Lecture notes for “Analysis of Algorithms”: Minimum Spanning Trees
نویسنده
چکیده
We present a general framework for obtaining efficient algorithms for computing minimum spanning trees. We use this framework to derive the classical algorithms of Prim, Kruskal and Bor̊uvka. We then describe the randomized linear-time algorithm of Karger, Klein and Tarjan. The algorithm of Karger, Klein and Tarjan uses deterministic linear-time implementations of a verification algorithm of Komlós. 1 Minimum Spanning Trees Let G = (V,E,w) be a weighted undirected graph, where w : E → R is a weight (or cost) function defined on its edges. A subgraph T = (V,ET ) of G which is a tree is said to be a spanning tree of G. The weight of a spanning tree T = (V,ET ) of G is defined to be w(T ) = ∑ e∈ET w(e). In the Minimum Spanning Tree (MST) problem we are asked to find a spanning tree of minimum weight of a given connected input graph G = (V,E).
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We present a deterministic linear time algorithm for the Tree Path Maxima problem. As a corollary, we obtain a deterministic linear time algorithm for checking whether a given spanning tree T of a weighted graph is a minimum spanning tree.
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