Minimum Spanning Tree

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

Minimum Spanning Tree

Example of MST

Problem: Laying Telephone Wire Central office

Wiring: Naïve Approach Central office Expensive!

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

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

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

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

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

Prim’s MST algorithm

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

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

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

Example II

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.

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

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

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

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

Minimum Spanning Tree

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