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: The structural properties of this Voronoi diagram are analyzed and it is shown that its combinatorial complexity is O(k(n-k), for non-crossing line segments, despite the presence of disconnected regions.
Abstract: Surprisingly, the order-$k$ Voronoi diagram of line segments had received no attention in the computational-geometry literature. It illustrates properties surprisingly different from its counterpart for points; for example, a single order-$k$ Voronoi region may consist of $\Omega(n)$ disjoint faces. We analyze the structural properties of this diagram and show that its combinatorial complexity for $n$ non-crossing line segments is $O(k(n-k))$, despite the disconnected regions. The same bound holds for $n$ intersecting line segments, when $k\geq n/2$. We also consider the order-$k$ Voronoi diagram of line segments that form a planar straight-line graph, and augment the definition of an order-$k$ Voronoi diagram to cover non-disjoint sites, addressing the issue of non-uniqueness for $k$-nearest sites. Furthermore, we enhance the iterative approach to construct this diagram. All bounds are valid in the general $L_p$ metric, $1\leq p\leq \infty$. For non-crossing segments in the $L_\infty$ and $L_1$ metrics, we show a tighter $O((n-k)^2)$ bound for $k>n/2$.
19 citations
Cites methods from "New applications of random sampling..."
...Instead, we use the abstract framework presented in [11,12,29]....
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TL;DR: An algorithm that efficiently counts all intersecting triples among a collection T of triangles in R^3 in nearly quadratic time is presented and it is proved that this counting problem belongs to the 3sum-hard family, and thus the algorithm is likely to be nearly optimal in the worst case.
Abstract: We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R^3 in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. Using a variant of the technique, one can represent the set of all @k triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly quadratic construction time and storage. Our approach also applies to any collection of planar objects of constant description complexity in R^3, with the same performance bounds. We also prove that this counting problem belongs to the 3sum-hard family, and thus our algorithm is likely to be nearly optimal in the worst case.
19 citations
TL;DR: A randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses is given, and it is demonstrated with a performance comparison that for segments that cross many edges, the algorithm is faster than gift-wrapping.
Abstract: The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a ? ( k n 2 ) worst-case running time, where n is the number of input vertices and k is the number of input segments.We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses. We demonstrate with a performance comparison that for segments that cross many edges, our algorithm is faster than gift-wrapping. We also show that a simple algorithm for segment location, which precedes segment insertion, is fast enough never to be a bottleneck in CDT construction. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O ( n log ? n + n log 2 ? k ) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O ( n log ? n ) time, incremental CDT construction is easier to program and competitive in practice.Lastly, we partly extend the analysis (albeit not the linear-time insertion algorithm) to randomized incremental CDT construction in three dimensions.
19 citations
28 Jan 1996
TL;DR: It is shown that linear programming in IRd can be solved deterministically in O(logn(loglogn)d-l) time using linear work in the PRAM model of computation, for any fixed constant d.
Abstract: We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in O(logn(loglogn)d-l) time using linear work on an EREW PRAM. A key component in these algorithms is a new, efficient parallel method for constructing c-nets and c-approximations (which have wide applicability in computational geometry). In addition, we introduce a new deterministic set approximation for range spaces with finite VC-exponent, which we call the b-relative c-approtimation, and we show how such approximations can be efficiently constructed in parallel.
19 citations
Cites result from "New applications of random sampling..."
...The study of random sampling in the design of efficient computational geometry methods really began in earnest with some outstanding early work of Clarkson [ 19 ], Haussler and Welzl [34], and Clarkson and Shor [al]....
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17 Aug 1989
19 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