Applications of Parametric Searching in Geometric Optimization

Agarwal, Pankaj K.; Sharir, Micha; Toledo, Sivan

doi:10.1006/jagm.1994.1038

Cited by 142 publications

(126 citation statements)

References 29 publications

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“…One of the most general approaches for reducing geometric optimization problems to their decision problems is parametric search, invented by Megiddo [54] (see [1], [4], [7], [12], [15], [17], [26], [34], [53], [59], [61], and [64] for just a partial list of examples). The basic idea is to simulate the decision algorithm-compare the optimum with twith the parameter t being the unknown optimum itself.…”

Section: Previous Approachesmentioning

confidence: 99%

Geometric applications of a randomized optimization technique

Chan

1998

Proceedings of the Fourteenth Annual Symposium on Computational Geometry - SCG '98

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Abstract. We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under translation, computing rectilinear p-centers and discrete 1-centers, and solving linear programs with k violations.

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Section: Previous Approachesmentioning

confidence: 99%

Geometric applications of a randomized optimization technique

Chan

1998

Proceedings of the Fourteenth Annual Symposium on Computational Geometry - SCG '98

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“…It seems likely that the time for these steps can be reduced using techniques for optimization in overlays of two planar diagrams [2], however this would not reduce the complexity of our overall algorithm.…”

Section: Lemma 2 the Algorithm Described Above Finds The Optimal Valmentioning

confidence: 99%

Optimized color gamuts for tiled displays

Bern¹,

Eppstein

2003

Proceedings of the Nineteenth Annual Symposium on Computational Geometry

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Abstract. We consider the problem of finding a large color space that can be generated by all units in multiprojector tiled display systems. Viewing the problem geometrically as one of finding a large parallelepiped within the intersection of multiple parallelepipeds, and using colorimetric principles to define a volume-based objective function for comparing feasible solutions, we develop an algorithm for finding the optimal gamut in time O(n 3 ), where n denotes the number of projectors in the system. We also discuss more efficient quasiconvex programming algorithms for alternative objective functions based on maximizing the quality of the color space extrema.

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“…This 1-dimensional piercing problem can be solved in O(n log p) time. 2 So if there is a vertical (or horizontal) line intersecting all rectangles, the problem can be solved in linear time for any xed p.…”

Section: Rectilinear P-piercingmentioning

confidence: 99%

“…Clearly, us depends only on s and s 0 (and the orientations , 0 ), and sv depends only on s and s 00 (and the orientations , 00 ), which readily implies the lemma. 2 We next x three edge orientations , 0 , 00 , x the point s as de ned above (there are c choices for this point), and construct a matrix M, whose rows correspond to the points s 0 2 S, sorted in increasing order of 1 (s ; s 0 ), as de ned above, and whose columns correspond to the points s 00 2 S, sorted in increasing order of 2 (s ; s 00 ). We put M ij = We remark that more general polygonal 2-center problems can be solved in O(n polylog n) time, using the parametric searching technique 36], as was also done in Section 3.2, for the general rectilinear 4-and 5-center problems.…”

Section: Polygonal 2-center Problemsmentioning

confidence: 99%

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Rectilinear and polygonal p-piercing and p-center problems

Sharir

Welzl

1996

Proceedings of the Twelfth Annual Symposium on Computational Geometry - SCG '96

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In the p-piercing problem, we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or near-linear algorithms for small values of p in cases where the given regions are either axis-parallel rectangles or convex c-oriented polygons in the plane (i.e., convex polygons with sides from a xed nite set of directions).We also investigate the planar rectilinear (and polygonal) p-center problem, in which we are given a set S of n points in the plane, and wish to nd p axis-parallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems.New results are a linear-time solution for the rectilinear 3-center problem (by showing that this problem can be formulated as an LP-type problem and by exhibiting a relation to Helly numbers). We give O(n logn)-time solutions for 4-piercing of translates of a square, as well as for the rectilinear 4-center problem; this is worst-case optimal. We give O(n polylog n)-time solutions for 4-and 5-piercing of axis-parallel rectangles, for more general rectilinear 4-center problems, and for rectilinear 5-center problems. 2-pierceability of a set of n convex c-oriented polygons can be decided in time O(c 2 n log n), and the 2-center problem for a convex c-gon can be solved in O(c 5 n logn) time. The rst solution is worst-case optimal when c is xed.

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Applications of Parametric Searching in Geometric Optimization

Cited by 142 publications

References 29 publications

Geometric applications of a randomized optimization technique

Geometric applications of a randomized optimization technique

Optimized color gamuts for tiled displays

Rectilinear and polygonal p-piercing and p-center problems

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