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OutputSensitive Results on Convex Hulls, Extreme Points, and Related Problems
, 1996
"... . We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) ..."
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Cited by 65 (13 self)
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. We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 11/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 11/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in ddimen...
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 56 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
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Cited by 46 (3 self)
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Random Sampling, Halfspace Range Reporting, and Construction of (≤k)Levels in Three Dimensions
 SIAM J. COMPUT
, 1999
"... Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the co ..."
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Cited by 33 (7 self)
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Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the orderk Voronoi diagram in O(n log n + nk log k) expected time.
A Bound on Local Minima of Arrangements that implies the Upper Bound Theorem
 Geom
, 1993
"... This paper shows that the ilevel of an arrangement of hyperplanes in E d has at most \Gamma i+d\Gamma1 d\Gamma1 \Delta local minima. This bound follows from ideas previously used to prove bounds on (k)sets. Using linear programming duality, the Upper Bound Theorem is obtained as a corollary, ..."
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Cited by 15 (0 self)
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This paper shows that the ilevel of an arrangement of hyperplanes in E d has at most \Gamma i+d\Gamma1 d\Gamma1 \Delta local minima. This bound follows from ideas previously used to prove bounds on (k)sets. Using linear programming duality, the Upper Bound Theorem is obtained as a corollary, giving yet another proof of this celebrated bound on the number of vertices of a simple polytope in E d with n facets. 1 Introduction We will need some terminology for arrangements, similar to that in Edelsbrunner 's text[3]. Let A(H) be a simple arrangement of a set H of n hyperplanes in E d . For h 2 H, let the upper halfspace h + be the open halfspace bounded by h that contains (1; 0; : : : ; 0), and let the lower halfspace h \Gamma be the other open halfspace bounded by h. Say that x 2 E d is above h 2 H if x 2 h + , and below h if x 2 h \Gamma . The ilevel of A(H) is the boundary of the set of points that are below no more than i hyperplanes of H. Thus for example the 0...
OutputSensitive Construction Of Convex Hulls
, 1995
"... The construction of the convex hull of a finite point set in a lowdimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates efficient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point ..."
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Cited by 3 (0 self)
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The construction of the convex hull of a finite point set in a lowdimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates efficient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point set and the size of the output polytope. Two new, simple, optimal, outputsensitive algorithms are presented in two dimensions and a simple, optimal, outputsensitive algorithm is presented in three dimensions. In four dimensions, we give the first outputsensitive algorithm that is within a polylogarithmic factor of optimal. In higher fixed dimensions, we obtain an algorithm that is optimal for sufficiently small output sizes and is faster than previous methods for sublinear output sizes; this result is further improved in even dimensions. Although the focus of the thesis is on the convex hull problem, applications of our techniques to many related problems in computational geometry are al...