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Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE?
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change
Q: Shortest path between any pair of vertices does not change
- a)P only
- b)Q only
- c)Neither P nor Q
- d)Both P and Q
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let G be a weighted connected undirected graph with distinct positive ...
The shortest path may change. The reason is, there may be different number of edges in different paths from s to t. For example, let shortest path be of weight 15 and has 5 edges. Let there be another path with 2 edges and total weight 25. The weight of the shortest path is increased by 5*10 and becomes 15 + 50. Weight of the other path is increased by 2*10 and becomes 25 + 20. So the shortest path changes to the other path with weight as 45. The Minimum Spanning Tree doesn't change. Remember the Kruskal's algorithm where we sort the edges first. IF we increase all weights, then order of edges won't change.
Most Upvoted Answer
Let G be a weighted connected undirected graph with distinct positive ...
Explanation:
Let's assume that every edge weight is increased by the same value 'x'. Now we will analyze the impact of this increase on the minimum spanning tree and shortest path between any pair of vertices of the graph G.
P: Minimum spanning tree of G does not change
Minimum spanning tree (MST) is a tree that spans all the vertices of a connected undirected graph with the minimum possible total edge weight. Since the edge weights are increased by the same value 'x', the relative differences in their weights remain the same. This means that the edge with the minimum weight in the original graph G will still have the minimum weight in the modified graph. Therefore, the MST of the modified graph will have the same set of edges as the MST of the original graph. Hence, the statement P is TRUE.
Q: Shortest path between any pair of vertices does not change
Shortest path between any pair of vertices is the path with the minimum possible total edge weight. Since every edge weight is increased by the same value 'x', the relative differences in their weights remain the same. This means that the edge with the minimum weight in the original graph G will still have the minimum weight in the modified graph. Therefore, the shortest path between any pair of vertices in the modified graph will have the same set of edges as the shortest path in the original graph. Hence, the statement Q is also TRUE.
Therefore, the correct answer is option 'A' i.e., P only.
Let's assume that every edge weight is increased by the same value 'x'. Now we will analyze the impact of this increase on the minimum spanning tree and shortest path between any pair of vertices of the graph G.
P: Minimum spanning tree of G does not change
Minimum spanning tree (MST) is a tree that spans all the vertices of a connected undirected graph with the minimum possible total edge weight. Since the edge weights are increased by the same value 'x', the relative differences in their weights remain the same. This means that the edge with the minimum weight in the original graph G will still have the minimum weight in the modified graph. Therefore, the MST of the modified graph will have the same set of edges as the MST of the original graph. Hence, the statement P is TRUE.
Q: Shortest path between any pair of vertices does not change
Shortest path between any pair of vertices is the path with the minimum possible total edge weight. Since every edge weight is increased by the same value 'x', the relative differences in their weights remain the same. This means that the edge with the minimum weight in the original graph G will still have the minimum weight in the modified graph. Therefore, the shortest path between any pair of vertices in the modified graph will have the same set of edges as the shortest path in the original graph. Hence, the statement Q is also TRUE.
Therefore, the correct answer is option 'A' i.e., P only.
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Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE?P: Minimum spanning tree of G does not changeQ: Shortest path between any pair of vertices does not changea)P onlyb)Q onlyc)Neither P nor Qd)Both P and QCorrect answer is option 'A'. Can you explain this answer?
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