Lecture10 - Minimum Spanning Tree CSE 421 Algorithms...

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1 CSE 421 Algorithms Richard Anderson Lecture 10 Minimum Spanning Trees Minimum Spanning Tree a b c s e g f 9 2 13 6 4 11 5 7 20 14 t u v 15 10 1 8 12 16 22 17 3 Undirected Graph G=(V,E) with edge weights Greedy Algorithms for Minimum Spanning Tree •[ P r im ] Extend a tree by including the cheapest out going edge • [Kruskal] Add the cheapest edge that joins disjoint components • [ReverseDelete] Delete the most expensive edge that does not disconnect the graph 4 11 5 7 20 8 22 a b c d e Why do the greedy algorithms work? • For simplicity, assume all edge costs are distinct Edge inclusion lemma • Let S be a subset of V, and suppose e = (u, v) is the minimum cost edge of E, with u in S and v in V-S • e is in every minimum spanning tree of G – Or equivalently, if e is not in T, then T is not a minimum spanning tree SV - S e Proof • Suppose T is a spanning tree that does not contain e • Add e to T, this creates a cycle • The cycle must have some edge e 1 = (u 1
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Lecture10 - Minimum Spanning Tree CSE 421 Algorithms...

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