Use Warshall's algorithm to compute the transitive closure of the graph with the following adjacency matrix: 0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 Choose the correct values for the unknown variables after each of the four steps for the algorithm. After the first step: 0 0 1 1 1 a1 b1 1 0 0 0 0 c1 1 1 0 [ Select ] After the second step: 0 0 1 1 1 a2 b2 1 0 0 0 0 c2 1 1 1 [ Select ] After the third step: 0 0 1 1 1 a3 b3 1 0 0 0 0 c3 1 1 1 [ Select ] After the fourth and final step: 1 1 1 1 1 a4 b4 1 0 0 0 0 c4 1 1 1 [ Select ]
Question
Use Warshall's algorithm to compute the transitive closure of the graph with the following adjacency matrix:
0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0
Choose the correct values for the unknown variables after each of the four steps for the algorithm.
After the first step:
0 0 1 1 1 a1 b1 1 0 0 0 0 c1 1 1 0
[ Select ]
After the second step:
0 0 1 1 1 a2 b2 1 0 0 0 0 c2 1 1 1
[ Select ]
After the third step:
0 0 1 1 1 a3 b3 1 0 0 0 0 c3 1 1 1
[ Select ]
After the fourth and final step:
1 1 1 1 1 a4 b4 1 0 0 0 0 c4 1 1 1
[ Select ]
Solution
Warshall's algorithm is used to find the transitive closure of a graph. It is an iterative method that updates the adjacency matrix of the graph. The algorithm works by considering each vertex and updating paths between all pairs of vertices in the graph.
Let's start with the given adjacency matrix:
0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0
After the first step, we consider the first vertex. There are no changes in the adjacency matrix because there are no vertices that can be reached from the first vertex that cannot already be reached directly. So, a1 = 0, b1 = 0, and c1 = 0.
After the second step, we consider the second vertex. The second vertex can reach the fourth vertex, so any vertex that can reach the second vertex can also reach the fourth vertex. This means that the first vertex can now reach the fourth vertex, so a2 = 1. There are no other changes, so b2 = 0 and c2 = 0.
After the third step, we consider the third vertex. The third vertex can reach the first and fourth vertices, so any vertex that can reach the third vertex can also reach the first and fourth vertices. However, no vertices can reach the third vertex, so there are no changes. Therefore, a3 = 1, b3 = 0, and c3 = 0.
After the fourth and final step, we consider the fourth vertex. The fourth vertex can reach the second and third vertices, so any vertex that can reach the fourth vertex can also reach the second and third vertices. This means that the first vertex can now reach the second and third vertices, so a4 = 1 and b4 = 1. The fourth vertex can also reach itself, so c4 = 1.
So, the final transitive closure of the graph is:
1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1
Similar Questions
Dynamic Programming: Warshall’s algorithmApply Warshall’s algorithm to find the transitive closure of the digraph defined by thefollowing adjacency matrix:[[0,0,0,1],[0,0,1,0],[0,0,0,1],[0,0,0,0]]
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The Warshall algorithm uses ____ nested loops to compute the transitive closure.
Which of the following pseudocode correctly represents the initialization step of the Warshall algorithm for finding the transitive closure of a graph?Marks : 1Negative Marks : 0Answer hereProcedure WarshallInit(Graph G): For each vertex v in G: For each vertex u in G: If u is adjacent to v: Set distance[u][v] to 1 Else: Set distance[u][v] to 0Procedure WarshallInit(Graph G): For each vertex u in G: For each vertex v in G: If u is adjacent to v: Set distance[u][v] to 1 Else: Set distance[u][v] to infinityProcedure WarshallInit(Graph G): For each vertex v in G: For each vertex u in G: If u is adjacent to v: Set distance[v][u] to 0 Else: Set distance[v][u] to 0Procedure WarshallInit(Graph G): For each vertex u in G: For each vertex v in G: If u is adjacent to v: Set distance[v][u] to 1 Else: Set distance[v][u] to infinity
What is the formula to compute the transitive closure of a graph? Options tij(k) = tij(k-1) AND (tik(k-1) OR tkj(k-1)) tij(k) = tij(k-1) OR (tik(k-1) AND tkj(k-1)) tij(k) = tij(k-1) AND (tik(k-1) AND tkj(k-1)) tij(k) = tij(k-1) OR (tik(k-1) OR tkj(k-1))
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