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: A survey of theoretical results and the main techniques in geometric range searching is presented, which can be used as subroutines in solutions to many seemingly unrelated problems.
Abstract: In geometric range searching, algorithmic problems of the following type are considered. Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Similar questions can be asked for point sets in higher dimensions, with triangles replaced by simplices or by more complicated shapes. Algorithms of this type are of crucial importance in computational geometry, as they can be used as subroutines in solutions to many seemingly unrelated problems, which are often motivated by practical applications, for instance in computer graphics (ray tracing, hidden-surface removal etc.). We present a survey of theoretical results and the main techniques in geometric range searching.
153 citations
Cites methods from "New applications of random sampling..."
...In the context of partition trees such a construction was introduced in Dobkin and Edelsbrunner [1987]. Recently it has been used quite frequently, e....
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...In the context of partition trees such a construction was introduced in Dobkin and Edelsbrunner [1987]. Recently it has been used quite frequently, e.g., see Overmars et al. [1990], Chazelle et al....
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TL;DR: The first nontrivial general upper bound for this problem is shown, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2λq(n) for some constantq depending on the shape and degree of the surfaces.
Abstract: We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d?3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(nd?1+?), for any ?>0; the constant of proportionality depends on ?, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2?q(n)) for some constantq depending on the shape and degree of the surfaces (where ?q(n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n2+?), and give several applications of the new bounds.
152 citations
Cites background from "New applications of random sampling..."
...Since no cell of ~'~r intersects any surface of ~ in its interior, and since each such cell is defined in terms of only a constant number of surfaces of Z, it follows, using standard probabilistic arguments [10], [23], that, with high probability, no cell of J [ ) intersects in its interior more than O((n/r) log r) surfaces of E....
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21 Oct 2007
TL;DR: It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
Abstract: For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or "approximately balanced," formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
149 citations
11 Jan 2004
TL;DR: The first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a non-degenerate point set in the plane is presented, using an interesting variant of the author's randomized optimization technique.
Abstract: We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a non-degenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fiexed dimension d ≥ 3, the expected time bound is O(nd-1), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linear-programming-type problems; some other applications of this technique are briefly mentioned.
142 citations
Cites methods from "New applications of random sampling..."
...A simple random sampling algorithm was suggested by Clarkson [11]....
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01 Jan 1993
TL;DR: The theme of this chapter is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry, which is to analyze a randomized algorithm as if it were running backwards in time, from output to input.
Abstract: The theme of this chapter is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as “analyze a randomized algorithm as if it were running backwards in time, from output to input.” We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others.
141 citations
Cites background from "New applications of random sampling..."
...In the mid 80's Ken Clarkson started to create and apply his random sampling technique, which in the mean time has developed into a surprising general framework with numerous applications [15, 16, 17, 18]....
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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