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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST -based clustering Caiming Zhong Pasi Franti

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Outline Minimum spanning tree (MST) MST-based clustering K-MST K-MST-based clustering Fast approximate MST MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Minimum Spanning Tree Spanning tree Given graph Spanning tree Non- Spanning tree MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Minimum Spanning Tree Minimize the sum of weights (Kruskal, Prim’s Algorithm) Given graph G=(V,E) MST T MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST-based clustering The most used Method1: removing long MST-edges MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST-based clustering Removing long MST-edges doesn’t always work MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST-based clustering The most used Method2: edge inconsistent Tree edge AB, whose weight W(AB) is significantly larger than the average of nearby edge weights on both sides of the edge AB, should be deleted. MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST What is K-MST? –Let G = (V,E) denote the complete graph –Let MST 1 denote the MST of G, and it is computed as MST 1 = mst(V, E). –Then, MST 2 denote the second round of MST of G, MST 2 = mst(V, E- MST 1 ). –MST k = mst(V, E- MST 1 -…-MST k-1 ). MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST K-MST-based graph MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST Typical clustering problems –Separated problems and touching problems. –Separated problems includes distance- separated problems and density-separated problems. MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST-based clustering Definition of edge weight for separated problems MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Three good features: (1) Weights of inter-cluster edges are quite larger than those of intra-cluster edges. (2) The inter- cluster edges are approximately equally distributed to T1 and T2. (3) Except inter- cluster edges, most of edges with large weights come from T2.

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi K-MST-based clustering Touching problems MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Partition(cut1) and Partition(cut3) are similar ; Partition(cut2) and Partition(cut3) are similar.

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Fast approximate MST (FAMST) Traditional MST algorithms take O(N 2 ) time, not favored by large data sets. In practical application, generally FAMST has as same result as exact MST Find a FAMST in O(N 1.55 ) MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Fast approximate MST (FAMST) Scheme: Divide-and-Conquer MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi Fast approximate MST (FAMST) Performance MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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University of Joensuu Dept. of Computer Science P.O. Box 111 FIN- 80101 Joensuu Tel. +358 13 251 7959 fax +358 13 251 7955 www.cs.joensuu.fi MST MST-based clustering K-MST K-MST-based clustering Fast approximate MST

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