Floyd-Warshall algorithm utilizes __________ to solve the all-pairs shortest paths problem on a directed graph in __________ time.
Question
Floyd-Warshall algorithm utilizes __________ to solve the all-pairs shortest paths problem on a directed graph in __________ time.
Solution
The Floyd-Warshall algorithm utilizes dynamic programming to solve the all-pairs shortest paths problem on a directed graph in O(n^3) time.
Here are the steps:
-
Initialization: Create a matrix, call it dist[][], such that dist[i][j] is initialized as the weight of the edge between i and j for every edge in the graph. If there is no edge between i and j, then dist[i][j] is initialized as infinity. Also, initialize dist[i][i] as 0 for every vertex i.
-
Iteration: The algorithm then performs three nested loops, each one iterating through the vertices of the graph. The outermost loop goes through each vertex k, the middle loop goes through each vertex i, and the innermost loop goes through each vertex j.
-
Relaxation: In each iteration of the innermost loop, the algorithm checks if the shortest path from vertex i to vertex j can be improved by going through vertex k, i.e., if dist[i][j] > dist[i][k] + dist[k][j]. If this is the case, then it updates dist[i][j] to dist[i][k] + dist[k][j].
-
Result: After all iterations, dist[i][j] for all pairs of vertices i and j will be the shortest possible distance between them in the graph.
This algorithm runs in O(n^3) time because it performs a constant amount of work for each pair of vertices for each vertex in the graph, and there are n^3 such combinations if n is the number of vertices in the graph.
Similar Questions
Write and explain the Floyd Warshall algorithm to find the all pair shortest path.
Floyd Warshall Algorithm used to solve the shortest path problem has a time complexity of __________
The shortest path problem can be solved using:A. Depth-first searchB. Breadth-first searchC. Dijkstra's algorithmD. Floyd-Warshall algorithm
What is the best time complexity we can achieve to precompute all-pairs shortest paths in a weighted graph?
What procedure is being followed in Floyd Warshall Algorithm?Marks : 1Negative Marks : 0Answer hereTop downBottom upBig bangSandwich
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.