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Journal ArticleDOI

New applications of random sampling in computational geometry

Kenneth L. Clarkson1
01 Jun 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 195-222
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].

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Citations
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Proceedings ArticleDOI
01 Sep 1991
TL;DR: An optimal algorithm for computing hyperplane cuttings results in a new kind of cutting, which enjoys all the properties of the previous ones and, in addition, can be refined by composition.
Abstract: An optimal algorithm for computing hyperplane cuttings is given. It results in a new kind of cutting, which enjoys all the properties of the previous ones and, in addition, can be refined by composition. An optimal algorithm for computing the convex hull of a finite point set in any fixed dimension is also given. >

82 citations

01 Apr 2003
TL;DR: The final author version and the galley proof are versions of the publication after peer review and the final published version features the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

80 citations

Journal ArticleDOI
TL;DR: The algorithm is applied to the problem of finding the distance between two $n$-vertex (or $n-facet) convex polyhedra in $d$-space, and to the computation of the smallest ball containing n points in £d-space; for both problems the first subexponential bounds in the arithmetic model of computation are given.
Abstract: An abstract optimization problem (AOP) is a triple $(H,<, \Phi)$ where $H$ is a finite set, $<$ a total order on $2^{H}$ and $\Phi$ an oracle that, for given $F\subseteq G\subseteq H$, either reports that $F=\min_{<}\{F'\mid F'\subseteq G\}$ or returns a set $F'\subseteq G$ with $F' $$e^{2\sqrt{n}+O(\sqrt[4]{n}\ln n)}$$ oracle calls, $n=|H|$. In contrast, any deterministic algorithm needs to make $2^{n}-1$ oracle calls in the worst case. The algorithm is applied to the problem of finding the distance between two $n$-vertex (or $n$-facet) convex polyhedra in $d$-space, and to the computation of the smallest ball containing $n$ points in $d$-space; for both problems we give the first subexponential bounds in the arithmetic model of computation.

76 citations


Cites background from "New applications of random sampling..."

  • ...A flrst nontrivial randomized time bound of O(nbd=2c) was given by Clarkson [ 3 ] (provided the polyhedra are specifled by n points)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors proved full Hausdorff dimension in a variant of the Kakeya problem involving circles in the plane, and also sharp estimates for the relevant maximal function.
Abstract: We prove full Hausdorff dimension in a variant of the Kakeya problem involving circles in the plane, and also sharp estimates for the relevant maximal function. These results can also be formulated in terms of the wave equation in two space variables. A novelty in our approach is the use of ideas from computational geometry.

75 citations


Cites background from "New applications of random sampling..."

  • ...2 is valid with high probability (this argument is from [2])....

    [...]

  • ...j j (1+j j) for any fixed multiindex , uniformly in t 2 [1, 2]....

    [...]

  • ...Next, Theorem 1 may be interpreted in terms of Sobolev spaces: on t 2 [1, 2], kmj f jkL3t L(1)x C kfk3, for any f and any fixed > 0....

    [...]

Proceedings ArticleDOI
07 Jun 1998
TL;DR: A simple, general, randomized technique to reduce certain geo- metric optimization problems to their corresponding decision problems, which increases the expected time complexity by only a constant factor and eliminates extra log- arithmic factors in previous approaches.
Abstract: We propose a simple, general, randomized technique to reduce certain geo- metric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra log- arithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under trans- lation, computing rectilinear p-centers and discrete 1-centers, and solving linear programs with k violations.

75 citations

References
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Book
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.

17,939 citations

01 Jan 1985
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."

6,525 citations


"New applications of random sampling..." refers background in this paper

  • ...(In fact the mapping γ is not unique in this regard: see [13, 23, 2]....

    [...]

Book ChapterDOI
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


"New applications of random sampling..." refers background in this paper

  • ...Vapnik and Chervonenkis [27] have derived general conditions under which several probabilities may be uniformly estimated using one random sample....

    [...]

Book
01 Jan 1978
TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."

3,419 citations