Tile & Merge: Distributed Delaunay Triangulations for Cloud Computing

Abstract

Motivated by the needs of a scalable out-of-core surface reconstruction algorithm available on the cloud, this paper addresses the computation of distributed Delaunay triangulations of massive point sets. The proposed algorithm takes as input a point cloud and first partitions it across multiple processing elements into tiles of relatively homogeneous point sizes. The distributed computation and communication between processing elements is orchestrated so that each one discovers the Delaunay neighbors of its input points within the theoretical overall Delaunay triangulation of all points and computes locally a partial view of this triangulation. This approach prevents memory limitations by never materializing the global triangulation. This efficiency is due to our proposed uncentralized model to represent, manage and locally construct the triangulation corresponding to each tile. The point set is first partitioned into non-overlapping tiles, then we construct within each tile the Delaunay triangulation of the local points and a minimal set of replicated foreign points in order to capture the simplices spanning multiple tiles. Inspired by the star splaying approach for Delaunay triangulation computation/repair, communication is limited to exchanging points of potential Delaunay neighbors across tiles. Therefore, our method is guaranteed to reconstruct, within each tile, a triangulation that contains the star of its local points, as though it were computed within the Delaunay triangulation of all points. The proposed algorithm is implemented with Spark for the scheduling and C ++ for the geometric computations. This allows both an optimal scheduling on multiple machines and efficient low-level computation. The results show the efficiency of our algorithm in terms of speedup and strong scaling on a classical Spark configuration with both synthetic and real use case datasets.

Provably Consistent Distributed Delaunay Triangulation

“…In this paper, we extend the preliminary work of [Caraffa et al, 2019] on distributed Delaunay triangulation computation, by providing a (i) theoretical proof of the consistency for the resulting Distributed Delaunay Triangulation. In addition, we also discuss (ii) implementation details on the Spark architecture and propose (iii) a deeper scalability analysis of the overall algorithm in the results section.…”

“…It is a common approach to tackle large data structures by operating on a spatial partitioning of them, for instance for combinatorial maps [Damiand et al, 2018] or 3D triangular meshes [Cabiddu, Attene, 2015]. The Tile & Merge approach [Caraffa et al, 2019] works similarly by decomposing the overall Delaunay Triangulation into a set of DT local to each input tile. However, contrary to other approaches, it is not a proper partition as cells (triangles in 2D) across tile boundaries are replicated in the triangulations of their neighboring tiles and points are replicated in the tiles of all their Delaunay neighbors.…”

“…In addition, we also discuss (ii) implementation details on the Spark architecture and propose (iii) a deeper scalability analysis of the overall algorithm in the results section. After providing an overview of the method proposed by [Caraffa et al, 2019], we review their algorithm and propose a theoretical proof of consistency in section 2. Then, section 3 presents the implementation details, followed by the evaluation 4.…”

“…Figure 3 and algorithm 1 [Caraffa et al, 2019] describe the Tile & Merge approach. Pi are point sets available after tiling the input data.…”

“…A further feature is that tile triangulations are simplified by removing foreign points that are not adjacent to any local points in order to limit the growth of the tile triangulations resulting from receiving candidate Delaunay neighbors that are eventually not Delaunay neighbors of their local point at convergence. Algorithm 1: Tile and merge [Caraffa et al, 2019] Input: Point set P Output: Distributed DT of P :…”