Bottom-Up Construction and 2:1 Balance Refinement of Linear Octrees in Parallel

Abstract

In this article, we propose new parallel algorithms for the construction and 2:1 balance refinement of large linear octrees on distributed memory machines. Such octrees are used in many problems in computational science and engineering, e.g., object representation, image analysis, unstructured meshing, finite elements, adaptive mesh refinement, and N-body simulations. Fixed-size scalability and isogranular analysis of the algorithms using an MPI-based parallel implementation was performed on a variety of input data and demonstrated good scalability for different processor counts (1 to 1024 processors) on the Pittsburgh Supercomputing Center's TCS-1 AlphaServer. The results are consistent for different data distributions. Octrees with over a billion octants were constructed and balanced in less than a minute on 1024 processors. Like other existing algorithms for constructing and balancing octrees, our algorithms have O(N log N ) work and O(N ) storage complexity. Under reasonable assumptions on the distribution of octants and the work per octant, the parallel time complexity is O( N np log( N np ) + np log np), where N is the size of the final linear octree and np is the number of processors.

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

“…Therefore, all we need to prove is that the near-field graph can be bounded by a constant for any µ-nonuniform distribution in order to extend the communication complexity of Lashuk et al to a µ-nonuniform case. The 2:1 balance refinement of octrees by Sundar et al [31] will yield precisely such a O(1) bound on the near-field graph. Therefore, the communication complexity O √ P (N/P ) 2/3 of Lashuk et al is valid for µ-nonuniform distributions if the hypercube reduce and scatter communication and 2:1 refinement are used.…”

“…Such cases do not exist for a uniform distribution so it is easy to prove that this factor disappears in this case. Furthermore, we will show in section 2.2 that it is still possible to bound the neighbors in the near-field list to O(1) for a nonuniform distribution by using a 2:1 refinement constraint [31] during the tree construction. We now focus on the two discrepancies between the communication complexity of Lashuk et al O √ P (N/P ) 2/3 and Ibeid et al O log P + (N/P ) 2/3 .…”

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

“…In scientific simulation software, the combination of tree-structured grids and space-filling curves has been used in several ways, for example augmented by hashing [41], or for partial differential equation solvers with cache-optimised data administration [42,5]. Octor [43] and Dendro [6] are two examples of parallel octree libraries that have been scaled to 62,000 [44] and 32,000 [45] cores, operating on parent-child pointers and a linearised octant storage, respectively.…”

“…They are constructed by recursively refining mesh elements into a fixed number of equally sized and equally shaped child elements. They are very efficient, in particular in terms of memory requirements and memory access patterns, if represented in a linearised form according to a depth-first traversal of the underlying tree and an ordering of the children prescribed by a space-filling curve [5,6,7]. Depending on the aggressiveness of the encoding, however, this may impede traversal of the mesh or parts thereof in an arbitrary order.…”

2D Adaptive Grid-Based Image Analysis Approach for Biological Networks

“…1) Quadtree: A quadtree is a well-known tree-based data structure used in the adaptive grid concepts, where every internal node has four points and every node represents a quadrant(square or rectangle) [10]- [14]. Recursive quadtree partitioning captures the features in a computational problem domain by setting the minimum and maximum resolution of the computational grid.…”