Parallelism in dynamic well-spaced point sets

Parallel algorithms and dynamic algorithms possess an interesting duality property: compared to sequential algorithms, parallel algorithms improve run-time while preserving work, while dynamic algorithms improve work but typically offer no parallelism. Although they are often considered separately, parallel and dynamic algorithms employ similar design techniques. They both identify parts of the computation that are independent of each other. This suggests that dynamic algorithms could be parallelized to improve work efficiency while preserving fast parallel run-time.

In this paper, we parallelize a dynamic algorithm for well-spaced point sets, an important problem related to mesh refinement in computational geometry. Our parallel dynamic algorithm computes a well-spaced superset of a dynamically changing set of points, allowing arbitrary dynamic modifications to the input set. On an EREW PRAM, our algorithm processes batches of k modifications such as insertions and deletions in O(k log Δ) total work and in O(log Δ) parallel time using k processors, where Δ is the geometric spread of the data, while ensuring that the output is always within a constant factor of the optimal size. EREW PRAM model is quite different from actual hardware such as modern multiprocessors. We therefore describe techniques for implementing our algorithm on modern multi-core computers and provide a prototype implementation. Our empirical evaluation shows that our algorithm can be practical, yielding a large degree of parallelism and good speedups.