ClassS04CS141/MinimumSpanningTreeExercises

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Minimum spanning tree exercises

Review the proof of correctness for Kruskal's algorithm: MinimumSpanningTreesByKruskals .


(i) Prim's algorithm:

 0. Input: weighted graph G=(V,E).  Output: MST S.
 1. Choose any "start" vertex s and let C = {s}.
 2. Let S={}.
 3. Until C == V do
 4.    From all the edges leaving C, choose a minimum-weight one (u,w).
 5.    Add w to C.
 6.    Add (u,w) to S.
 7. Return S.

claim: Prim's algorithm is correct. That is, given any connected weighted graph, the algorithm returns a minimum spanning tree.

exercise: prove or disprove the claim.

stepping stone: prove that the first edge chosen by the algorithm is in some MST


(ii) Another algorithm:

 1. While the graph contains cycles do:
 2.   Find a cycle C. 
 3.   Find a maximum-weight edge e on C.
 4.   Delete e from the graph.
 5. Return the tree formed by the remaining edges.

claim: This algorithm is correct. That is, given any connected weighted graph, the algorithm returns a minimum spanning tree.

exercise: prove or disprove the claim.

stepping stone: prove that there is an MST that does not contain the first deleted edge


(iii) A problem:

claim: In any connected, weighted graph with distinct edge weights (no two edges have the same weight), there is exactly one minimum spanning tree.

exercise: prove or disprove the claim.

stepping stone: prove that if the min-wt edge is unique, then every MST contains it


References


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Edited May 28, 2004 7:12 pm by Neal (diff)
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