Linear Programming in Linear Time When the Dimension Is Fixed

Megiddo, Nimrod

doi:10.1145/2422.322418

Cited by 562 publications

(283 citation statements)

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“…In Step 3, computing the upper and the lower tangents reduces to 2-dimensional linear programming (as in Kirkpatrick and Seidel [15]), which can be solved in O(n) time [17]. Also it is straightforward to find P L and P R in O(n) time.…”

Section: Algorithm: Convexhull(p)mentioning

confidence: 99%

Adaptive Algorithms for Planar Convex Hull Problems

Ahn

Okamoto

2011

IEICE Trans. Inf. & Syst.

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SUMMARYWe study problems in computational geometry from the viewpoint of adaptive algorithms. Adaptive algorithms have been extensively studied for the sorting problem, and in this paper we generalize the framework to geometric problems. To this end, we think of geometric problems as permutation (or rearrangement) problems of arrays, and define the "presortedness" as a distance from the input array to the desired output array. We call an algorithm adaptive if it runs faster when a given input array is closer to the desired output, and furthermore it does not make use of any information of the presortedness. As a case study, we look into the planar convex hull problem for which we discover two natural formulations as permutation problems. An interesting phenomenon that we prove is that for one formulation the problem can be solved adaptively, but for the other formulation no adaptive algorithm can be better than an optimal outputsensitive algorithm for the planar convex hull problem. To further pursue the possibility of adaptive computational geometry, we also consider constructing a kd-tree. key words: adaptive algorithms, convex hulls, computational geometry

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Section: Algorithm: Convexhull(p)mentioning

confidence: 99%

Adaptive Algorithms for Planar Convex Hull Problems

Ahn

Okamoto

2011

IEICE Trans. Inf. & Syst.

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“…Standard combinatorial algorithms for linear programming, including algorithms for linear programming in small dimensions [8,11,18,19,29,30] as well as the simplex method (cf., [7]), can generate a basic infeasible subsystem given an infeasible linear program. The details of finding a basic infeasible subsystem using linear programming are discussed further in Appendix A.…”

Section: An Algorithm For Points In R Dmentioning

confidence: 99%

Output-sensitive algorithms for Tukey depth and related problems

Bremner

Chen

Iacono³

et al. 2008

Stat Comput

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The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.

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“…First, using e.g. Megiddo's linear time linear programming method [8] we check whether either vy ≤ α or vy ≥ α holds for all y ∈ R by maximizing/minimizing the function vy over the set R. If one of these holds we return with "≤" or "≥" respectively and with the dual solution of the corresponding linear program as a certificate. (Note that Megiddo's algorithm can also compute the the dual variables or the certificate of emptiness if the linear program is infeasible.…”

Section: Claim 22 (Comparing Subroutine)mentioning

confidence: 99%