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

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

1,* , 2 and 3
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.
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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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