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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TL;DR: It is shown that the minimum possible size of an ε-net for point objects and line (or rectangle)-ranges in the plane is (slightly) bigger than linear in $\frac{1}{\epsilon}$.
Abstract: We show that the minimum possible size of an e-net for point objects and line (or rectangle)-ranges in the plane is (slightly) bigger than linear in $\frac{1}{\epsilon}$. This settles a problem raised by Matousek, Seidel and Welzl (Proc. 6th Annu. ACM Sympos. Comput. Geom., pp. 16–22, 1990).
55 citations
Cites background from "New applications of random sampling..."
...A linear (in 1/ ) upper bound for the size of -nets has been established for several special geometric cases, such as point objects and halfspace ranges in two and three dimensions, and point objects and disk or pseudo-disk ranges in the plane; see [11], [1], [28], [10], [24],[22] and the survey [14] for some earlier results on the subject....
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TL;DR: In this article, a general lemma on the existence of (1/r)-cuttings of geometric objects in Ed that satisfy certain properties was proved, and the authors used this lemma to construct small cuttings for arrangements of line segments in the plane and arrangements of triangles in 3-space.
Abstract: We prove a general lemma on the existence of (1/r)-cuttings of geometric objects in Ed that satisfy certain properties. We use this lemma to construct (1/r)-cuttings of small size for arrangements of line segments in the plane and arrangements of triangles in 3-space; for line segments in the plane we obtain a cutting of size O(r+Ar2/n2), and for triangles in 3-space our cutting has size O(r2+c+Ar3/n3). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3-space.
55 citations
08 Jun 2009
TL;DR: This work considers the following combinatorial problem, and obtains a randomized polynomial time algorithm that gives an O(log log log k)-approximation for the problem of covering k points by the smallest subset of a given set of triangles.
Abstract: We consider the following combinatorial problem: given a set of n objects (for example, disks in the plane, triangles), and an integer L ≥ 1, what is the size of the smallest subset of these n objects that covers all points that are in at least L of the objects? This is the classic question about the size of an L/n-net for these objects. It is well known that for fairly general classes of geometric objects the size of an L/n-net is O(n/L log n/L). There are some instances where this general bound can be improved, and this improvement is usually due to bounds on the combinatorial complexity (size) of the boundary of the union of these objects. Thus, the boundary of the union of m disks has size O(m), and this translates to an O(n/L) bound on the size of an L/n-net for disks. For m fat triangles, the size of the union boundary is O(m log log m), and this yields L/n-nets of size O(n/L log log n/L). Improved nets directly translate into an upper bound on the ratio between the optimal integral solution and the optimal fractional solution for the corresponding geometric set cover problem. Thus, for covering k points by disks, this ratio is O(1); and for covering k points by fat triangles, this ratio is O(log log k). This connection to approximation algorithms for geometric set cover is a major motivation for attempting to improve bounds on nets. Our main result is an argument that in some cases yields nets that are smaller than those previously obtained from the size of the union boundary. Thus for fat triangles, for instance, we obtain nets of size O(n/L log log log n). We use this to obtain a randomized polynomial time algorithm that gives an O(log log log k)-approximation for the problem of covering k points by the smallest subset of a given set of triangles.
54 citations
TL;DR: A (Las Vegas) randomized algorithm for linear programming in a fixed dimension for which the expected computation time is O(d^{(3 + \varepsilon _d )d} n)$$, where limd→∞εd = 0.5 improves the corresponding worst-case complexity.
Abstract: We give a (Las Vegas) randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is\(O(d^{(3 + \varepsilon _d )d} n)\), where limd→∞ed = 0. This improves the corresponding worst-case complexity,\(O(3^{d^2 } n)\). The method is based on a recent idea of Clarkson. Two variations on the algorithm are examined briefly.
54 citations
Cites background or methods from "New applications of random sampling..."
...In [2], Clarkson discusses the applicat ion of r andom sampling to some problems in Computa t iona l Geometry ....
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...The approach taken here to the search problem is somewhat different from [8] and [4] and is in fact adapted from [2]....
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...Dyer [4] reduced this to O(3a2n) (as did Clarkson [1] independent ly) , and showed that this approach could not be expected to lead to algorithms with complexi ty better than O ( d !n)....
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...The following method (also suggested in [2]) suits us....
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...The essential idea is due to Clarkson [2]....
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01 Jun 1991
TL;DR: By using a duatity transform that is of interest in its own right, it is shown that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space.
Abstract: We present a simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duality transform that is of interest in its own right, we show that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space. In particular, the order-k Voronoi diagram for n sites can be computed in time and optimal space by an on-line randomized incremental algorithm. The time bound can be improved by a logarithmic factor without losing much simplicity. For k≥log2 n, this is optimal for a randomized incremental construction; we show that the expected number of structural changes during the construction is ⊝(nk2). Finally, by going back to primal space, we obtain a dynamic data structure that supports k-nearest neighbor queries, insertions, and deletions in a planar set of sites. The structure promises easy implementation, exhibits a satisfactory expected performance, and occupies no more storage than the current order-k Voronoi diagram.
53 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