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

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 $\mathcal{O}(N\log N)$ work and $\mathcal{O}(N)$ storage complexity. Under reasonable assumptions on the distribution of octants and the work per octant, the parallel time complexity is $\mathcal{O}(\frac{N}{n_p}\log(\frac{N}{n_p})+n_p\log n_p)$, where $N$ is the size of the final linear octree and $n_p$ is the number of processors.