Proceedings ArticleDOI
A deterministic view of random sampling and its use in geometry
Bernard Chazelle,Joel Friedman +1 more
- pp 539-549
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It is shown how to compute, in polynomial time, a simplicial packing of size O(r/sup d/) that covers d-space, each of whose simplices intersects O(n/r) hyperplanes.Abstract:
A number of efficient probabilistic algorithms based on the combination of divide-and-conquer and random sampling have been recently discovered. It is shown that all those algorithms can be derandomized with only polynomial overhead. In the process. results of independent interest concerning the covering of hypergraphs are established, and various probabilistic bounds in geometry complexity are improved. For example, given n hyperplanes in d-space and any large enough integer r, it is shown how to compute, in polynomial time, a simplicial packing of size O(r/sup d/) that covers d-space, each of whose simplices intersects O(n/r) hyperplanes. It is also shown how to locate a point among n hyperplanes in d-space in O(log n) query time, using O(n/sup d/) storage and polynomial preprocessing. >read more
Citations
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New lower bounds for ε-nets
Abstract: Following groundbreaking work by Haussler and Welzl (1987), the use of small e-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest e-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in double-struck R4 by a family of half-spaces such that the size of any e-net for them is Ω(1/e log 1/e) (Pach and Tardos). The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in double-struck Rd, for any d ≥ 4, to show that the general upper bound, O(d/e log 1/e), of Haussler and Welzl for the size of the smallest e-nets is tight. © Andrey Kupavskii, Nabil H. Mustafa, and Janos Pach.
Proceedings ArticleDOI
Optimizing wireless networks for heterogeneous spatial loads
Balaji Rengarajan,G. de Veciana +1 more
TL;DR: A grid-based approximation algorithm to compute the placement of access points that minimizes the number of access Points required while ensuring that the received SNR at each location is sufficient to meet the offered load at that location.
(Extended Abstract) AND THE COMPLEXITY OF THE V-C DIMENSION
TL;DR: This work characterize precisely the complex- ity of several natural computational problems in NP, which have been proposed but not categorized sat- isfactorily in the literature: Computing the Vapnik- Chervonenkis dimension of a 0-1 matrix; finding the minimum dominating set of a tournament mv; satisfying a Boolean expression by perturbing the de- fault truth assignment.
Time-Space Trade-os for Voronoi Diagrams
TL;DR: A randomized s-workspace algorithm for VD( S) in expected time O((n 2 =s) logs + n logs log s).
Posted Content
On Planar Visibility Counting Problem.
TL;DR: In this article, a randomized algorithm was proposed to compute the exact answer of the visibility counting problem for any constant number, where 0 0 is an arbitrary constant number and n is the number of disjoint line segments.
References
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Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
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Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
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Combinatorial problems and exercises
TL;DR: In this article, the authors present a dictionary of combinatorial phrases and concepts used in graph theory, including the sieve, the sieving, and the graph sieve.
Journal ArticleDOI
On the ratio of optimal integral and fractional covers
TL;DR: It is shown that the ratio of optimal integral and fractional covers of a hypergraph does not exceed 1 + log d, where d is the maximum degree and this theorem may replace probabilistic methods in certain circumstances.
Proceedings ArticleDOI
Applications of random sampling in computational geometry, II
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.