Algorithm 1 The Floyd-Warshall’s shortest path algorithm

Input:

The initial weights of the graph: $w\left(i,j\right)$

The number of vertices: $n$

Output:

The final matrix of the shortest paths: $dis\left(i,j\right)$

1: for $i=1$ to $n$ do

2:   for $j=1$ to $n$ do

3:     $dis\left(i,j\right)=w\left(i,j\right)$;

4:   end for

5: end for

6: for $k=1$ to $n$ do

7:   for $i=1$ to $n$ do

8:     for $j=1$ to $n$ do

9:      $dis\left(i,j\right)=\min \left(dis\left(i,j\right),dis\left(i,k\right)+dis\left(k,j\right)\right)$;

10:     end for

11:   end for

12: end for

13: return $dis$;