GATE CS 2016 Set 1 — Question 48
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Algorithms → Graph Algorithms → Dijkstra's Algorithm
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Question
Consider the weighted undirected graph with 4 vertices, where the weight of edge is given by the entry in the matrix .The largest possible integer value of , for which at least one shortest path between some pair of vertices will contain the edge with weight is _________.
Correct answer
12 to 12
Solution
Let the vertices be 1, 2, 3, 4. The edge weights are:
We want to find the largest integer such that the edge is part of a shortest path between some pair of vertices. The most likely candidate pair is itself.Calculate the shortest path between 3 and 4 using only the other edges:
For the direct edge with weight to be a shortest path, we must have .If , the edge is a shortest path (length 12). If , the path is strictly shorter, so would not be used.Check if can be larger for other pairs:
We want to find the largest integer such that the edge is part of a shortest path between some pair of vertices. The most likely candidate pair is itself.Calculate the shortest path between 3 and 4 using only the other edges:
- Path :
- Path :
- Path :
For the direct edge with weight to be a shortest path, we must have .If , the edge is a shortest path (length 12). If , the path is strictly shorter, so would not be used.Check if can be larger for other pairs:
- For pair , shortest path is via 2 (length ). Via : . Requires .
- For pair , shortest path is via 1 (length ). Via : . Requires .
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