GATE CS 2015 Set 1 — Question 63
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Algorithms → Greedy Algorithms → Kruskal's MST
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Question
The graph shown below has 8 edges with distinct integer edge weights. The minimum spanning tree (MST) is of weight 36 and contains the edges: {(A, C), (B, C), (B, E), (E, F), (D, F)}. The edge weights of only those edges which are in the MST are given in the figure shown below. The minimum possible sum of weights of all 8 edges of this graph is ___________.
Correct answer
69 to 69
Solution
The graph has 8 edges. The MST has 5 edges with weights: . Sum = 36.
The 3 non-MST edges are , , and .
For the MST to be valid, every non-MST edge must create a cycle where is the strictly heaviest edge (since weights are distinct).
Total sum = .
The 3 non-MST edges are , , and .
For the MST to be valid, every non-MST edge must create a cycle where is the strictly heaviest edge (since weights are distinct).
1.Edge (A,B): Forms cycle A-C-B. Path in MST is A-C (9) - C-B (2). Max weight is 9. Thus, . Minimum integer is 10.
2.Edge (C,D): Forms cycle C-B-E-F-D. Path in MST is C-B (2) - B-E (15) - E-F (4) - F-D (6). Max weight is 15. Thus, . Minimum integer is 16.
3.Edge (D,E): Forms cycle D-F-E. Path in MST is D-F (6) - F-E (4). Max weight is 6. Thus, . Minimum integer is 7.
We must ensure all weights are distinct. The existing weights are . The proposed new weights are . All are distinct and valid.Total sum = MST Weight + Total sum = .
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