on the parallel complexity of minimum sum of diameters clustering

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On the Parallel Complexity of Minimum Sum

of Diameters Clustering

Nopadon Juneam 1,2 and Sanpawat Kantabutra 2

1Department of Computer Science, Chiang Mai University, Chiang Mai, Thailand2The Theory of Computation Group, Department of Computer Engineering

Chiang Mai University, Chiang Mai, Thailand

Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 1 / 19

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Outline

1 IntroductionMinimum Sum of Diameters Clustering Problem(MSDCP)

2 Basic Parallel Complexity TheoryBasic Concepts

3 Our ContributionsParallel ComplexityMore Practical Parallel Algorithm


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Outline





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Background

Conceptually, a cluster is a collection of similar entities, whereentities within the same cluster are more similar to each other thanto those in different clusters.

A basic problem of clustering is to partition a given finite set ofentities into clusters.

Applications of clustering are known in a variety of areas such asdata mining, machine learning, pattern recognition, bioinformaticsand social networking.


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Basic Problem of Clustering

Given a set of points on the plane.(e.g. climate data such as GPS locations)


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Basic Problem of Clustering (cont.)

The set of points partitioned into three clusters.


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Minimum Sum of Diameters ClusteringProblem (MSDCP)

Objective: Partition a given set of entities into clusters such that the sumof clusters’ diameters is minimized.


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Minimum Sum of Diameters ClusteringProblem (MSDCP) (cont.)

“In this work, we studied the complexity of the MSDCPon a parallel computer.”


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Outline





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Key Concepts

Researches in computational complexity theory are centered on thequestion how difficult it is to solve a computational problem in theterm of some computational resources.

On a parallel computer, important computational resources are thecomputational time and the number of processors.

Problems can obtain some speed improvement while being solvedon a parallel computer. However, a significant speed up may notpossible for every problem.


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Time Order

source: https://apelbaum.wordpress.com/2011/05/05/big-o/


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Theory of P-Completeness [2]

The parallel complexity class NC contains all the problems thathave a polylogarithmic-time parallel algorithm using a polynomialnumber of processors.

Such a parallel algorithm is considered as a highly parallel solution,or efficient parallel algorithm for the problem.

In contrast, the parallel complexity class P-complete contains allthe problems for which no highly parallel solution is known; up tosome technicality.


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Research Question

“Whether the MSDCP is in NC or P-complete? If it is in NC,then how good its highly parallel solution can be?

in the use of parallel algorithms in practice.”


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Outline





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Previous Studies

For the case of a partition into three or more clustersIn 1978, Brucker showed that the MSDCP is NP-hard [1]

The problem is extremely difficult.

An exact solution to the problem requires at least exponential time.

For the case of a partition into two clustersIn 1987, Hansen and Jaumard gave an algorithm in O(n3 log n)time [3].

The problem becomes easy.


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Our Contributions

For the case of a partition into two clusters, we show that theMSDCP is in class NC. That is, the problem has highly parallelsolutions.

1 The problem can be fasteastly solved in O(log n) parallel timeand n6 processors on the Common CRCW PRAM modelof parallel computer.

2 A more practical NC algorithm can be implemented inO(log3 n) parallel time using n3.376 processors on the EREWPRAM model of parallel computer.

In addition, these results will be published in [4].


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References

P. Brucker.On the complexity of clustering problems.In R. Henn, B. Korte, and W. Oettli, editors, Optimization andOperations Research, volume 157 of Lecture Notes in Economics andMathematical Systems, pages 45–54. Springer Berlin Heidelberg, 1978.

R. Greenlaw, H. J. Hoover, and W. L. Ruzzo.Limits to Parallel Computation: P-completeness Theory.Oxford University Press, Inc., New York, NY, USA, 1995.

P. Hansen and B. Jaumard.Minimum sum of diameters clustering.Journal of Classification, 4(2):215–226, 1987.

N. Juneam and S. Kantabutra.On the parallel complexity of minimum sum of diameters clustering.Journal of Internet Technology, 2017 (in press).


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Acknowledgment

Financial support from the Thailand Research Fund throughthe Royal Golden Jubilee Ph.D. Program

(Grant No. PHD/0267/2552) is acknowledged.


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Thanks [email protected]
