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We study scalable parallel computational geometry algorithms for the coarse grained multicomputer model: p processors solving a problem on n data items, were each processor has O(n/p)≫O(1) local memory and all processors are connected via some arbitrary interconnection network (e.g. mesh, hypercube, fat tree). We present O(Tsequential/p+Ts(n, p)) time scalable parallel algorithms for several computational geometry problems. Ts(n, p) refers to the time of a global sort operation. Our results are independent of the multicomputer’s interconnection network. Their time complexities become optimal when Tsequential/p dominates Ts(n, p) or when Ts(n, p) is optimal. This is the case for several standard architectures, including meshes and hypercubes, and a wide range of ratios n/p that include many of the currently available machine configurations. Our methods also have some important practical advantages: For interprocessor communication, they use only a small fixed number of one global routing operation, global sort, and all other programming is in the sequential domain. Furthermore, our algorithms use only a small number of very large messages, which greatly reduces the overhead for the communication protocol between processors. (Note however, that our time complexities account for the lengths of messages.) Experiments show that our methods are easy to implement and give good timing results.

The subtree prune and regraft distance (dSPR) between phylogenetic trees is important both as a general means of comparing phylogenetic tree topologies as well as a measure of lateral gene transfer (LGT). Although there has been extensive study on the computation of dSPR and similar metrics between rooted trees, much less is known about SPR distances for unrooted trees, which often arise in practice when the root is unresolved. We show that unrooted SPR distance computation is NP-Hard and verify which techniques from related work can and cannot be applied. We then present an efficient heuristic algorithm for this problem and benchmark it on a variety of synthetic datasets. Our algorithm computes the exact SPR distance between unrooted tree, and the heuristic element is only with respect to the algorithm’s computation time. Our method is a heuristic version of a fixed parameter tractability (FPT) approach and our experiments indicate that the running time behaves similar to FPT algorithms. For real data sets, our algorithm was able to quickly compute dSPR for the majority of trees that were part of a study of LGT in 144 prokaryotic genomes. Our analysis of its performance, especially with respect to searching and reduction rules, is applicable to computing many related distance measures.

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