Abstract
We present a new algorithm for the two-line center problem (also called unweighted orthogonal L ∞-fit problem): “Given a set S of n points in the real plane, find two closed strips whose union contains all of the points and such that the width of the wider strip is minimized.” An almost quadratic O(n 2 log2 n) solution is given. The previously best known algorithm for this problem has time complexity O(n 2log5 n) and uses a parametric search methodology. Our solution applies a new geometric structure, anchored lower and upper chains, and is based on examining several constraint versions of the problem. The anchored lower and upper chain structure is of interest by itself and provides a fast response to queries that involve planar configurations of points. The algorithm does not assume the general position of the input data points.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
P. K. Agarwal and M. Sharir. Planar Geometric Location Problems and Maintaining the Width of a Planar Set, Proc. of the Second ACM-SIAM Symp. on Discr. Alg., (1991), pp. 449–458
H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer Verlag (1987).
R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set, Inform. Process. Lett., 1 (1972), pp. 132–133.
M. E. Houle, H. Imai, K. Imai, J.-M. Robert. Weighted orthogonal linear L ∞-approximation and applications, Lect. Notes Computer Sci., 382 (1989), pp. 183–191.
J. Hershberger and S. Suri. Offline Maintenance of Planar Configurations, Proc. of the Second ACM-SIAM Symp. on Discr. Alg., (1991), pp. 32–41
J. W. Jaromczyk and M. Kowaluk. Coordinate-system independent covering of point sets in R 2 with pairs of rectangles or optimal squares, Manuscript, (1995).
N. M. Korneenko and H. Martini. Hyperplane Approximation and Related Topics, New Trends in Discrete and Computational Geometry (Janos Pach Ed.), (1993), pp. 135–161
D. T. Lee and Y. T. Ching. The power of geometric duality revised, Inform. Process. Lett., 21 (1985), pp. 117–122.
N. Megiddo and A. Tamir. On the complexity of location linear facilities in the plane, Oper. Res. Lett., 1 (1982), pp. 194–197.
N. Megiddo and A. Tamir. Finding least-distances lines, SIAM J. Algebr. Discrete Math., 4 (1983), pp. 207–211.
M. Overmars and J. van Leeuwen. Maintenance of configurations in the plane, Journal of Computer and System Sciences, 23 (1981), pp. 166–204.
F. Preparata, M. I. Shamos. Computational Geometry. An Introduction. Springer Verlag (1985).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jaromczyk, J.W., Kowaluk, M. (1995). The two-line center problem from a polar view: a new algorithm and data structure. In: Akl, S.G., Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1995. Lecture Notes in Computer Science, vol 955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60220-8_47
Download citation
DOI: https://doi.org/10.1007/3-540-60220-8_47
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60220-0
Online ISBN: 978-3-540-44747-4
eBook Packages: Springer Book Archive