Computational geometry: an introduction
Computational geometry: an introduction
Line transversals of balls and smallest enclosing cylinders in three dimensions
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Deterministic algorithms for 3-D diameter and some 2-D lower envelopes
Proceedings of the sixteenth annual symposium on Computational geometry
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
Proceedings of the sixteenth annual symposium on Computational geometry
Approximation algorithms for projective clustering
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Cylindrical hierarchy for deforming necklaces
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
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In this paper, we study a 3D geometric problem originated from computing neural maps in the computational biology community: Given a set S of n points in 3D, compute k cylindrical segments (with different radii, orientations, lengths and no segment penetrates another) enclosing S such that the sum of their radii is minimized. There is no known result in this direction except when k = 1. The general problem is strongly NP-hard and we obtain a polynomial time approximation scheme (PTAS) for any fixed k 1 in O(n3k-2/驴4k-3) time by returning k cylindrical segments with sum of radii at most (1 + 驴) of the corresponding optimal value. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k = 1.