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Tirgul 14

Tirgul 14. (and more) sample questions. (reminder?) Kruskal’s MST Algorithm. Q. Suppose that all edges weights in a graph are integers in the range of 1 to | V |. How fast can you make Kruskal’s algorithm run?

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Tirgul 14

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  1. Tirgul 14 • (and more) • sample questions DAST 2005

  2. (reminder?)Kruskal’s MST Algorithm DAST 2005

  3. Q. Suppose that all edges weights in a graph are integers in the range of 1 to |V|. How fast can you make Kruskal’s algorithm run? A. Sorting |E| elements with values in the range of 1 to |V| takes O(V+E) (counting sort), find-set using an efficient union-find takes log*(E) (very fast) – total complexity of O(E log*(E)). (another reminder) DAST 2005

  4. Q. Let G be a connected undirected graph with the property that every edge has a different weight. Prove that there is only one minimum spanning tree for G. A. We assume there are two such trees and show there is a contradiction: exchange lemma: Let T and T’ be spanning trees in G(V,E). Given any there exists an edge such that is also a spanning tree proof (outline): adding e’ to T results in a cycle. Removing an edge e along this cycle restores the spanning tree Now suppose there are two MSTs to G, let e’ be the lightest edge in T’ -T, we will add e’ to T, but get a contradiction when trying to remove an edge from the resulting cycle. DAST 2005

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