Load-Balancing for Parallel Delaunay Triangulations

Computing the Delaunay triangulation (DT) of a given point set in $\mathbb{R}^D$ is one of the fundamental operations in computational geometry. Recently, Funke and Sanders (2017) 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.Comment: Short version submitted to EuroPar 201

One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

Algorithms for Triangles, Cones & Peaks

Three different geometric objects are at the center of this dissertation: triangles, cones and peaks. In computational geometry, triangles are the most basic shape for planar subdivisions. Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc. We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set. Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of $k$ cones drawn around it. They are used to aid the construction of Euclidean minimum spanning trees or in wireless networks for topology control and routing. We present the first implementation of an optimal $\mathcal{O}(n \log n)$-time sweepline algorithm to construct Yao graphs. One metric to quantify the importance of a mountain peak is its isolation. Isolation measures the distance between a peak and the closest point of higher elevation. Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms. We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes

Efficient Generating And Processing Of Large-Scale Unstructured Meshes

Unstructured meshes are used in a variety of disciplines to represent simulations and experimental data. Scientists who want to increase accuracy of simulations by increasing resolution must also increase the size of the resulting dataset. However, generating and processing a extremely large unstructured meshes remains a barrier. Researchers have published many parallel Delaunay triangulation (DT) algorithms, often focusing on partitioning the initial mesh domain, so that each rectangular partition can be triangulated in parallel. However, the comproblems for this method is how to merge all triangulated partitions into a single domain-wide mesh or the significant cost for communication the sub-region borders. We devised a novel algorithm --Triangulation of Independent Partitions in Parallel (TIPP) to deal with very large DT problems without requiring inter-processor communication while still guaranteeing the Delaunay criteria. The core of the algorithm is to find a set of independent} partitions such that the circumcircles of triangles in one partition do not enclose any vertex in other partitions. For this reason, this set of independent partitions can be triangulated in parallel without affecting each other. The results of mesh generation is the large unstructured meshes including vertex index and vertex coordinate files which introduce a new challenge \-- locality. Partitioning unstructured meshes to improve locality is a key part of our own approach. Elements that were widely scattered in the original dataset are grouped together, speeding data access. For further improve unstructured mesh partitioning, we also described our new approach. Direct Load which mitigates the challenges of unstructured meshes by maximizing the proportion of useful data retrieved during each read from disk, which in turn reduces the total number of read operations, boosting performance

DeWall: A fast divide and conquer Delaunay triangulation algorithm in Ed

The paper deals with Delaunay Triangulations (DT) in Ed space. This classic computational geometry problem is studied from the point of view of the efficiency, extendibility to any dimensionality, and ease of implementation. A new solution to DT is proposed, based on an original interpretation of the well-known Divide and Conquer paradigm. One of the main characteristics of this new algorithm is its generality: it can be simply extended to triangulate point sets in any dimension. The technique adopted is very efficient and presents a subquadratic behaviour in real applications in E', although its computational complexity does not improve the theoretical bounds reported in the literature. An evaluation of the performance on a number of datasets is reported, together with a comparison with other DT algorithms.

Algorithm Libraries for Multi-Core Processors

By providing parallelized versions of established algorithm libraries, we ease the exploitation of the multiple cores on modern processors for the programmer. The Multi-Core STL provides basic algorithms for internal memory, while the parallelized STXXL enables multi-core acceleration for algorithms on large data sets stored on disk. Some parallelized geometric algorithms are introduced into CGAL. Further, we design and implement sorting algorithms for huge data in distributed external memory

Theoretically-Efficient and Practical Parallel DBSCAN

The DBSCAN method for spatial clustering has received significant attention due to its applicability in a variety of data analysis tasks. There are fast sequential algorithms for DBSCAN in Euclidean space that take $O(n\log n)$ work for two dimensions, sub-quadratic work for three or more dimensions, and can be computed approximately in linear work for any constant number of dimensions. However, existing parallel DBSCAN algorithms require quadratic work in the worst case, making them inefficient for large datasets. This paper bridges the gap between theory and practice of parallel DBSCAN by presenting new parallel algorithms for Euclidean exact DBSCAN and approximate DBSCAN that match the work bounds of their sequential counterparts, and are highly parallel (polylogarithmic depth). We present implementations of our algorithms along with optimizations that improve their practical performance. We perform a comprehensive experimental evaluation of our algorithms on a variety of datasets and parameter settings. Our experiments on a 36-core machine with hyper-threading show that we outperform existing parallel DBSCAN implementations by up to several orders of magnitude, and achieve speedups by up to 33x over the best sequential algorithms