A PTAS for the k-Consensus Structures Problem Under Euclidean Squared Distance

Authors:
Shuai Cheng Li;Yen Kaow Ng;Louxin Zhang
Affiliations:
David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada ON N2L 3G1;Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka, Japan 819-0395;Department of Mathematics, National University of Singapore, Singapore 117543
Venue:
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
Year:
2008

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Abstract

In this paper we consider a basic clustering problem that has uses in bioinformatics. A structural fragmentis a sequence of 驴 points in a 3D space, where 驴 is a fixed natural number. Two structural fragments f1and f2are equivalent iff under some rotation and translation . We consider the distancebetween two structural fragments to be the sum of the Euclidean squared distance between all corresponding points of the structural fragments. Given a set of nstructural 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, ..., fnto its nearest structural fragment in g1, ..., gk. In this paper we show a PTAS for the problem through a simple sampling strategy.