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Unformatted text preview: CS 310 Unit 19 Minimum Spanning Trees Furman Haddix Ph.D. Assistant Professor Minnesota State University, Mankato Spring 2008 Unit 19 Minimum Spanning Trees Objectives • Minimum Spanning Tree Problem • Minimum Spanning Tree Examples (After Kruskal) • Kruskal’s Algorithm and Analysis • Minimum Spanning Tree Example (After Jarnik) • Jarnik’s Algorithm and Analysis • Text Chapter 23.1, 23.2 Minimum Spanning Tree Problem • A spanning tree is a tree which connects n vertices with n1 edges. • In a graph with unweighted or equal weight edges, any spanning tree is a minimum spanning tree. • In a graph with weighted edges a minimum spanning tree is the spanning tree with the minimum weight for the included edges totalWeight(MST) = Σ (u, v) ∈ MST w(u, v) • If an edge is added to a MST, a cycle will result. If an edge of the MST is replaced with another edge totalWeight(T) ≥ totalWeight(MST) A Greedy Minimum Spanning Tree Algorithm • Sort edges in order of decreasing weight • Assign each vertex to be its own leader • Consider edges one by one in order • If leader of one vertex is different from leader of other, add vertex to the spanning tree • Else discard edge Minimum Spanning Tree Example 1 • Initially, a forest of trees • Sort edges by weight • Ties don’t matter 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 3 4 1 5 6 7 8 3 11 12 9 13 14 15 10 11 15 17 7 edge weight node ID leader Minimum Spanning Tree Example 1 • Consider each edge, lowest weight edge first • If edge does not connect to vertices with same leader, include in minimum spanning tree 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 3 4 1 5 6 7 8 3 11 12 9 13 14 15 10 11 15 17 7 edge weight node ID leader Minimum Spanning Tree Example 1 • Consider each edge, lowest weight edge first • If edge does not connect to vertices with same leader, include in minimum spanning tree 1 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 3 4 1 5 6 7 8 3 11 12 9 13 14 15 10 11 15 17 7 edge weight node ID leader Minimum Spanning Tree Example 1 • Consider each edge, lowest weight edge first • If edge does not connect to vertices with same leader, include in minimum spanning tree 1 2 3 4 4 5 5 6 6 7 7 8 8 9 9 3 4 1 5 6 7 8 3 11 12 9 13 14 15 10 11 15 17 7 edge weight node ID leader Minimum Spanning Tree Example 1 • Consider each edge, lowest weight edge first • If edge does not connect to vertices with same leader, include in minimum spanning tree 1 2 3 4 5 5 6 6 7 7 8 8 9 9 3 4 1 5 6 7 8 3 11 12 9 13 14 15 10 11 15 17 7 edge weight node ID leader Minimum Spanning Tree Example 1 • Consider each edge, lowest weight edge first • If edge does not connect to vertices with same leader,...
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 Spring '08
 FurmanHaddix
 Graph Theory, LG, Kruskal, Spanning Tree Example

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