DAST 2005 Tirgul 14 (and more) sample questions. DAST 2005 (reminder?) Kruskal’s MST Algorithm.

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Presentation transcript:

DAST 2005 Tirgul 14 (and more) sample questions

DAST 2005 (reminder?) Kruskal’s MST Algorithm

DAST 2005 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 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.