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Unformatted text preview: 65 f 30 d 29 c 25 a, b 75% = A63% = B42% = Ce. Why does a greedy strategy work here? Suppose we’ve already added some edges, and they are a part of an MST, which edges can we add next? Cut property: Suppose edges X < E are part of some MST T. pick any subset of nodes S in V such that X has no edges between S and V – S. Let e be the lightest edge across cut (S, V – S). Then X U {e} is part of some MST T’ e.i. If e is in T: we’re done. Otherwise, we will show that there iiis a different MST T’ that contains e e.ii. Ad e to T T + e: contains a cycle, the cycle must cross the cut with another edge e’.look at e.iii. Look at T’ = T + e – e’ T is connected (removing a cycle edge cannot disconnect a graph) T’ has V – 1 edges Therefore, T’ is a tree Weight(T’) = weight(T) + w e – w e’ <= weight(T) Weight(T’) = weight(T) and T’ is an MST. e.iv....
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 Spring '08
 staff
 Algorithms, Graph Theory, Greedy algorithm, node Sort edges, Makeset

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