1. Minimum Spanning Tree Neil Tang 3/25/2010
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Class Overview
The minimum spanning tree problem
Applications
Prim’s algorithm
Kruskal’s algorithm
CS223 Advanced Data Structures and Algorithms

3. Minimum 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.
CS223 Advanced Data Structures and Algorithms

4. Minimum Spanning Tree Problem
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Applications
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.
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Prim’s Algorithm
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Prim’s Algorithm
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Prim’s Algorithm
CS223 Advanced Data Structures and Algorithms

9. CS223 Advanced Data Structures and Algorithms
Prim’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.
CS223 Advanced Data Structures and Algorithms

10. CS223 Advanced Data Structures and Algorithms
Dijkstra’s Algorithm
Need to be changed:
CS223 Advanced Data Structures and Algorithms

11. CS223 Advanced Data Structures and Algorithms
Prim’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|)
CS223 Advanced Data Structures and Algorithms

12. CS223 Advanced Data Structures and Algorithms
Kruskal’s Algorithm
CS223 Advanced Data Structures and Algorithms

13. CS223 Advanced Data Structures and Algorithms
Kruskal’s Algorithm
CS223 Advanced Data Structures and Algorithms

14. CS223 Advanced Data Structures and Algorithms
Kruskal’s Algorithm
O(|E|)
O(|E|log|E|)
O(|E|log|V|)
O(|E|)
Time complexity: O(|E|log|E|) = O (|E|log|V|)
CS223 Advanced Data Structures and Algorithms