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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06 Jan 2013
TL;DR: The results successfully explain the variation from the nearest-site to the farthest-site geodesic Voronoi diagrams, i.e., from k = 1 to k = n -- 1, and also illustrate the formation of a disconnected Vor onoi region, which does not occur in many commonly used distance metrics, such as the Euclidean, L1, and city metrics.
Abstract: We investigate the higher-order Voronoi diagrams of n point sites with respect to the geodesic distance in a simple polygon with h > 0 polygonal holes and c corners. Given a set of n point sites, the kth-order Voronoi diagram partitions the plane into several regions such that all points in a region share the same k nearest sites. The nearest-site (first-order) geodesic Voronoi diagram has already been well-studied, and its total complexity is O(n+c). On the other hand, Bae and Chwa [3] recently proved that the total complexity of the farthest-site ((n -- 1)st-order) geodesic Voronoi diagram and the number of faces in the diagram are Θ(nc) and Θ(nh), respectively. It is of high interest to know what happens between the first-order and the (n -- 1)st-order geodesic Voronoi diagrams. In this paper we prove that the total complexity of the kth-order geodesic Voronoi diagram is Θ(k(n -- k) + kc), and the number of faces in the diagram is Θ(k(n -- k) + kh). Our results successfully explain the variation from the nearest-site to the farthest-site geodesic Voronoi diagrams, i.e., from k = 1 to k = n -- 1, and also illustrate the formation of a disconnected Voronoi region, which does not occur in many commonly used distance metrics, such as the Euclidean, L1, and city metrics. We show that the kth-order geodesic Voronoi diagram can be computed in O(k2(n+c) log(n+c)) time using an iterative algorithm.
12 citations
Cites background from "New applications of random sampling..."
...Furthermore, there are randomized algorithms [1, 7, 16], on-line algorithms [2, 4], and higherdimensional results [8, 9]....
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TL;DR: Borders are given for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.
Abstract: We use the probabilistic method to obtain versions of the colourful Caratheodory theorem and Tverberg's theorem with tolerance.In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.We prove a bound N = rt + O() for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.
12 citations
Cites background from "New applications of random sampling..."
...Among some notable examples are the crossing number theorem [1, 35], the cutting lemma [13, 14] and the existence of epsilon-nets for families of sets with bounded VC-dimension [20]....
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TL;DR: The analysis uses cuttings, combined with the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ.
Abstract: We show that the combinatorial complexity of the union of n “fat” tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n2+e), for any e > 0;the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3, having arbitrary side lengths, is O(n2+e), for any e > 0 (again, significantly extending the result of Pach et al. [2003]). Finally, our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in R3.
12 citations
Cites background from "New applications of random sampling..."
...The e-net theory [Clarkson 1987; Haussler and Welzl 1987] implies that, with high probability, each simplex of the resulting decomposition is crossed by at most n/r (planes containing) facets of F....
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...The ε-net theory [8, 17] implies that, with high probability, each simplex of the resulting decomposition is crossed by at most n/r (planes containing) facets of F ....
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01 May 1990
TL;DR: The jirst optimal parallel algorithm computing arrangements of hyperplanes in Ed is proposed and computes the arrangement of n hyperplanes within expected logarithmic time on a CRCW-PRAM with O(nd/ log n) processors.
Abstract: We propose the jirst optimal parallel algorithm computing arrangements of hyperplanes in Ed (d 2 2). The algorithm is randomized and computes the arrangement of n hyperplanes within expected logarithmic time on a CRCW-PRAM with O(nd/ log n) processors.
12 citations
01 Mar 1991
TL;DR: This p~per gives a partitioning scheme for the geometric, planar traveling salesman problem, under the Euclidean metric: given a set S of n points in the Diane, given a shortest tour of S, the scheme employs randomization, and gives a tour that can be expected to be short.
Abstract: This p~per gives a partitioning scheme for the geometric, planar traveling salesman problem, under the Euclidean metric: given a set S of n points in the Diane. find a shortest ;losed tour (path) v;siting all the “point’s. The scheme employs randomization, and gives a tour that can be expected to be short, if S satisfies the condition that a random subset R C S has o]: average a tour much shorter than an optimal tour of S. ‘~his condition holds for points independently, identiccdly distributed in the plane, for example, for which a tour within 1 -I-c of shortest can be found in expected time nk22k, whew k = O(log log n)3/e2. One algorithm employed in the scheme is of interest in its own right: when given a simple polygon P, it finds a Steiner triangulation of the interior of P. If P has n sides and perimeter LP, the edges of the triangulation have total length LP O(log n). If this algorithm is applied to a simple polygon induced by a minimum spanning tree of a point set, the result is a Steiner triangulation of the set with total length within a factor of O(log n) of the minimum possible.
12 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]....
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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