Results 1  10
of
32
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 43 (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.
Shape Fitting with Outliers
 SIAM J. Comput
, 2003
"... we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the k ..."
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Cited by 33 (9 self)
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we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the klevel of the arrangement of the subset approximates that of the original arrangement. Using this algorithm, we propose ecient approximation algorithms for shape tting with outliers for various shapes. This is the rst algorithms to handle outliers eciently for the shape tting problems considered.
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 22 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
Fast Algorithms for Computing the Smallest kEnclosing Disc
 In Proc. 11th Annu. European Sympos. Algorithms, volume 2832 of Lect. Notes in Comp. Sci
, 2003
"... We consider the problem of nding, for a given n point set P in the plane and an integer k n, the smallest circle enclosing at least k points of P . We present a randomized algorithm that computes in O(nk) expected time such a circle, improving over previously known algorithms. ..."
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Cited by 22 (4 self)
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We consider the problem of nding, for a given n point set P in the plane and an integer k n, the smallest circle enclosing at least k points of P . We present a randomized algorithm that computes in O(nk) expected time such a circle, improving over previously known algorithms.
On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of ddimensional spheres
"... In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in ..."
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Cited by 20 (8 self)
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In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in dimension d; (2) the convex hull of a set of ddimensional spheres. In particular, given a set of n spheres in dimension d, we show that the worst case complexity of both a single additively weighted Voronoi cell and the convex hull of the set of spheres is d ⌈ Θ(n 2 ⌉). The equivalence between additively weighted Voronoi cells and convex hulls of spheres permits us to compute a single additively weighted Voronoi cell in did ⌈ mension d in worst case optimal time O(n log n+n 2 ⌉).
Efficient Detection of Patterns in 2D Trajectories of Moving Points
 IN: GEOINFORMATICA
, 2007
"... Moving point object data can be analyzed through the discovery of patterns in trajectories. We consider the computational efficiency of detecting four such spatiotemporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., Finding REMO—detecting relative mot ..."
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Cited by 13 (4 self)
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Moving point object data can be analyzed through the discovery of patterns in trajectories. We consider the computational efficiency of detecting four such spatiotemporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., Finding REMO—detecting relative motion patterns in geospatial lifelines, 201–214, (2004). These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns.
Preprocessing Chains for Fast Dihedral Rotations Is Hard or Even Impossible
, 2002
"... We examine a computational geometric problem concerning the structure of polymers. Wemodel a polymer as a polygonal chain in three dimensions. Each edge splits the polymer into two subchains, and a dihedral rotation rotates one of these subchains rigidly about the edge.The problem is to determine, g ..."
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Cited by 12 (1 self)
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We examine a computational geometric problem concerning the structure of polymers. Wemodel a polymer as a polygonal chain in three dimensions. Each edge splits the polymer into two subchains, and a dihedral rotation rotates one of these subchains rigidly about the edge.The problem is to determine, given a chain, an edge, and an angle of rotation, if the motion can be performed without causing the chain to selfintersect. An \Omega (n log n) lower bound on thetime complexity of this problem is known. We prove that preprocessing a chain of n edges and answering n dihedral rotation queriesis 3sumhard, giving strong evidence that \Omega (n2) preprocessing is required to achieve sublinearquery time in the worst case. For dynamic queries, which also modify the chain if the requested dihedral rotation is feasible, we show that answering n queries is by itself 3sumhard, suggesting that sublinear query time is impossible after any amount of preprocessing.
On the Least Median Square Problem
, 2003
"... We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in IR and a parameter k, the problem is equivalent to computing the ..."
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Cited by 8 (2 self)
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We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in IR and a parameter k, the problem is equivalent to computing the slab bounded by two parallel hyperplanes of minimum separation that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds on the computational complexity of these problems.