A deterministic view of random sampling and its use in geometry
24 Oct 1988-pp 539-549
TL;DR: 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. >
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06 Jan 1988
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.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.
1,163 citations
01 Jan 2007
TL;DR: This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area.
Abstract: About ten years ago, the eld of range searching, especially simplex range searching, was wide open. At that time, neither e cient algorithms nor nontrivial lower bounds were known for most range-searching problems. A series of papers by Haussler and Welzl [161], Clarkson [88, 89], and Clarkson and Shor [92] not only marked the beginning of a new chapter in geometric searching, but also revitalized computational geometry as a whole. Led by these and a number of subsequent papers, tremendous progress has been made in geometric range searching, both in terms of developing e cient data structures and proving nontrivial lower bounds. From a theoretical point of view, range searching is now almost completely solved. The impact of general techniques developed for geometric range searching | "-nets, 1=rcuttings, partition trees, multi-level data structures, to name a few | is evident throughout computational geometry. This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area. Range searching arises in a wide range of applications, including geographic information systems, computer graphics, spatial databases, and time-series databases. Furthermore, a variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R , and let
428 citations
Cites background from "A deterministic view of random samp..."
...Chazelle and Friedman [74] improved the size bound to O(rd), which is optimal in the worst case....
[...]
...Chazelle and Friedman [74] improved the size bound to O(r), which is optimal in the worst case....
[...]
Book•
15 Jun 2011TL;DR: This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.
Abstract: Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.
410 citations
TL;DR: A deterministic algorithm for computing the convex hull of n points inEd in optimalO(n logn+n⌞d/2⌟) time and a by-product of this result is an algorithm for Computing the Voronoi diagram ofn points ind-space in optimal O(nLogn+ n⌜d/ 2⌝) time.
Abstract: We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n?d/2?) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n?d/2?) time.
387 citations
TL;DR: It is shown that M(k/n,V) >= (cn/(k+d)log(n/k) for some constant c so that any two distinct vectors in W differ on at least k indices.
371 citations
References
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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.
Abstract: 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. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
Book•
01 Jan 1987TL;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.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations
"A deterministic view of random samp..." refers background in this paper
...Details can be found in (Clarkson [2], Edelsbrunner [ 7 ])....
[...]
Book•
01 Jan 1979TL;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.
Abstract: Basic enumeration. The sieve. Permutations. Two classical enumeration problems in graph theory. Parity and duality. Connectivity. Factors of graphs. Independent sets of points. Chromatic number. Extremal problems for graphs. Spectra of graphs and random walks. Automorphisms of graphs. Hypergraphs. Ramsey Theory. Reconstruction. Dictionary of the combinatorial phrases and concepts used. Notation. Index of the abbreviations of textbooks and monographs. Subject index. Author index.
1,626 citations
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.
1,227 citations
06 Jan 1988
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.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.
1,163 citations