GATE CS 2025 Set 2 — Question 37
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Question
Let be an edge-weighted undirected graph with positive edge weights. Suppose a positive constant is added to the weight of every edge.Which ONE of the following statements is TRUE about the minimum spanning trees (MSTs) and shortest paths (SPs) in before and after the edge weight update?
Correct answer
(C) Every MST remains an MST, and SPs need not remain SPs.
Solution
1.Minimum Spanning Tree (MST): Algorithms like Kruskal's and Prim's determine the MST based on the relative order of edge weights. If a constant is added to every edge weight, the inequality becomes . The relative order of edges remains unchanged. Therefore, the set of edges forming the MST remains the same.
2.Shortest Path (SP): The shortest path depends on the sum of edge weights. Let be a path with edges and total weight , and be a path with edges and total weight . Suppose is the shortest path initially, so . After adding to each edge, the new weights are and . If (the shortest path has more edges), it is possible that for a large enough , , making the new shortest path. Thus, shortest paths can change.
Conclusion: Every MST remains an MST, but SPs need not remain SPs.Continue learning with Success Tracker
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