The idea and perfection of the Warshall algorithm

The Warshall-Floyd algorithm is based essentially on the idea: to use a link between a problem and its simpler, not a lesser, version. Warshall and Floyd published their algorithms without mentioning dynamic programming. However, algorithms certainly have a dynamic programming flavor and have come to be seen as an application of this technique.

ALGORITHM Warshall(A[1..n, 1..n])
    //ImplementsWarshall’s algorithm for computing the transitive closure
    //Input: The adjacency matrix A of a digraph with n vertices
    //Output: The transitive closure of the digraph
    R(0) ←A
    for k←1 to n do
        for i ←1 to n do
            for j ←1 to n do
                R(k)[i, j ]←R(k−1)[i, j ] or (R(k−1)[i, k] and R(k−1)[k, j])
    return R(n)

      

        
        
        
      

    

We can speed up the above implementation of Warshalls algorithm for some inputs by restructuring its innermost loop

My question above the text is as follows

• What does the author mean by the idea: "to use the link between the problem and its simpler, not a smaller version." Please elobaorate.

• How can we improve the speed like the author mentioned in the above implementation.