Algorithm 2 The improved 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:     $t=dis\left(i,k\right)$

9:      for $j=1$ to $i$ do

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

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

12:      end for

13:   end for

14: end for

15: return $dis$;