On the parallel complexity of minimum sum of diameters clustering

Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This paper considers the problem called Minimum Sum of Diameters Clustering Problem, where a partition of the set of entities into k clusters such that the sum of the diameters of these clusters is min...

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Published in:2015 International Computer Science and Engineering Conference (ICSEC) pp. 1 - 6
Main Authors: Juneam, Nopadon, Kantabutra, Sanpawat
Format: Conference Proceeding
Language:English
Published: IEEE 01.11.2015
Subjects:
application in social networking
clustering
Clustering algorithms
Complexity theory
Computational modeling
minimum sum of diameters
Parallel algorithms
parallel complexity
Phase change random access memory
PRAM algorithm
Program processors
Social network services
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Abstract Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This paper considers the problem called Minimum Sum of Diameters Clustering Problem, where a partition of the set of entities into k clusters such that the sum of the diameters of these clusters is minimized. Brucker showed that the complexity of the problem is NP-hard, when k ≥ 3 [1]. For the case of k = 2, Hansen and Jaumard gave an O(n 3 log n) algorithm [2], which Ramnath later improved the running time to O(n 3 ) [3]. This paper discusses the parallel complexity of the Minimum Sum of Diameters Clustering Problem. For the case of k = 2, we show that the problem in parallel in fact belongs in class NC 1 In particular, we show that the parallel complexity of the problem is O(log n) parallel time and n 7 processors on the Common CRCW PRAM model. Additionally, we propose the parallel algorithmic technique which can be applied to improve the processor bound by a factor of n. As a result, we show that the problem can be quickly solved in O(log n) parallel time using n 6 processors on the Common CRCW PRAM model. In addition, regarding the issue of high processor complexity, we also propose a more practical NC algorithm which can be implemented in O(log 3 n) parallel time using n 3.376 processors on the EREW PRAM model.
AbstractList Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This paper considers the problem called Minimum Sum of Diameters Clustering Problem, where a partition of the set of entities into k clusters such that the sum of the diameters of these clusters is minimized. Brucker showed that the complexity of the problem is NP-hard, when k ≥ 3 [1]. For the case of k = 2, Hansen and Jaumard gave an O(n 3 log n) algorithm [2], which Ramnath later improved the running time to O(n 3 ) [3]. This paper discusses the parallel complexity of the Minimum Sum of Diameters Clustering Problem. For the case of k = 2, we show that the problem in parallel in fact belongs in class NC 1 In particular, we show that the parallel complexity of the problem is O(log n) parallel time and n 7 processors on the Common CRCW PRAM model. Additionally, we propose the parallel algorithmic technique which can be applied to improve the processor bound by a factor of n. As a result, we show that the problem can be quickly solved in O(log n) parallel time using n 6 processors on the Common CRCW PRAM model. In addition, regarding the issue of high processor complexity, we also propose a more practical NC algorithm which can be implemented in O(log 3 n) parallel time using n 3.376 processors on the EREW PRAM model.
Author Kantabutra, Sanpawat
Juneam, Nopadon
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  givenname: Sanpawat
  surname: Kantabutra
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  organization: Theor. of Comput. Group, Chiang Mai Univ., Chiang Mai, Thailand
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Snippet Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This paper considers the problem called Minimum Sum of...
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SubjectTerms application in social networking
clustering
Clustering algorithms
Complexity theory
Computational modeling
minimum sum of diameters
Parallel algorithms
parallel complexity
Phase change random access memory
PRAM algorithm
Program processors
Social network services
Title On the parallel complexity of minimum sum of diameters clustering
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