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K -MST -based clustering. Caiming Zhong Pasi Franti. 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. Minimum Spanning Tree. Spanning tree. - PowerPoint PPT Presentation
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University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
K-MST -based clustering
Caiming Zhong
Pasi Franti
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.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
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Minimum Spanning Tree
• Spanning tree
Given graph
Spanning tree
Non-
Spanning tree
MSTMST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Minimum Spanning Tree• Minimize the sum of weights (Kruskal, Pri
m’s Algorithm)
Given graph
G=(V,E)
MST
T
),(
),()(Tvu
vuwTwMSTMST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST-based clustering
• The most used Method1: removing
long MST-edgesMST
MST-based MST-based
clusteringclustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST
MST-based MST-based
clusteringclustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST-based clustering
• Removing long MST-edges doesn’t
always workMST
MST-based MST-based
clusteringclustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.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 MST-based
clusteringclustering
K-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
K-MST
• What is K-MST?
– Let G = (V,E) denote the complete graph
– Let MST1 denote the MST of G, and it is
computed as MST1 = mst(V, E).
– Then, MST2 denote the second round of
MST of G, MST2 = mst(V, E- MST1).
– MSTk = mst(V, E- MST1-…-MSTk-1).
MST
MST-based
clustering
KK-MST-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST
MST-based
clustering
KK-MST-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
K-MST
• K-MST-based graph
MST
MST-based
clustering
KK-MST-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.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
KK-MST-MST
K-MST-based
clustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
K-MST-based clustering• Definition of edge weight for separated
problems
MST
MST-based
clustering
K-MST
KK-MST-based -MST-based
clusteringclustering
Fast approximate
MST
)(
})){(}),{(min(1)(
ab
abbabaab
e
eEavgeEavgew
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.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.
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST
MST-based
clustering
K-MST
KK-MST-based -MST-based
clusteringclustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST
MST-based
clustering
K-MST
KK-MST-based -MST-based
clusteringclustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
K-MST-based clustering• Touching problems
MST
MST-based
clustering
K-MST
KK-MST-based -MST-based
clusteringclustering
Fast approximate
MST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Partition(cut1) and Partition(cut1) and
Partition(cut3) are similar ;Partition(cut3) are similar ;
Partition(cut2) and Partition(cut2) and
Partition(cut3) are similar .Partition(cut3) are similar .
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Fast approximate MST (FAMST)
• Traditional MST algorithms take
O(N2) time, not favored by large data
sets.
• In practical application, generally
FAMST has as same result as exact
MST
• Find a FAMST in O(N1.55)
MST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate Fast approximate
MSTMST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Fast approximate MST (FAMST)
• Scheme: Divide-and-Conquer
MST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate Fast approximate
MSTMST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
Fast approximate MST (FAMST)
• Performance
MST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate Fast approximate
MSTMST
University of JoensuuDept. of Computer ScienceP.O. Box 111FIN- 80101 JoensuuTel. +358 13 251 7959fax +358 13 251 7955www.cs.joensuu.fi
MST
MST-based
clustering
K-MST
K-MST-based
clustering
Fast approximate Fast approximate
MSTMST