Merging free trees in parallel for efficient voronoi diagram construction

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

doi:10.1007/bfb0032049

Cited by 27 publications

(17 citation statements)

References 23 publications

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“…Researchers have also shown that this divide-and-conquer technique can be applied in designing asymptotically efficient parallel algorithms [1,6]. In this case, however, the approach does not appear to be practical.…”

Section: Parallel Delaunay Triangulationmentioning

confidence: 99%

A Parallel Dynamic-Mesh Lagrangian Method for Simulation of Flows With Dynamic Interfaces

Antaki¹,

Blelloch²,

Ghattas³

et al. 2000

ACM/IEEE SC 2000 Conference (SC'00)

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Abstract. Many important phenomena in science and engineering, including our motivating problem of microstructural blood flow, can be modeled as flows with dynamic interfaces. The major challenge faced in simulating such flows is resolving the interfacial motion. Lagrangian methods are ideally suited for such problems, since interfaces are naturally represented and propagated. However, the material description of motion results in dynamic meshes, which become hopelessly distorted unless they are regularly regenerated. Lagrangian methods are particularly challenging on parallel computers, because scalable dynamic mesh methods remain elusive. Here, we present a parallel dynamic mesh Lagrangian method for flows with dynamic interfaces. We take an aggressive approach to dynamic meshing by triangulating the propagating grid points at every timestep using a scalable parallel Delaunay algorithm. Contrary to conventional wisdom, we show that the costs of the geometric components (triangulation, coarsening, refinement, and partitioning) can be made small relative to the flow solver.1. Motivation. Flows with dynamic interfaces arise in many fluid-solid and fluid-fluid interaction problems, and are among the most difficult computational problems in continuum mechanics. Examples abound in the aerospace, automotive, biomedical, chemical, marine, materials, and wind engineering sciences. These include large-amplitude vibrations of such flexible aerodynamic components as high aspect ratio wings and blades; flows of mixtures and slurries; wind-induced deformation of towers, antennas, and lightweight bridges; hydrodynamic flows around offshore structures; interaction of biofluids with elastic vessels; and materials phase transition problems.One of our target problems is modeling the flow of blood at the microstructural level. Blood is a mixture of deformable cellular bodies (primarily red blood cells) suspended in an essentially Newtonian fluid (plasma). The cells themselves are composed of a fluid gel (hemoglobin) contained within a solid membrane. The flow of the fluid-solid mixture is illustrated in Figure 1.1a, which depicts a snapshot of blood flow through a 12m arteriole. Because of the computational difficulties of resolving numerous dynamically deforming cellular interfaces, as illustrated in Figure 1.1b, no one to date has simulated realistic blood flows at the microstructural level. Yet such simulations are necessary in order to gain a better understanding of blood damage-which is central to improved artificial organ design-and for the development of more rational macroscopic blood models.We are specifically interested in simulating the flow of blood in artificial heart assist devices, such as the axial flow blood pump being developed at the University of Pittsburgh Medical Center [16]. Such pumps have regions in which the length scales of interest are of the order of tens of cell diameters, for example in bearing journals and near blade tips. Significant blood damage can occur in these "hot spots." Here, macroscopic ...

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Section: Parallel Delaunay Triangulationmentioning

confidence: 99%

A Parallel Dynamic-Mesh Lagrangian Method for Simulation of Flows With Dynamic Interfaces

Antaki¹,

Blelloch²,

Ghattas³

et al. 2000

ACM/IEEE SC 2000 Conference (SC'00)

View full text Add to dashboard Cite

show abstract

“…Its boundary is a convex polygon whose corners are sites of S. The distance |x − y| between two points in the plane is Euclidean distance. For any point x, the distance of 9 The distinct pairs could have one site in common, so this implies no three sites are collinear. x to a nonempty set T of points is…”

Section: The Convex Hull H (S)mentioning

confidence: 99%

“…24 As noted previously, this was one of the main difficulties of our paper. In the earlier version [9], not just a ruling for the beachline but the full beachline was precomputed, with n log(n) processors overall (CREW). 24 Solution to the recurrence F(2k) = 3F(k).…”

Section: The Beachline Description Is Next Written To An Array G Of Smentioning

confidence: 99%

“…Earlier algorithms [1], [9] were presented for CREW 5 machines. The algorithm in [1] used n processors and took O(log 2 (n)) parallel time; that in [2] used n log(n) processors and took O(log(n) log log(n)) parallel time.…”

Section: Introductionmentioning

confidence: 99%

“…In [9] the beachlines were all precomputed independently in O(log(n)) parallel steps using fractional cascading [3]. This allowed the algorithm to proceed without difficulty but was processor-inefficient, since the precomputed structures were built using n log(n) processors.…”

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confidence: 99%

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A nearly optimal deterministic parallel Voronoi diagram algorithm

1996

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Abstract.We describe an n-processor, O(log(n) log log(n))-time CRCW algorithm to construct the Voronoi diagram for a set of n point-sites in the plane.Key Words. Voronoi diagram, Parallel algorithm.1. Introduction. Outline of the Algorithm. The Voronoi diagram is a geometric structure of great computational interest: see [5] for a useful survey. This paper addresses the problem of constructing the diagram in parallel, given as input a set of n points ("sites") in the plane. The Voronoi diagram for a set of sites is the locus of points equidistant from two closest sites: Figure 1 illustrates a diagram with 32 sites.The model of parallelism we assume is a CRCW PRAM, a system of independent processors accessing a shared random-access memory, where the same memory cell can be read by several processors simultaneously (concurrent read) and written by several processors simultaneously (concurrent write). Write-conflicts are resolved arbitrarily: the model of computation is an ARBITRARY CRCW PRAM.Each processor is assumed capable of exact rational and integer arithmetic in unit time.Earlier algorithms [1], [9] were presented for CREW 5 machines. The algorithm in [1] used n processors and took O(log 2 (n)) parallel time; that in [2] used n log(n) processors and took O(log(n) log log(n)) parallel time. Our algorithm reduces the overall work (parallel time × number of processors) to O(n log(n) log log(n)), while maintaining a runtime of O(log(n) log log(n)). Both of these figures are within the factor log log(n)

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Ultrafast Expected Time Parallel Algorithms

MacKenzie

Stout

1998

Journal of Algorithms

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It has been shown previously that sorting n items into n locations with a polynomial number of processors requires Ω(log n/ log log n) time. We sidestep this lower bound with the idea of Padded Sorting, or sorting n items into n + o(n) locations. Since many problems do not rely on the exact rank of sorted items, a Padded Sort is often just as useful as an unpadded sort. Our algorithm for Padded Sort runs on the Tolerant CRCW PRAM and takes Θ(log log n/ log log log n) expected time using n log log log n/ log log n processors, assuming the items are taken from a uniform distribution. Using similar techniques we solve some computational geometry problems, including Voronoi Diagram, with the same processor and time bounds, assuming points are taken from a uniform distribution in the unit square. Further, we present an Arbitrary CRCW PRAM algorithm to solve the Closest Pair problem in constant expected time with n processors regardless of the distribution of points. All of these algorithms achieve linear speedup in expected time over their optimal serial counterparts.

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Merging free trees in parallel for efficient voronoi diagram construction

Cited by 27 publications

References 23 publications

A Parallel Dynamic-Mesh Lagrangian Method for Simulation of Flows With Dynamic Interfaces

A Parallel Dynamic-Mesh Lagrangian Method for Simulation of Flows With Dynamic Interfaces

A nearly optimal deterministic parallel Voronoi diagram algorithm

Ultrafast Expected Time Parallel Algorithms

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