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Minimum Spanning Tree Dr. Bernard Chen Ph.D. University of Central Arkansas Fall 2008

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Minimum Spanning Tree

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Example of MST

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Problem: Laying Telephone Wire Central office

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Wiring: Naïve Approach Central office Expensive!

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Wiring: Better Approach Central office Minimize the total length of wire connecting the customers

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Growing an MST: general idea GENERIC-MST(G,w) 1. A {} 2. while A does not form a spanning tree 3. do find an edge (u,v) that is safe for A 4. A A U {(u,v)} 5. return A

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Tricky part How do you find a safe edge? This safe edge is part of the minimum spanning tree

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Algorithms for MST Prim’s Grow a MST by adding a single edge at a time Kruskal’s Choose a smallest edge and add it to the forest If an edge is formed a cycle, it is rejected

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Prim’s greedy algorithm Start from some (any) vertex. Build up spanning tree T, one vertex at a time. At each step, add to T the lowest-weight edge in G that does not create a cycle. Stop when all vertices in G are touched

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Prim’s MST algorithm

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Example A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge Pick a root A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 = in heap

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Min Edge = 1 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 2 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 6 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Example II

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Kruskal’s Algorithm Choose the smallest edge and add it to a forest Keep connecting components until all vertices connected If an edge would form a cycle, it is rejected.

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Example A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 1 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 2 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Now have 2 disjoint components: ABFG and CH

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Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Two components now merged into one.

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Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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Min Edge = 5 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Rejected due to a cycle BFGB

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Min Edge = 6 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

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