NDLI: An Expander-Based Approach to Geometric Optimization

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An Expander-Based Approach to Geometric Optimization

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Katz, Matthew J. Sharir, Micha
Copyright Year	1997
Abstract	We present a new approach to problems in geometric optimization that are traditionally solved using the parametric-searching technique of Megiddo [J. ACM, 30 (1983), pp. 852--865]. Our new approach is based on expander graphs and range-searching techniques. It is conceptually simpler, has more explicit geometric flavor, and does not require parallelization or randomization. In certain cases, our approach yields algorithms that are asymptotically faster than those currently known (e.g., the second and third problems below) by incorporating into our (basic) technique a subtechnique that is equivalent to (though much more flexible than) Cole's technique for speeding up parametric searching [J. ACM, 34 (1987), pp. 200--208]. We exemplify the technique on three main problems---the slope selection problem, the planar distance selection problem, and the planar {\em two-line center} problem. For the first problem we develop an $O(n\log^3 n)$ solution, which, although suboptimal, is very simple. The other two problems are more typical examples of our approach. Our solutions have running time $O(n^{4/3}\log^2n)$ and $O(n^2 \log^4 n)$, respectively, slightly better than the previous respective solutions of [Agarwal et al., Algorithmica, 9 (1993), pp. 495--514], [Agarwal and Sharir, Algorithmica, 11 (1994), pp. 185--195]. We also briefly mention two other problems that can be solved efficiently by our technique.In solving these problems, we also obtain some auxiliary results concerning batched range searching, where the ranges are congruent discs or annuli. For example, we show that it is possible to compute deterministically a compact representation of the set of all point-disc incidences among a set of n congruent discs and a set of m points in the plane in time $O((m^{2/3} n^{2/3}+m+n)\log n)$, again slightly better than what was previously known.
Starting Page	1384
Ending Page	1408
Page Count	25
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539794268649
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5 (Special Section on Proballistic Proof Systems)
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	parametric searching computational geometry Graph theory geometric optimization slope selection range searching Discrete location and assignment expander graphs facility location
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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