1. Minimum Spanning Trees and Clustering By Swee-Ling Tang April 20, 2010 04/20/20101

2. Spanning Tree A spanning tree of a graph is a tree and is a subgraph that contains all the vertices. A graph may have many spanning trees; for example, the complete graph on four vertices has sixteen spanning trees: 04/20/20102

3. Spanning Tree – cont. 04/20/20103

4. Minimum Spanning Trees Suppose that the edges of the graph have weights or lengths. The weight of a tree will be the sum of weights of its edges. Based on the example, we can see that different trees have different lengths. The question is: how to find the minimum length spanning tree? 04/20/20104

5. Minimum Spanning Trees The question can be solved by many different algorithms, here is three classical minimum- spanning tree algorithms : – Boruvka's Algorithm – Kruskal's Algorithm – Prim's Algorithm 04/20/20105

6. 6 Kruskal's Algorithm Joseph Bernard Kruskal, Jr Kruskal Approach: – Select the minimum weight edge that does not form a cycle Kruskal's Algorithm: sort the edges of G in increasing order by length keep a subgraph S of G, initially empty for each edge e in sorted order if the endpoints of e are disconnected in S add e to S return S

7. Kruskal's Algorithm - Example 04/20/20107

8. Kruskal's Algorithm - Example 04/20/20108

9. 9 Prim’s Algorithm Robert Clay Prim Prim Approach: – Choose an arbitrary start node v – At any point in time, we have connected component N containing v and other nodes V-N – Choose the minimum weight edge from N to V-N Prim's Algorithm: let T be a single vertex x while (T has fewer than n vertices) { find the smallest edge connecting T to G-T add it to T }

10. 04/20/201010 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

11. 04/20/201011 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

12. 04/20/201012 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

13. 04/20/201013 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

14. 04/20/201014 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

15. 04/20/201015 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

16. 04/20/201016 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

17. 04/20/201017 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

18. 04/20/201018 Prim's Algorithm - Example 13 50 11 7 2 812 9 10 14 40 3 1 6 20

19. 04/20/201019 Boruvka's Algorithm Otakar Borůvka – Inventor of MST – Czech scientist – Introduced the problem – The original paper was written in Czech in 1926. – The purpose was to efficiently provide electric coverage of Bohemia.

20. 04/20/201020 Boruvka’s Algorithm Boruvka Approach: – Prim “in parallel” – Repeat the following procedure until the resulting graph becomes a single node. For each node u, mark its lightest incident edge. Now, the marked edges form a forest F. Add the edges of F into the set of edges to be reported. Contract each maximal subtree of F into a single node.

21. 04/20/201021 Boruvka’s Algorithm - Example 2.1 1.3 2.3 1.2 2.2 3.1 2.4 3 1 1.5 1.4 2.6 2.7 2.5 3.2 5 3.3 4 4.1 5.1

22. Usage of Minimum Spanning Trees Network design: – telephone, electrical, hydraulic, TV cable, computer, road Approximation algorithms for NP-hard (non- deterministic polynomial-time hard) problems: – traveling salesperson problem Cluster Analysis 04/20/201022

23. Clustering Definition – Clustering is “the process of organizing objects into groups whose members are similar in some way”. – A cluster is therefore a collection of objects which are “similar” between them and are “dissimilar” to the objects belonging to other clusters. – A data mining technique 04/20/201023

24. Why Clustering? Unsupervised learning process Pattern detection Simplifications Useful in data concept construction 04/20/201024

25. The use of Clustering Data Mining Information Retrieval Text Mining Web Analysis Marketing Medical Diagnostic Image Analysis Bioinformatics 04/20/201025

26. Image Analysis – MST Clustering 04/20/201026 An image file before/after color clustering using HEMST (Hierarchical EMST clustering algorithm) and SEMST (Standard EMST clustering algorithm).

27. References Minimum Spanning Tree Based Clustering Algorithms - http://www4.ncsu.edu/~zjorgen/ictai06.pdf http://www4.ncsu.edu/~zjorgen/ictai06.pdf Minimal Spanning Tree based Fuzzy Clustering - http://www.waset.org/journals/waset/v8/v8-2.pdf http://www.waset.org/journals/waset/v8/v8-2.pdf Minimum Spanning Tree – Wikipedia - http://en.wikipedia.org/wiki/Minimum_spanning_tree http://en.wikipedia.org/wiki/Minimum_spanning_tree Kruskal's algorithm - http://en.wikipedia.org/wiki/Kruskal%27s_algorithm http://en.wikipedia.org/wiki/Kruskal%27s_algorithm Prim's Algorithm – Wikipedia - http://en.wikipedia.org/wiki/Prim%27s_algorithm http://en.wikipedia.org/wiki/Prim%27s_algorithm Design and Analysis of Algorithms - http://www.ics.uci.edu/~eppstein/161/960206.html http://www.ics.uci.edu/~eppstein/161/960206.html 04/20/201027