Define transitive closure. Write Warshall's algorithm to compute transitive closure. Mention itstime efficiency
Question
Define transitive closure. Write Warshall's algorithm to compute transitive closure. Mention itstime efficiency
Solution
Transitive Closure: In graph theory, the transitive closure of a graph is a graph which contains an edge (i, j) if there is a path from i to j in the original graph. In other words, if a node j is reachable from node i, then the edge (i, j) is in the transitive closure of the graph.
Warshall's Algorithm: Warshall's algorithm is a method to compute the transitive closure of a graph. The algorithm is named after Stephen Warshall, who first described it in 1962. Here is a simple implementation of Warshall's algorithm:
- Start with an adjacency matrix A of the graph, where A[i][j] is 1 if there is an edge from i to j, and 0 otherwise.
- For each pair of vertices i and j, check for every other vertex k if there is a path from i to k and a path from k to j. If such a vertex k exists, then there is a path from i to j, so set A[i][j] to 1.
- Repeat step 2 for all pairs of vertices until no more updates are made to the matrix A.
Here is a pseudo-code representation of the algorithm:
function Warshall(A)
let n = number of vertices in A
for k from 1 to n
for i from 1 to n
for j from 1 to n
A[i][j] = A[i][j] or (A[i][k] and A[k][j])
return A
Time Efficiency: The time complexity of Warshall's algorithm is O(n^3), where n is the number of vertices in the graph. This is because the algorithm contains three nested loops, each of which can iterate up to n times.
Similar Questions
The Warshall algorithm uses ____ nested loops to compute the transitive closure.
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 ]
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
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]]
If a graph has 'n' vertices, what is the maximum size of the matrix required to represent its transitive closure using the Warshall algorithm?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.