Effective Use of Geometric Information for Clustering and Related Topics

Tetsuo ASANO

Effective Use of Geometric Information for Clustering and Related Topics

Tetsuo ASANO

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This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k 3 as far as k is a fixed constant. For the case k=2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.

Publication: IEICE TRANSACTIONS on Information Vol.E83-D No.3 pp.418-427

Publication Date: 2000/03/25

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Type of Manuscript: INVITED SURVEY PAPER

Category: Algorithms for Geometric Problems

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Tetsuo ASANO, "Effective Use of Geometric Information for Clustering and Related Topics" in IEICE TRANSACTIONS on Information, vol. E83-D, no. 3, pp. 418-427, March 2000, doi: .
Abstract: This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k 3 as far as k is a fixed constant. For the case k=2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.
URL: https://global.ieice.org/en_transactions/information/10.1587/e83-d_3_418/_p

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@ARTICLE{e83-d_3_418,
author={Tetsuo ASANO, },
journal={IEICE TRANSACTIONS on Information},
title={Effective Use of Geometric Information for Clustering and Related Topics},
year={2000},
volume={E83-D},
number={3},
pages={418-427},
abstract={This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k 3 as far as k is a fixed constant. For the case k=2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.},
keywords={},
doi={},
ISSN={},
month={March},}

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TY - JOUR
TI - Effective Use of Geometric Information for Clustering and Related Topics
T2 - IEICE TRANSACTIONS on Information
SP - 418
EP - 427
AU - Tetsuo ASANO
PY - 2000
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E83-D
IS - 3
JA - IEICE TRANSACTIONS on Information
Y1 - March 2000
AB - This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k 3 as far as k is a fixed constant. For the case k=2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.
ER -