Parallel Cover Trees and their Applications

The cover tree is the canonical data structure that efficiently maintains a dynamic set of points on a metric space and supports nearest and k-nearest neighbor searches. For most real-world datasets with reasonable distributions (constant expansion rate and bounded aspect ratio mathematically), single-point insertion, single-point deletion, and nearest neighbor search (NNS) only cost logarithmically to the size of the point set. Unfortunately, due to the complication and the use of depth-first traversal order in the cover tree algorithms, we were unaware of any parallel approaches for these cover tree algorithms. This paper shows highly parallel and work-efficient cover tree algorithms that can handle batch insertions (and thus construction) and batch deletions. Assuming constant expansion rate and bounded aspect ratio, inserting or deleting m points into a cover tree with n points takes O(m log n) expected work and polylogarithmic span with high probability. Our algorithms rely on some novel algorithmic insights. We model the insertion and deletion process as a graph and use a maximal independent set (MIS) to generate tree nodes without conflicts. We use three key ideas to guarantee work-efficiency: the prefix-doubling scheme, a careful design to limit the graph size on which we apply MIS, and a strategy to propagate information among different levels in the cover tree. We also use path-copying to make our parallel cover tree a persistent data structure, which is useful in several applications. Using our parallel cover trees, we show work-efficient (or near-work-efficient) and highly parallel solutions for a list of problems in computational geometry and machine learning, including Euclidean minimum spanning tree (EMST), single-linkage clustering, bichromatic closest pair (BCP), density-based clustering and its hierarchical version, and others. To the best of our knowledge, many of them are the first solutions to achieve work-efficiency and polylogarithmic span assuming constant expansion rate and bounded aspect ratio.