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 Collins' classical quantifier elimination procedure contains most of the ingredients for an efficient point location algorithm in higher-dimensional space, which leads to a polynomial-size data structure that allows us to locate a point among a collection of real algebraic varieties of constant maximum degree.
38 citations
TL;DR: A new randomized incremental algorithm for computing a cutting in an arrangement of n lines in the plane that generates small cuttings whose size is guaranteed to be close to the best known upper bound of J. Matou{s}ek.
Abstract: We present several variants of a new randomized incremental algorithm for computing a cutting in an arrangement of n lines in the plane. The algorithms produce cuttings whose expected size is O(r2), and the expected running time of the algorithms is O(nr). Both bounds are asymptotically optimal for nondegenerate arrangements. The algorithms are also simple to implement, and we present empirical results showing that they perform well in practice. We also present another efficient algorithm (with slightly worse time bound) that generates small cuttings whose size is guaranteed to be close to the best known upper bound of J. Matou{s}ek [Discrete Comput. Geom., 20 (1998), pp. 427--448].
38 citations
05 Jan 1997
TL;DR: A randomized algorithm is presented that computes an {epsilon}-approximation of size O(c{sup 2} log{Sup 2} c) in O(n {sup 2+{delta}} + c{sup 3} log {Sup 3}c log n/c) expected time, where c is the size of the {Epsilon]- approximation with the minimum number of vertices and {delta} is any arbitrarily small positive
Abstract: Given a set S of n points in {Re}{sup 3}, sampled from an unknown bivariate function f (x, y) (i.e., for each point p {element_of} S, z{sub p} = f (x{sub p}, y{sub p})), a piecewise-linear function g(x, y) is called an {epsilon}-approximation of f (x, y) if for every p {element_of} S, {vert_bar}f (x, y) - g (x, y){vert_bar} {le} {epsilon}. The problem of computing an {epsilon}-approximation with the minimum number of vertices is NP-Hard. We present a randomized algorithm that computes an {epsilon}-approximation of size O(c{sup 2} log{sup 2} c) in O(n{sup 2+{delta}} + c{sup 3} log{sup 2}c log n/c) expected time, where c is the size of the {epsilon}-approximation with the minimum number of vertices and {delta} is any arbitrarily small positive number. Under some reasonable assumptions, the size of the output is close to O(c log c) and the expected running time is O(n{sup 2+{delta}}). We have implemented a variant of this algorithm and include some empirical results.
36 citations
01 Sep 1991
TL;DR: It is shown that if for any m-point subset Y contained in X the number of distinct subsets induced by R on Y is bounded by O(m/sup d/) for a fixed integer d, then there is a coloring with discrepancy at most O(n/sup 1/2-1/2d/ n), implying improved upper bounds on the size of in -approximations for (X, R.
Abstract: Let (X, R) be a set system on an n-point set X. For a two-coloring on X, its discrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in R. It is shown that if for any m-point subset Y contained in X the number of distinct subsets induced by R on Y is bounded by O(m/sup d/) for a fixed integer d is a coloring with discrepancy bounded by O(n/sup 1/2-1/2d/ (log n)/sup 1+1/2d/). Also, if any subcollection of m sets of R partitions the points into at most O(m/sup d/) classes, then there is a coloring with discrepancy at most O(n/sup 1/2-1/2d/ n). These bounds imply improved upper bounds on the size of in -approximations for (X, R). All of the bounds are tight up to polylogarithmic factors in the worst case. The results allow the generalization of several results of J. Beck (1984) bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure. >
36 citations
Book•
28 Aug 2011TL;DR: A recognition algorithm (RAST) that works efficiently even when no correspondence or grouping information is given; that is, it works in the presence of large amounts of clutter and with very primitive features form the basis for a simple, efficient, and robust approach to the geometric aspects of 3D object recognition from 2D image.
Abstract: Systems (artificial or natural) for visual object recognition are faced with three fundamental problems: the correspondence problem, the problem of representing 3D shape, and the problem of defining a robust similarity measure between images and views of objects.
In this thesis, I address each of these problems: (1) I present a recognition algorithm (RAST) that works efficiently even when no correspondence or grouping information is given; that is, it works in the presence of large amounts of clutter and with very primitive features. (2) I discuss representations of 3D objects as collections of 2D views for the purposes of visual object recognition. Such representations greatly simplify the problems of model acquisition and representing complex shapes. I present theoretical and empirical evidence that this "view-based approximation" is an efficient, robust, and reliable approach to 3D visual object recognition. (3) I present Bayesian and MDL approaches to the similarity problem that may help us build more robust recognition systems.
These results form the basis for a simple, efficient, and robust approach to the geometric aspects of 3D object recognition from 2D image. The results presented in this thesis also strongly suggest that future research directed towards building reliable recognition systems for real world environments must focus on the non-geometric aspects of visual object recognition, such as statistics and scene interpretation. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
36 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•
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....
[...]
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