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 Sep 1995
TL;DR: This work shows that the combinatorial complexity of the vertical decomposition of the ${\le}k-level of the arrangement of the bivariate algebraic functions of constant maximum degree is $O(k^{3+\varepsilon}\psi({n/k}))', which implies the existence of shallow cuttings of small size in arrangements of bivariategebraic functions.
Abstract: Let ${\cal F}$ be a collection of $n$ bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ${\le}k$-level of the arrangement ${\cal A}({\cal F})$ is $O(k^{3+\varepsilon}\psi({n/k}))$, for any $\varepsilon<0$, where $\psi (r)$ is the maximum complexity of the lower envelope of a subset of at most $r$ functions of ${\cal F}$. This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) %dynamic data structures for maintaining bichromatic %closest pairs under a fairly general distance function, and for %maintaining minimum spanning trees of a set of points under an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.
97 citations
Cites background or methods from "New applications of random sampling..."
...A similar algorithm was developed by Clarkson [23] for computing the (≤k)-level in an arrangement of hyperplanes....
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...The nearest-neighbor searching problem, also known as the post-office problem, has been widely studied because of its numerous applications [4, 6, 23, 53], but most of the work to date deals with the case where δ is the L1, L2, or L∞ metric and where S does not change dynamically....
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TL;DR: In this article, the authors present quasi-optimal upper bounds for simplex range searching, where the problem is to preprocess a setP ofn points in the query space so that, given any query simplexq, the points inP źq can be counted or reported efficiently.
Abstract: This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in źd so that, given any query simplexq, the points inP źq can be counted or reported efficiently. Ifm units of storage are available (n 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.
92 citations
Cites methods from "New applications of random sampling..."
...The latter can be done in O(n d+~) expected preprocessing time, using the O(n a +~)-size data structure of Clarkson [ 11 ] (for any fixed e > 0). The first part of the preprocessing involves computing the cumulative weight of the planes lying above each vertex of the arrangement....
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...A typical such application (as in the previous sections) involves the technique of e-nets developed by Haussler and Welzl [25], or the related random sampling technique of Clarkson and Shor [ 11 ], [14]....
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TL;DR: This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(nϵ)-time method by Agarwal and Matoušek a decade ago.
Abstract: We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log3n) expected amortized time, deletions take O(log6n) expected amortized time, and extreme-point queries take O(log2n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(nϵ)-time method by Agarwal and Matousek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).
90 citations
Book•
17 Aug 2011
TL;DR: The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles and extend the analysis to three dimensions, where they concentrate on the case of spheres, which is relevant for the three-dimensional unit-distance problem.
Abstract: The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles. They also extend the analysis to three dimensions, where they concentrate on the case of spheres, which is relevant for the three-dimensional unit-distance problem. They obtain upper bounds for certain quantities. The authors believe that the techniques they use are of independent interest. >
89 citations
Cites methods from "New applications of random sampling..."
...dom sampling, akin to the e-net techniques of Haussler and Weld [24] and the related techniques of Clarkson [ 7 ,8]....
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31 May 2009
TL;DR: Improved approximation factors are obtained for the hitting set or the set cover problems associated with the corresponding range spaces for ε-nets of size O(1/ε log log log 1/ε) for planar point sets and axis-parallel rectangular ranges.
Abstract: We show the existence of e-nets of size O(1/e log log 1/e) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of e-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of sizeO(1/e log log log 1/e) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.
87 citations
Cites background from "New applications of random sampling..."
...Small-size e-Nets for Axis-Parallel Rectangles and Boxes* [Extended Abstract] Boris Aronov Esther Ezra Micha Sharir Department of Computer Department of Computer School of Computer Science Science and Engineering Science Tel Aviv University Polytechnic Institute of NYU Duke University, Durham Tel…...
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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