New applications of random sampling in computational geometry
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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01 Jan 2000
6 citations
Posted Content•
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TL;DR: In this paper, it was shown that the reduced Euler characteristic of the poset of faces equals zero whenever n odd, regardless of the position of the sites in the set S of n points in general position.
Abstract: Given a set S of n points in general position we consider all kth order Voronoi diagrams on S for k n simultaneously We deduce symmetry relations for the number of faces number of vertices and number of circles of certain orders These symmetry relations are independent of the position of the sites in S As a consequence we show that the reduced Euler characteristic of the poset of faces equals zero whenever n odd
6 citations
Cites background from "New applications of random sampling..."
...Given a set S of n points in general position we consider all k th order Voronoi diagrams on S for k n simultaneously We deduce symmetry relations for the number of faces number of vertices and number of circles of certain orders These symmetry relations are independent of the position of the sites…...
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08 Nov 1998
TL;DR: It is shown how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k and the first optimal randomized algorithm for the construction of the (/spl les/k)-level in an arrangement of n planes in three dimensions is obtained.
Abstract: Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the (/spl les/k)-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk/sup 2/) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the order-k Voronoi diagram in O(n log n+nk log k) expected time.
6 citations
13 Dec 2000
TL;DR: The study of discrepancy theory predates complexity theory and a wealth of mathematical techniques can be brought to bear to prove nontrivial derandomization results, which constitutes the discrepancy method.
Abstract: In 1935, van der Corput asked the following question: Given an infinite sequence of reals in [0, 1], define D(n) = sup0≤x≤1||Sn ∩ [O, x]| - nx|, where Sn consists of the first n elements in the sequence Is it possible for D(n) to stay in O(1)? Many years later, Schmidt proved that D(n) can never be in o(log n) In other words, there are limitations on how well the discrete distribution, x → |Sn ∩ [0, x]|, can simulate the continuous one, x → nx The study of this intriguing phenomenon and its numerous variants related to the irregularities of distributions has given rise to discrepancy theory The relevance of the subject to complexity theory is most evident in the study of probabilistic algorithms Suppose that we feed a probabilistic algorithm not with a perfectly random sequence of bits (as is usually required) but one that is only pseudorandom or even deterministic Should performance necessarily suffer? In particular, suppose that one could trade an exponential-size probability space for one of polynomial size without letting the algorithm realize the change This form of derandomization can be expressed by saying that a very large distribution can be simulated by a small one for the purpose of the algorithm Put differently, there exists a measure with respect to which the two distributions have low discrepancy The study of discrepancy theory predates complexity theory and a wealth of mathematical techniques can be brought to bear to prove nontrivial derandomization results The pipeline of ideas that flows from discrepancy theory to complexity theory constitutes the discrepancy method We give a few examples in this survey A more thorough treatment is given in our book [15] We also briefly discuss the relevance of the discrepancy method to complexity lower bounds
6 citations
Cites background from "New applications of random sampling..."
...Cuttings are among the most useful, versatile tools in computational geometry, as they lay the grounds for efficient divide-and-conquer [1,2,20,26,27]....
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25 Oct 2008
TL;DR: The combinatorial complexity of the union of n infinite cylinders in R3, having arbitrary radii, is O(n2+epsiv), for any epsiv >0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir, who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders.
Abstract: We show that the combinatorial complexity of the union of n infinite cylinders in R3, having arbitrary radii, is O(n2+epsiv), for any epsiv >0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir, who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir, in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in. Finally, we extend our technique to the case of "cigars'' of arbitrary radii (that is, Minkowski sums of line-segments and balls), and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in for the restricted case where all cigars are (nearly) equal-radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders, and is significantly simpler than the proof presented in [3].
6 citations
Cites background or methods from "New applications of random sampling..."
...We use a divide-and-conquer approach, based on (1/r)cuttings[ 9 , 8]. Specifically, we projectall the cylindersin K onto the xy-plane, thereby obtaining a set K0 of n infinite strips....
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...ory [ 9 , 16] implies that, with high probability, each simplex of the resulting decomposition is crossed by at most n/r0 lines of L. We pick one sample R0 for which this property holds, and fix it in the throughout analysis....
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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]....
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Book•
01 Jan 1973
4,243 citations
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....
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Book•
01 Jan 1978TL;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