’ indices, describing shortest paths between them.
Path reconstruction
In order to reconstruct the path from vertex u
to vertex v
, you need follow the elements of matrix R
:
List<Int32> Path = new List<Int32>();
while (start != end)
{
Path.Add(start);
start = R[start, end];
}
Path.Add(end);
Summary
The whole code could be wrapped in a couple of methods:
using System;
using System.Collections.Generic;
public class FloydWarshallPathFinder {
private int N;
private int[,] D;
private int[,] R;
public FloydWarshallPathFinder(int NumberOfVertices, int[,] EdgesLengths) {
N = NumberOfVertices;
D = EdgesLengths;
R = null;
}
public int[,] FindAllPaths() {
R = new int[N, N];
for (int i = 0; i < N; i++)
{
for (int t = 0; t < N; t++)
{
R[i, t] = t;
}
}
for (int k = 0; k < N; k++)
{
for (int v = 0; v < N; v++)
{
for (int u = 0; u < N; u++)
{
if (D[u, k] + D[k, v] < D[u, v])
{
D[u, v] = D[u, k] + D[k, v];
R[u, v] = R[u, k];
}
}
}
}
return R;
}
public List<Int32> FindShortestPath(int start, int end) {
if (R == null) {
FindAllPaths();
}
List<Int32> Path = new List<Int32>();
while (start != end)
{
Path.Add(start);
start = R[start, end];
}
Path.Add(end);
return Path;
}
}
public class MainClass
{
public static void Main()
{
int N = 4;
int[,] D = new int[N, N];
for (int i = 0; i < N; i++) {
for (int t = 0; t < N; t++) {
if (i == t) {
D[i, t] = 0;
} else {
D[i, t] = 9999;
}
}
}
D[0, 2] = 1;
D[1, 0] = 7;
D[1, 2] = 6;
D[2, 3] = 5;
D[3, 1] = 2;
FloydWarshallPathFinder pathFinder = new FloydWarshallPathFinder(N, D);
int start = 0;
int end = 1;
Console.WriteLine("Path: {0}", String.Join(" > ", pathFinder.FindShortestPath(start, end).ToArray()));
}
}
You can read ‘bout this algorithm on wikipedia and get some data structures generated automatically here