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A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance
David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada N2L 3G1
Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 819-0395, Japan
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
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, ... , gk, so as to minimize the sum of the distances between each of f1, f2, ... , fn to its nearest structural fragment in g1, ... , gk. 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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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.
Li SC, Ng YK, Zhang L. A PTAS For The k-Consensus Structures Problem Under Squared Euclidean Distance. Algorithms. 2008; 1(2):43-51.
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.