Abstract
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(n 3k− 2 /δ4k−3) time by returning k cylindrical segments with sum of radii at most (1 + δ) of the corresponding optimal value. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k = 1.
This research is partially supported by Hong Kong RGC CERG grant CityU1103/ 99E, NSF CARGO grant DMS-0138065 and a MONTS grant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Agarwal, B. Aronov and M. Sharir. Line transversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Symp on Discrete Algorithms (SODA’97), New Orleans, LA, pages 483–492, Jan, 1997.
P. Agarwal and C. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Symp on Discrete Algorithms (SODA’00), pages 538–547, 2000.
T. Chan. Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. In Proc. 16th ACM Symp on Computational Geometry (SCG’00), Hong Kong, pages 300–309, June, 2000.
M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco, CA, 1979.
S. Har-Peled and K. Varadarajan. Projective clustering in high dimensions using core-sets. In Proc. 18th ACM Symp on Computational Geometry (SCG’02), to appear, 2002.
G. Jacobs and F. Theunissen. Functional organization of a neural map in the cricket cercal sensory system. J. of Neuroscience, 16(2):769–784, 1996.
G. Jacobs and F. Theunissen. Extraction of sensory parameters from a neural map by primary sensory interneurons. J. of Neuroscience, 20(8):2934–2943, 2000.
J. Matoušek, M. Sharir and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16:498–516, 1992.
N. Megiddo and A. Tamir. On the complexity of locating linear facilities in the plane. Operation Research Letters, 1(5):194–197, 1982.
S. Paydar, C. Doan and G. Jacobs. Neural mapping of direction and frequency in the cricket cercal sensory system. J. of Neuro science, 19(5):1771–1781, 1999.
F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
E. Ramos. Deterministic algorithms for 3-D diameter and some 2-D lower envelopes. In Proc. 16th ACM Symp on Computational Geometry (SoCG’00), pages 290–299, June, 2000.
E. Schömer, J. Sellen, M. Teichmann and C.K. Yap. Smallest enclosing cylinders. Algorithmica, 27:170–186, 2000.
E. Welzl. Smallest enclosing disks (balls and ellipsoids). In New results and new trends in computer science, LNCS 555, pages 359–370, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhu, B. (2002). Approximating 3D Points with Cylindrical Segments. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_45
Download citation
DOI: https://doi.org/10.1007/3-540-45655-4_45
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43996-7
Online ISBN: 978-3-540-45655-1
eBook Packages: Springer Book Archive