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Which algorithm can be used to find the shortest path between all pairs of vertices in a directed graph?
- a)Dijkstra's algorithm
- b)Bellman–Ford algorithm
- c)Floyd Warshall algorithm
- d)Kruskal's algorithm
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Which algorithm can be used to find the shortest path between all pair...
The Floyd Warshall algorithm can find the shortest path between all pairs of vertices in a directed graph.
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Which algorithm can be used to find the shortest path between all pair...
Introduction
In graph theory, finding the shortest path between all pairs of vertices is a crucial task. The Floyd-Warshall algorithm is particularly effective for this purpose in directed graphs.
Floyd-Warshall Algorithm Overview
- It is a dynamic programming algorithm that computes shortest paths between all pairs of vertices in a graph.
- It can handle both positive and negative edge weights, but it does not work with graphs containing negative cycles.
Algorithm Steps
1. Initialization:
- Create a distance matrix where each entry (i, j) holds the weight of the edge from vertex i to vertex j. If no edge exists, set it to infinity. The diagonal entries are set to zero.
2. Updating Distances:
- For each vertex k, update the distance matrix. For every pair of vertices (i, j), check if the path from i to j through k is shorter than the current known distance. If so, update it.
3. Complexity:
- The time complexity is O(V^3), where V is the number of vertices, making it suitable for smaller graphs.
Advantages of Floyd-Warshall
- Simplicity: The algorithm is straightforward to implement.
- Comprehensive: It finds shortest paths for all pairs, rather than just a single source.
Conclusion
The Floyd-Warshall algorithm is the correct choice for finding the shortest paths between all pairs of vertices in a directed graph due to its effectiveness and ability to handle various edge weights.
In graph theory, finding the shortest path between all pairs of vertices is a crucial task. The Floyd-Warshall algorithm is particularly effective for this purpose in directed graphs.
Floyd-Warshall Algorithm Overview
- It is a dynamic programming algorithm that computes shortest paths between all pairs of vertices in a graph.
- It can handle both positive and negative edge weights, but it does not work with graphs containing negative cycles.
Algorithm Steps
1. Initialization:
- Create a distance matrix where each entry (i, j) holds the weight of the edge from vertex i to vertex j. If no edge exists, set it to infinity. The diagonal entries are set to zero.
2. Updating Distances:
- For each vertex k, update the distance matrix. For every pair of vertices (i, j), check if the path from i to j through k is shorter than the current known distance. If so, update it.
3. Complexity:
- The time complexity is O(V^3), where V is the number of vertices, making it suitable for smaller graphs.
Advantages of Floyd-Warshall
- Simplicity: The algorithm is straightforward to implement.
- Comprehensive: It finds shortest paths for all pairs, rather than just a single source.
Conclusion
The Floyd-Warshall algorithm is the correct choice for finding the shortest paths between all pairs of vertices in a directed graph due to its effectiveness and ability to handle various edge weights.
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Which algorithm can be used to find the shortest path between all pairs of vertices in a directed graph?a)Dijkstras algorithmb)Bellman–Ford algorithmc)Floyd Warshall algorithmd)Kruskals algorithmCorrect answer is option 'C'. Can you explain this answer?
Question Description
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