Efficient approximation algorithms for clustering point-sets
In this paper, we consider the problem of clustering a set of n finite point-sets in d-dimensional Euclidean space. Different from the traditional clustering problem (called points clustering problem where the to-be-clustered objects are points), the point-sets clustering problem requires that all p...
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| Vydáno v: | Computational geometry : theory and applications Ročník 43; číslo 1; s. 59 - 66 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
2010
|
| Témata: | |
| ISSN: | 0925-7721 |
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| Abstract | In this paper, we consider the problem of clustering a set of
n finite point-sets in
d-dimensional Euclidean space. Different from the traditional clustering problem (called points clustering problem where the to-be-clustered objects are points), the point-sets clustering problem requires that all points in a single point-set be clustered into the same cluster. This requirement disturbs the metric property of the underlying distance function among point-sets and complicates the clustering problem dramatically. In this paper, we use a number of interesting observations and techniques to overcome this difficulty. For the
k-center clustering problem on point-sets, we give an
O
(
m
+
n
log
k
)
-time 3-approximation algorithm and an
O
(
k
m
)
-time
(
1
+
3
)
-approximation algorithm, where
m is the total number of input points and
k is the number of clusters. When
k is a small constant, the performance ratio of our algorithm reduces to
(
2
+
ϵ
)
for any
ϵ
>
0
. For the
k-median problem on point-sets, we present a polynomial time
(
3
+
ϵ
)
-approximation algorithm. Our approaches are rather general and can be easily implemented for practical purpose. |
|---|---|
| AbstractList | In this paper, we consider the problem of clustering a set of
n finite point-sets in
d-dimensional Euclidean space. Different from the traditional clustering problem (called points clustering problem where the to-be-clustered objects are points), the point-sets clustering problem requires that all points in a single point-set be clustered into the same cluster. This requirement disturbs the metric property of the underlying distance function among point-sets and complicates the clustering problem dramatically. In this paper, we use a number of interesting observations and techniques to overcome this difficulty. For the
k-center clustering problem on point-sets, we give an
O
(
m
+
n
log
k
)
-time 3-approximation algorithm and an
O
(
k
m
)
-time
(
1
+
3
)
-approximation algorithm, where
m is the total number of input points and
k is the number of clusters. When
k is a small constant, the performance ratio of our algorithm reduces to
(
2
+
ϵ
)
for any
ϵ
>
0
. For the
k-median problem on point-sets, we present a polynomial time
(
3
+
ϵ
)
-approximation algorithm. Our approaches are rather general and can be easily implemented for practical purpose. |
| Author | Xu, Jinhui Xu, Guang |
| Author_xml | – sequence: 1 givenname: Guang surname: Xu fullname: Xu, Guang email: guangxu@cse.buffalo.edu – sequence: 2 givenname: Jinhui surname: Xu fullname: Xu, Jinhui email: jinhui@cse.buffalo.edu |
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| Cites_doi | 10.1145/1007352.1007400 10.1145/62212.62255 10.1007/s00453-005-1166-x 10.1145/331499.331504 10.1126/science.281.5382.1502 10.1007/BF02187718 10.1007/3-540-48481-7_33 10.1016/0304-3975(85)90224-5 10.1145/509943.509947 |
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| Keywords | K-center clustering Point-sets Core-sets Clustering K-median clustering |
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| References_xml | – reference: M. Bădoiu, K.L. Clarkson, Smaller core-sets for balls, in: Proc. 14th Annual ACM–SIAM Symposium on Discrete Algorithms, 2003, pp. 801–802 – reference: S. Har-Peled, S. Mazumdar, Coresets for – reference: A. Goel, P. Indyk, K.R. Varadarajan, Reductions among high dimensional proximity problems, in: Proc. 12th ACM–SIAM Sympos. Discrete Algorithms, 2001, pp. 769–778 – reference: T. Feder, D.H. Greene, Optimal algorithm for approximation clustering, in: Proc. 20th ACM Symp. Theory of Computing, 1988, pp. 434–444 – reference: P.K. Agarwal, C.M. Procopiuc, Exact and approximation algorithms for clustering, in: Proc. 9th ACM–SIAM Sympos. Discrete Algorithms, 1998, pp. 658–667 – reference: D. Eppstein, Faster construction of planar two-centers, in: Proc. 8th ACM–SIAM Sympos. Discrete Algorithms, 1997 – year: 1997 ident: bib005 article-title: Approximation algorithms for geometric problems publication-title: Approximation Algorithms for NP-Hard Problems – year: 2001 ident: bib006 article-title: Pattern Classification – volume: vol. 2 year: 1994 ident: bib012 article-title: Geometric Algorithms and Combinatorial Optimization publication-title: Algorithm and Combinatorics – year: 2001 ident: bib018 article-title: Approximation Algorithms – volume: 31 start-page: 264 year: 1999 end-page: 323 ident: bib015 article-title: Data clustering: A review publication-title: ACM Computing Surveys – reference: S.G. Kolliopoulos, S. Rao, A nearly linear-time approximation scheme for the Euclidean – reference: -median problem, in: Proc. 7th Annu. European Sympos. Algorithms, 1999, pp. 378–389 – year: 2000 ident: bib009 article-title: Data Mining – volume: 38 start-page: 293 year: 1985 end-page: 306 ident: bib011 article-title: Clustering to minimize the maximum intercluster distance publication-title: Theoretical Computer Science – reference: -median clustering and their applications, in: Proc. 36th ACM Symposium on Theory of Computing, 2004, pp. 291–300 – year: 1995 ident: bib014 publication-title: Approximation Algorithms for NP-Hard Problems – volume: 4 start-page: 101 year: 1989 end-page: 115 ident: bib017 article-title: An publication-title: Discrete and Computational Geometry – volume: 42 start-page: 221 year: 2005 end-page: 230 ident: bib002 article-title: Algorithms for a publication-title: Algorithmica – reference: M. Bădoiu, S. Har-Peled, P. Indyk, Approximate clustering via core-sets, in: Proceedings of the 34th ACM Symposium on Theory of Computing, 2002, pp. 250–257 – reference: -means and – volume: 281 start-page: 1502 year: 1998 end-page: 1506 ident: bib019 article-title: Segregation of transcription and replication sites into higher order domains publication-title: Sciences – ident: 10.1016/j.comgeo.2007.12.002_bib013 doi: 10.1145/1007352.1007400 – ident: 10.1016/j.comgeo.2007.12.002_bib008 doi: 10.1145/62212.62255 – year: 1995 ident: 10.1016/j.comgeo.2007.12.002_bib014 – volume: 42 start-page: 221 issue: 3–4 year: 2005 ident: 10.1016/j.comgeo.2007.12.002_bib002 article-title: Algorithms for a k-line center publication-title: Algorithmica doi: 10.1007/s00453-005-1166-x – volume: 31 start-page: 264 year: 1999 ident: 10.1016/j.comgeo.2007.12.002_bib015 article-title: Data clustering: A review publication-title: ACM Computing Surveys doi: 10.1145/331499.331504 – volume: 281 start-page: 1502 year: 1998 ident: 10.1016/j.comgeo.2007.12.002_bib019 article-title: Segregation of transcription and replication sites into higher order domains publication-title: Sciences doi: 10.1126/science.281.5382.1502 – year: 2000 ident: 10.1016/j.comgeo.2007.12.002_bib009 – volume: 4 start-page: 101 year: 1989 ident: 10.1016/j.comgeo.2007.12.002_bib017 article-title: An O(nlogn) algorithm for the all-nearest-neighbors problem publication-title: Discrete and Computational Geometry doi: 10.1007/BF02187718 – ident: 10.1016/j.comgeo.2007.12.002_bib016 doi: 10.1007/3-540-48481-7_33 – ident: 10.1016/j.comgeo.2007.12.002_bib010 – year: 2001 ident: 10.1016/j.comgeo.2007.12.002_bib018 – volume: vol. 2 year: 1994 ident: 10.1016/j.comgeo.2007.12.002_bib012 article-title: Geometric Algorithms and Combinatorial Optimization – year: 2001 ident: 10.1016/j.comgeo.2007.12.002_bib006 – ident: 10.1016/j.comgeo.2007.12.002_bib007 – year: 1997 ident: 10.1016/j.comgeo.2007.12.002_bib005 article-title: Approximation algorithms for geometric problems – volume: 38 start-page: 293 year: 1985 ident: 10.1016/j.comgeo.2007.12.002_bib011 article-title: Clustering to minimize the maximum intercluster distance publication-title: Theoretical Computer Science doi: 10.1016/0304-3975(85)90224-5 – ident: 10.1016/j.comgeo.2007.12.002_bib003 – ident: 10.1016/j.comgeo.2007.12.002_bib001 – ident: 10.1016/j.comgeo.2007.12.002_bib004 doi: 10.1145/509943.509947 |
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n finite point-sets in
d-dimensional Euclidean space. Different from the traditional clustering... |
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| SubjectTerms | Clustering Core-sets K-center clustering K-median clustering Point-sets |
| Title | Efficient approximation algorithms for clustering point-sets |
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