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Consider a directed graph with six vertices. The initial adjacency matrix is given below. After applying the Warshall algorithm, how many entries in the resulting adjacency matrix will be '1'?Initial Adjacency Matrix:  | 1 2 3 4 5 6----------------1 | 0 1 0 1 0 02 | 0 0 1 0 0 03 | 0 0 0 0 1 14 | 0 0 0 0 0 05 | 0 0 0 1 0 06 | 0 0 0 0 0 0

Question

Consider a directed graph with six vertices. The initial adjacency matrix is given below. After applying the Warshall algorithm, how many entries in the resulting adjacency matrix will be '1'?Initial Adjacency Matrix:  | 1 2 3 4 5 6----------------1 | 0 1 0 1 0 02 | 0 0 1 0 0 03 | 0 0 0 0 1 14 | 0 0 0 0 0 05 | 0 0 0 1 0 06 | 0 0 0 0 0 0

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Solution

The Warshall algorithm is used to find the transitive closure of a directed graph. The transitive closure of a graph is a matrix M such that M[i][j] is 1 if there is a path from i to j, and 0 otherwise.

The initial adjacency matrix is:

1 2 3 4 5 6
1 0 1 0 1 0 0
2 0 0 1 0 0 0
3 0 0 0 0 1 1
4 0 0 0 0 0 0
5 0 0 0 1 0 0
6 0 0 0 0 0 0

After applying the Warshall algorithm, we get the following matrix:

1 2 3 4 5 6
1 0 1 1 1 1 1
2 0 0 1 0 1 1
3 0 0 0 0 1 1
4 0 0 0 0 0 0
5 0 0 0 1 0 0
6 0 0 0 0 0 0

So, there are 7 entries in the resulting adjacency matrix that are '1'.

This problem has been solved

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1/3

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