Minimal Spanning Trees - University of … · 2 Finding Minimal Spanning Trees 2.1 Prim’s...

Minimal Spanning TreesSection 9.4

Prof. Nathan Wodarz

Math 209 - Fall 2008

Contents1 Minimal Spanning Trees 2

1.1 Minimal Spanning Trees . . . . . . . . . . . . . . . . . . . . . . 2

2 Finding Minimal Spanning Trees 22.1 Prim’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Kruskal’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Minimal Spanning Trees

1.1 Minimal Spanning TreesMinimal Spanning Trees

• In a weighted graph, what is the cheapest way to connect all vertices?

• Must:

– Contain all vertices

– Be connected

– Have a unique simple path from any vertex to any other

• This must be a spanning tree

• Want sum of weights to be a minimum: a minimal spanning tree

2 Finding Minimal Spanning Trees

2.1 Prim’s AlgorithmPrim’s Algorithm

• Idea: Keep adding edges until minimal spanning tree is obtained

• Adds minimum-weight edge at each iteration

• To start:

– Weighted graph with n vertices

– A starting vertex s

• Repeat n − 1 times (once for each edge to add)

– List all edges with one end in the tree, and one end not in the tree

– Add the cheapest of these

• Another example of a greedy algorithm

• Prim’s Algorithm is correct - output tree is a minimal spanning tree

• This version examines Θ(n3) edges in worst case

• Possible to implement so that algorithm examinesΘ(n2) edges in worst case

Prim’s AlgorithmExample. Use Prim’s Algorithm to find a minimal spanning tree, starting with thevertex a

• • •

2.2 Kruskal’s AlgorithmKruskal’s Algorithm

• Idea: Keep adding edges until minimal spanning tree is obtained

• Adds minimum-weight edge at each iteration

• To start:

– Weighted graph with n vertices

– Include all vertices in beginning tree

• Repeat n − 1 times (once for each edge)

– List all edges that won’t complete a cycle

– Add the cheapest of these

• Yet another example of a greedy algorithm

• Kruskal’s Algorithm is also correct - output tree is a minimal spanning tree

Kruskal’s AlgorithmExample. Use Kruskal’s Algorithm to find a minimal spanning tree

• • •

SummarySummary

You should be able to:

• Be able to identify a minimal spanning tree

• Use Kruskal’s and Prim’s algorithms to find minimal spanning trees

Minimal Spanning Trees - University of … · 2 Finding Minimal Spanning Trees 2.1 Prim’s...

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