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

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

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 / Revised: 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, ... , 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. View Full-Text
Keywords: Clustering 3D point sequences; squared Euclidean distance; algorithm; polynomial-time approximation scheme Clustering 3D point sequences; squared Euclidean distance; algorithm; polynomial-time approximation scheme
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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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.

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