Load-Balancing for Parallel Delaunay Triangulations

Abstract

Computing the Delaunay triangulation (DT) of a given point set in R D is one of the fundamental operations in computational geometry. Recently, Funke and Sanders [18] presented a divide-and-conquer DT algorithm that merges two partial triangulations by re-triangulating a small subset of their vertices -the border vertices -and combining the three triangulations efficiently via parallel hash table lookups. The input point division should therefore yield roughly equal-sized partitions for good load-balancing and also result in a small number of border vertices for fast merging. In this paper, we present a novel divide-step based on partitioning the triangulation of a small sample of the input points. In experiments on synthetic and real-world data sets, we achieve nearly perfectly balanced partitions and small border triangulations. This almost cuts running time in half compared to non-data-sensitive division schemes on inputs exhibiting an exploitable underlying structure.Recently, we presented a novel divide-and-conquer (D&C) DT algorithm for arbitrary dimension [18] that lends itself equally well to shared and distributed memory parallelism and thus hybrid parallelization. While previous D&C DT algorithms suffer from a complex -often sequential -divide or merge step [12,24], our algorithm reduces the merging of two partial triangulations to re-triangulating a small subset of their vertices -the border vertices -using the same parallel algorithm and combining the three triangulations efficiently via hash table lookups. All steps required for the merging -identification of relevant vertices, triangulation and combining the partial DTs -are performed in parallel.The division of the input points in the divide-step needs to address a twofold sensitivity to the point distribution: the partitions need to be approximately equal-sized for good load-balancing, arXiv:1902.07554v1 [cs.DS]

AMST: Accelerated Minimum Spanning Tree for Scalable Data

“…Funke et al (6) created a divide-and-conquer algorithm for computing triangulations in two dimensions, which is improved by Nguyen et al (7) , Dinas et al (8) and Mikhailov et al (9) . In this algorithm, a line recursively partitions the nodes into two distinct subsets.…”