High-Performance Computation of Distributed-Memory Parallel 3D Voronoi and Delaunay Tessellation

Abstract

ABSTRACT. Computing a Voronoi or Delaunay tessellation from a set of points is a core part of the analysis of many simulated and measured datasets: N-body simulations, molecular dynamics codes, and LIDAR point clouds are just a few examples. Such computational geometry methods are common in data analysis and visualization; but as the scale of simulations and observations surpasses billions of particles, the existing serial and shared-memory algorithms no longer suffice. A distributed-memory scalable parallel algorithm is the only feasible approach. The primary contribution of this paper is a new parallel Delaunay and Voronoi tessellation algorithm that automatically determines which neighbor points need to be exchanged among the subdomains of a spatial decomposition. Other contributions include periodic and wall boundary conditions, comparison of our method using two popular serial libraries, and application to numerous science datasets.E-mail addresses: tpeterka@mcs.anl.gov, dmitriy@mrzv.org, cphillips@anl.gov. DISCLAIMER. This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California.

Analysis of Satellite Constellations for the Continuous Coverage of Ground Regions

“…It is also possible to first compute the Voronoi diagrams directly and then obtain the Delaunay triangulation. There are many efficient algorithms for computing either of the two and we do not elaborate on them here [34][35][36][37].…”

Block-parallel data analysis with DIY2

“…The first analysis code computes a Voronoi and Delaunay tessellation in parallel at large scale. We ported a parallel algorithm [31], originally implemented using DIY1, to DIY2. Our dataset contains 1024 3 dark matter tracer particles computed by the HACC cosmology code [17].…”

Efficient Delaunay Tessellation through K-D Tree Decomposition

“…The closest work to ours is Peterka et. al [32] who, based on a 3D regular domain decomposition, demonstrated parallel strong scaling efficiency of up to 90% for 2048 3 points uniformly distributed over 128Ki MPI processes. The efficiency dropped, however, to as low as 14% when dark matter particles from a late-stage cosmological simulation were used as input because of their unbalanced distribution.…”

“…Peterka et al [32] introduce Algorithm 1 to compute the distributed Delaunay tessellation. compute the Delaunay tessellation of the points…”

“…Delaunay edges are undirected: if a point p needs to be connected to a point q by a Delaunay edge, then q needs to be connected to p. This symmetry allows the above algorithm to exchange points directly, skipping the circumsphere exchange. As Peterka et al [32] show, the result of the above algorithm is a distributed Delaunay tessellation. This result is guaranteed to be correct under the following assumption.…”