Let be a weighted digraph with n vertices. Design a variation of Floyd-Warshall's algorithm for computing the lengths of the shortest paths from each vertex to every other vertex in O(n3) time.

Suppose we are given a directed graph with n vertices, and let M be the n× n adjacency matrix corresponding to .

a. Let the product of M with itself (M2) be defined, for 1 ≤ i, j ≤ n, as follows:M2(i, j) =M(i, 1)⊙M(l,j)⊕…⊕M(i,n)⊙M(n,j),where “w” is the Boolean or operator and “a” is Boolean and. Given this definition, what does M2(i, j) = 1 imply about the vertices i and j? What if M2(i, j) = 0?

b.Suppose M4 is the product of M2 with itself. What do the entries of M4 signify? How about the entries of M5 = (M4) (M) ? In , what information is contained in the matrix Mp?

c.

Now suppose that is weighted and assume the following:1:for 1 ≤ i ≤ n,M(i,i)=0.

2:for 1 ≤ i,j ≤ n, M(i, j) = weight(i, j) if (i, j) is in E.

3:for 1 ≤ i, j ≤ n, M(i, j) = ∞ if (i, j) is not in E.Also, let M2 be defined, for 1 ≤ i,j ≤ n, as follows:

M2(i, j) = min{M(i, 1) +M(1,j),… ,M(i,n) +M(n,j)}.

If M2(i, j) = k, what may we conclude about the relationship between vertices i and j

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