1,405 research outputs found
On the Combinatorial Complexity of Approximating Polytopes
Approximating convex bodies succinctly by convex polytopes is a fundamental
problem in discrete geometry. A convex body of diameter
is given in Euclidean -dimensional space, where is a constant. Given an
error parameter , the objective is to determine a polytope of
minimum combinatorial complexity whose Hausdorff distance from is at most
. By combinatorial complexity we mean the
total number of faces of all dimensions of the polytope. A well-known result by
Dudley implies that facets suffice, and a dual
result by Bronshteyn and Ivanov similarly bounds the number of vertices, but
neither result bounds the total combinatorial complexity. We show that there
exists an approximating polytope whose total combinatorial complexity is
, where conceals a
polylogarithmic factor in . This is a significant improvement
upon the best known bound, which is roughly .
Our result is based on a novel combination of both old and new ideas. First,
we employ Macbeath regions, a classical structure from the theory of convexity.
The construction of our approximating polytope employs a new stratified
placement of these regions. Second, in order to analyze the combinatorial
complexity of the approximating polytope, we present a tight analysis of a
width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering.
Finally, we use a deterministic adaptation of the witness-collector technique
(developed recently by Devillers et al.) in the context of our stratified
construction.Comment: In Proceedings of the 32nd International Symposium Computational
Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and
Computational Geometr
Approximate range searching☆☆A preliminary version of this paper appeared in the Proc. of the 11th Annual ACM Symp. on Computational Geometry, 1995, pp. 172–181.
AbstractThe range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ε>0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance εw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in Rd can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/ε)d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/ε)d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/ε)d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times
Economical Delone Sets for Approximating Convex Bodies
Convex bodies are ubiquitous in computational geometry and optimization theory. The high combinatorial complexity of multidimensional convex polytopes has motivated the development of algorithms and data structures for approximate representations. This paper demonstrates an intriguing connection between convex approximation and the classical concept of Delone sets from the theory of metric spaces. It shows that with the help of a classical structure from convexity theory, called a Macbeath region, it is possible to construct an epsilon-approximation of any convex body as the union of O(1/epsilon^{(d-1)/2}) ellipsoids, where the center points of these ellipsoids form a Delone set in the Hilbert metric associated with the convex body. Furthermore, a hierarchy of such approximations yields a data structure that answers epsilon-approximate polytope membership queries in O(log (1/epsilon)) time. This matches the best asymptotic results for this problem, by a data structure that both is simpler and arguably more elegant
Approximate Nearest-Neighbor Search for Line Segments
Approximate nearest-neighbor search is a fundamental algorithmic problem that
continues to inspire study due its essential role in numerous contexts. In
contrast to most prior work, which has focused on point sets, we consider
nearest-neighbor queries against a set of line segments in , for
constant dimension . Given a set of disjoint line segments in
and an error parameter , the objective is to
build a data structure such that for any query point , it is possible to
return a line segment whose Euclidean distance from is at most
times the distance from to its nearest line segment. We
present a data structure for this problem with storage and query time , where is the spread of the set of
segments . Our approach is based on a covering of space by anisotropic
elements, which align themselves according to the orientations of nearby
segments.Comment: 20 pages (including appendix), 5 figure
Approximation algorithm for the kinetic robust K-center problem
AbstractTwo complications frequently arise in real-world applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the k-center problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard k-center problem, the objective is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center, or equivalently, the smallest radius such that S can be covered by k disks of this radius. In the discrete k-center problem the disk centers are drawn from the points of S, and in the absolute k-center problem the disk centers are unrestricted.We generalize this problem in two ways. First, we assume that points are in continuous motion, and the objective is to maintain a solution over time. Second, we assume that some given robustness parameter 0<t⩽1 is given, and the objective is to compute the smallest radius such that there exist k disks of this radius that cover at least ⌈tn⌉ points of S. We present a kinetic data structure (in the KDS framework) that maintains a (3+ε)-approximation for the robust discrete k-center problem and a (4+ε)-approximation for the robust absolute k-center problem, both under the assumption that k is a constant. We also improve on a previous 8-approximation for the non-robust discrete kinetic k-center problem, for arbitrary k, and show that our data structure achieves a (4+ε)-approximation. All these results hold in any metric space of constant doubling dimension, which includes Euclidean space of constant dimension
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