Minimum Spanning Trees

Subgraph

• A graph G is a subgraph of graph H if – The vertices of G are a subset of the

vertices of H, and– The edges of G are a subset of the

edges of H.

Spanning Tree

• A spanning tree of a graph G is a subgraph of G that is a tree and includes all the vertices of G.

• Every connected graph has a spanning tree– Start with G. If G is a tree then it is a ST of G.

If not, G has a cycle. Remove any edge of the cycle. The result is a connected subgraph of G. Repeat. Process must terminate since the number of edges is finite.

Spanning Trees

Spanning Trees

a spanning tree

Spanning Trees

another spanning tree(can have many)

Weighted Graphs

• A weighted graph is a graph with a mapping of edges to R+ assigning a weight to each edge

• A minimum spanning tree of a weighted graph is a spanning tree of minimum total weight (sum of the weights of the edges in the tree)

Minimum Spanning Trees

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1. Choose any vertex

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2. Choose minimum weight edge connected to that vertex

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3. Choose minimum weight edge connected to chosen vertices

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4. Choose minimum weight edge connected to chosen vertices

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5. Choose minimum weight edge connected to chosen vertices without creating a cycle

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6. Choose minimum weight edge connected to chosen vertices without creating a cycle

Prim’s algorithm• To find a MST of a connected weighted

graph G, we build a tree one edge at a time– Start with T being any node of G– Pick a minimum weight edge that does not

create a cycle and that connects a node in T to a node not in T. Add that edge to T.

– Repeat until all nodes of G have been added to T.

• A “Greedy algorithm”

Proof that Prim’s algorithm works

• Invariant: At every stage, T is a subgraph of a MST

• True initially since T is just a single node• The algorithm runs until all nodes have

been added to T, so in the end the invariant says that T is a MST.

• So we just need to show that each stage of the algorithm preserves the invariant.

Proof that the algorithm maintains the invariant

• Proof by contradiction. Suppose there is a stage at which the invariant is true before and false after.

• Let e be the edge added when the invariant is first violated. Suppose e joins node v in T to node w not in T.

• So T is a subgraph of a MST but T+e is not.

• Let T’ be a MST of which T is a subgraph. • Edge e is not in T’.

Proof that the algorithm maintains the invariant

• Since T’ is a spanning tree, there is a path p from v to w in T’. Path p does not go through e since e is not part of T’. So p+e is a cycle consisting of two different paths from u to v.

• Path p must include an edge f which has one end in T and the other not in T.

• Edge f must have weight ≥ the weight of e, otherwise the algorithm would have chosen to add edge f to T instead of e.

• Removing f from T’ and adding e results in a spanning tree of weight ≤ the weight of T’ and including T as a subgraph, contradicting the assumption that T+e was not a subgraph of a MST. ✔

Finis