Geometric applications of a randomized optimization technique
Proceedings of the fourteenth annual symposium on Computational geometry
Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
Rectilinear Static and Dynamic Discrete 2-center Problems
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Dynamic algorithms for approximating interdistances
Nordic Journal of Computing
Proximity problems on line segments spanned by points
Computational Geometry: Theory and Applications
Covering point sets with two disjoint disks or squares
Computational Geometry: Theory and Applications
Proximity problems on line segments spanned by points
Computational Geometry: Theory and Applications
Dynamic algorithms for approximating interdistances
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Counting inversions, offline orthogonal range counting, and related problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Piecewise-linear approximations of uncertain functions
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Covering many or few points with unit disks
WAOA'06 Proceedings of the 4th international conference on Approximation and Online Algorithms
An efficient approximation algorithm for point pattern matching under noise
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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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.