Optimal Randomized Parallel Algorithms for Computational Geometry I
TL;DR: In this paper, the authors present parallel algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting, which have running time of O(logn)$ using $n$ processors, with very high probability.
Abstract: We present parallel algorithms for some fundamental problems in computational geometry which have running time of $O(logn)$ using $n$ processors, with very high probability (approaching 1 as $n~ \rightarrow~ \infty$). These include planar point location, triangulation and trapezoidal decomposition. We also present optimal algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on CREW PRAM model and have optimal processor-time product which improve on the previously best known algorithms of Atallah and Goodrich [3] for these problems. The crux of these algorithms is a useful data structure which emulates the plane sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk [22] Reif and Valiant [21] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.
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TL;DR: Techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time are surveyed.
28 citations
01 Sep 1991
TL;DR: The use of randomization in dynamic search structures by means of a technique called dynamic sampling is investigated and an efficient algorithm for dynamic point location in 3-D partitions induced by a set of possibly interesting polygons in R/sup 3/ is given.
Abstract: The use of randomization in dynamic search structures by means of a technique called dynamic sampling is investigated. In particular, an efficient algorithm for dynamic (logarithmic time) point location in 3-D partitions induced by a set of possibly interesting polygons in R/sup 3/ is given. The expected running time of the algorithm on a random sequence of updates is close to optimal. Efficient algorithms for dynamic nearest-k-neighbor queries and half space range queries in R/sup d/ are also given. >
27 citations
06 Jul 2020
TL;DR: A strong theoretical analysis is provided showing that for n points in any constant dimension, the standard incremental algorithm is inherently parallel, and it is shown that for problems where the size of the support set can be bounded by a constant, the depth of the configuration dependence graph is shallow.
Abstract: The randomized incremental convex hull algorithm is one of the most practical and important geometric algorithms in the literature. Due to its simplicity, and the fact that many points or facets can be added independently, it is also widely used in parallel convex hull implementations. However, to date there have been no non-trivial theoretical bounds on the parallelism available in these implementations. In this paper, we provide a strong theoretical analysis showing that the standard incremental algorithm is inherently parallel. In particular, we show that for n points in any constant dimension, the algorithm has O(log n) dependence depth with high probability. This leads to a simple work-optimal parallel algorithm with polylogarithmic span with high probability. Our key technical contribution is a new definition and analysis of the configuration dependence graph extending the traditional configuration space, which allows for asynchrony in adding configurations. To capture the "true" dependence between configurations, we define the support set of configuration c to be the set of already added configurations that it depends on. We show that for problems where the size of the support set can be bounded by a constant, the depth of the configuration dependence graph is shallow (O(log n) with high probability for input size n). In addition to convex hull, our approach also extends to several related problems, including half-space intersection and finding the intersection of a set of unit circles. We believe that the configuration dependence graph and its analysis is a general idea that could potentially be applied to more problems.
26 citations
TL;DR: A randomized algorithm for SBR that tries to sort the permutation by repeatedly performing a random oriented reversal, and gives empirical evidence that this process indeed succeeds with high probability on a random permutation.
21 citations
TL;DR: GPC techniques can be applied to a wide variety of geometric problems, including triangulation of point sets, two-set dominance counting, ECDF searching, finding two-and three-dimensional maximal points, the reconstruction of trees from their traversals, counting inversions in a permutation, and matching parentheses.
Abstract: We introduce a generic problem component that captures the most common, difficult “kernel” of many problems. This kernel involves general prefix computations (GPC). GPC's lower bound complexity of Ω(n logn) time is established, and we give optimal solutions on the sequential model inO(n logn) time, on the CREW PRAM model inO(logn) time, on the BSR (broadcasting with selective reduction) model in constant time, and on mesh-connected computers inO(√n) time, all withn processors, plus anO(log2
n) time solution on the hypercube model. We show that GPC techniques can be applied to a wide variety of geometric (point set and tree) problems, including triangulation of point sets, two-set dominance counting, ECDF searching, finding two-and three-dimensional maximal points, the reconstruction of trees from their traversals, counting inversions in a permutation, and matching parentheses.
21 citations
References
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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
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.
3,760 citations
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.
1,163 citations
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TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.
Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.
847 citations
TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.
Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.
810 citations