A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance

doi:10.3390/a1020043

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Algorithms 2008, 1(2), 43-51; doi:10.3390/a1020043

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A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance

Shuai Cheng Li^1,* , Yen Kaow Ng² and Louxin Zhang³

¹ David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada N2L 3G1 ² Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 819-0395, Japan ³ Department of Mathematics, National University of Singapore, Singapore 117543

* Author to whom correspondence should be addressed.

Received: 5 September 2008; in revised form: 1 September 2008 / Accepted: 9 October 2008 / Published: 9 October 2008

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Abstract: In this paper we consider a basic clustering problem that has uses in bioinformatics. A structural fragment is a sequence of l points in a 3D space, where l is a fixed natural number. Two structural fragments f₁ and f₂ are equivalent if and only if f₁ = f₂ x R + τ under some rotation R and translation τ . We consider the distance between two structural fragments to be the sum of the squared Euclidean distance between all corresponding points of the structural fragments. Given a set of n structural fragments, we consider the problem of finding k (or fewer) structural fragments g₁, g₂, ... , g_k, so as to minimize the sum of the distances between each of f₁, f₂, ... , f_n to its nearest structural fragment in g₁, ... , g_k. In this paper we show a polynomial-time approximation scheme (PTAS) for the problem through a simple sampling strategy.

Keywords: Clustering 3D point sequences; squared Euclidean distance; algorithm; polynomial-time approximation scheme

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Cite This Article

MDPI and ACS Style

Li, S.C.; Ng, Y.K.; Zhang, L. A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance. Algorithms 2008, 1, 43-51.

AMA Style

Li SC, Ng YK, Zhang L. A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance. Algorithms. 2008; 1(2):43-51.

Chicago/Turabian Style

Li, Shuai Cheng; Ng, Yen Kaow; Zhang, Louxin. 2008. "A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance." Algorithms 1, no. 2: 43-51.

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