CS223 Advanced Data Structures and Algorithms 1

Minimum Spanning Tree Minimum Spanning Tree

Neil TangNeil Tang3/25/20103/25/2010

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Class OverviewClass Overview

The minimum spanning tree problem

Applications

Prim’s algorithm

Kruskal’s algorithm

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Minimum Spanning Tree ProblemMinimum Spanning Tree Problem

The cost of a tree: The sum of the weights of all links on the tree.

The Minimum Spanning Tree (MST) problem: Given a weighted undirected graph G, find a minimum cost tree connecting all the vertices on the graph.

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Minimum Spanning Tree ProblemMinimum Spanning Tree Problem

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ApplicationsApplications

Broadcasting problem in computer networks: Find the minimum cost route to send packages from a source node to all the other nodes in the network.

Multicasting problem in computer networks: Find the minimum cost route to send packages from a source node to a subset of other nodes in the network.

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Prim’s AlgorithmPrim’s Algorithm

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Prim’s AlgorithmPrim’s Algorithm

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Prim’s AlgorithmPrim’s Algorithm

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Prim’s AlgorithmPrim’s Algorithm

Arbitrarily pick a vertex to start with.

Relaxation: dw=min(dw, cwv), where v is the newly marked vertex, w is one of its unmarked neighbors, cwv is the weight of edge (w,v) and dw indicates the current distance between w and one of the marked vertices.

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Dijkstra’s AlgorithmDijkstra’s Algorithm

Need to be changed:

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Prim’s AlgorithmPrim’s Algorithm

Trivial: O(|V|2 + |E|) = O(|V|2)

Heap: deleteMin |V| times + decreaseKey |E| times

O(|V|log|V| + |E|log|V|) = O (|E|log|V|)

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Kruskal’s AlgorithmKruskal’s Algorithm

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Kruskal’s AlgorithmKruskal’s Algorithm

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Kruskal’s AlgorithmKruskal’s Algorithm

O(|E|)

O(|E|log|E|)O(|E|log|V|)

O(|E|)

Time complexity: O(|E|log|E|) = O (|E|log|V|)