A nearly optimal deterministic parallel Voronoi diagram algorithm

Cole, Richard; Goodrich, Michael T.; Dúnlaing, Colm Ó

doi:10.1007/bf01944352

Cited by 15 publications

(7 citation statements)

References 20 publications

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“…However, such algorithms rely on a geometric decomposition of a stored global set of generators-a memory-intensive process for many scientific applications [17,28,18]. Numerous algorithms for direct Voronoi construction on shared-memory devices have been formulated, for instance [5].…”

Section: Problem Statementmentioning

confidence: 99%

A new parallel algorithm for constructing Voronoi tessellations from distributed input data

Starinshak

Owen

Johnson

2014

Computer Physics Communications

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We present a new parallel algorithm for generating consistent Voronoi diagrams from distributed input data for the purposes of simulation and visualization. The algorithm functions by building upon any serial Voronoi tessellation algorithm. The output of such a serial tessellator is used to determine the connectivity of the distributed domains without any assumptions about how points are distributed across those domains, and then in turn to build the portion of the global tessellation local to each domain using information from that domains neighbors. The result is a generalized methodology for adding distributed capabilities to serial tessellation packages. Results from several two-dimensional tests are presented, including strong and weak scaling of its current implementation.

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Section: Problem Statementmentioning

confidence: 99%

A new parallel algorithm for constructing Voronoi tessellations from distributed input data

Starinshak

Owen

Johnson

2014

Computer Physics Communications

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“…The parallel algorithms used to construct Voronoi diagrams for a point set in two‐dimensional space have been researched. For example, Cole and Goodrich provide n processors for a recursive method with a point set of n points. Given a vertical line, Cole's algorithm divides the point set in the plane into two groups of n /2 points, each of which is assigned n /2 processors.…”

Section: Parallel Algorithmmentioning

confidence: 99%

“…To date, there are relatively few approaches that have used parallel computing technology to construct Voronoi diagrams. Cole and Goodrich combined the divide‐and‐conquer algorithm with parallel computing, and Kühn. (2001) used a convex distance function to implement a randomized parallel algorithm .…”

Section: Introductionmentioning

confidence: 99%

A parallel algorithm for constructing Voronoi diagrams based on point‐set adaptive grouping

Wang

Cui

Rui

et al. 2013

Concurrency and Computation

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SUMMARYThis paper presents a parallel algorithm for constructing Voronoi diagrams based on point-set adaptive grouping. The binary tree splitting method is used to adaptively group the point set in the plane and construct sub-Voronoi diagrams for each group. Given that the construction of Voronoi diagrams in each group consumes the majority of time and that construction within one group does not affect that in other groups, the use of a parallel algorithm is suitable. After constructing the sub-Voronoi diagrams, we extracted the boundary points of the four sides of each sub-group and used to construct boundary site Voronoi diagrams. Finally, the sub-Voronoi diagrams containing each boundary point are merged with the corresponding boundary site Voronoi diagrams. This produces the desired Voronoi diagram. Experiments demonstrate the efficiency of this parallel algorithm, and its time complexity is calculated as a function of the size of the point set, the number of processors, the average number of points in each block, and the number of boundary points.

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“…Goodrich [18] has given an algorithm for three-dimensional convex hulls that does optimal work, but has O(log 2 n) run-time, and Amato and Preparata [3] have described an algorithm that runs in O(log n) time but uses O(n 1+ε ) processors for any ε > 0. More recently, Cole et al [13] gave an O(log n log log n)-time and O(n)-processor algorithm for constructing the Voronoi diagram of points.…”

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confidence: 99%

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Rajasekaran¹,

Ramaswami²

2002

Algorithmica

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We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is (n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich,Ó'Dúnlaing, and Yap, which runs in O(log 2 n) time using O(n) processors. We obtain this result by using a new "two-stage" random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size "blow-up" that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection). Introduction.Voronoi diagrams are elegant and versatile geometric structures which have applications for a wide range of problems in computational geometry and other areas. For example, given the Voronoi diagram of a set of line segments, one can easily compute the minimum weight spanning tree of the segments, or the nearest neighbor of each line segment. Voronoi diagrams are, in general, very useful tools for solving various proximity problems efficiently. Certain two-dimensional motion planning problems can also be solved quickly when the Voronoi diagram is available [30]. (Surveys on Voronoi diagrams and their applications can be found in [7] and [16].) In a number of these applications, exploiting parallelism to obtain faster solutions is a desirable goal because real-time solutions are critical. In this paper we develop an optimal parallel randomized algorithm on a PRAM (Parallel Random Access Machine) for constructing the Voronoi diagram of a set of nonintersecting, except possibly at endpoints, line segments in the plane. The first sequential algorithm for this problem was given by Lee and Drysdale [24], which ran in O(n log 2 n) time. This run-time was later improved to O(n log n) in numerous papers [22], [35], [15], which is optimal since constructing the Voronoi diagram takes (n log n) time. (This follows from a simple reduction from sorting.) The

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A nearly optimal deterministic parallel Voronoi diagram algorithm

Cited by 15 publications

References 20 publications

A new parallel algorithm for constructing Voronoi tessellations from distributed input data

A new parallel algorithm for constructing Voronoi tessellations from distributed input data

A parallel algorithm for constructing Voronoi diagrams based on point‐set adaptive grouping

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

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