Parallel Range and Segment Queries with Augmented Maps
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The range and segment query problems are fundamental problems in computational geometry, and have extensive applications in many domains. Despite the large body of theoretical work on parallel geometry algorithms, efficient implementations can be complicated. We know of very few practical implementations of the algorithms, and most implementations do not have tight theoretical bounds. In this paper we focus on simple and efficient parallel algorithms and implementations for range and segment queries. We propose using a simple framework (the augmented map) to model the problem. Based on the augmented map interface, we develop both multi-level tree structures and sweepline algorithms supporting range and segment queries in two dimensions. For the sweepline algorithms, we also propose a parallel paradigm and show corresponding cost bounds. All of our data structures are work-efficient to build in theory (all need O(n log n) work) and achieve a low parallel depth (poly-logarithmic for the multi-level tree structures, and O(n?) for sweepline algorithms). We have implemented all the data structures described in the paper using a parallel augmented map library. Based on the library each data structure only requires about 100 lines of C++ code. We test their performance on large data sets (up to 108 elements) and a machine with 72-cores (144 hyperthreads). The parallel construction times are all less than 8 seconds in parallel, achieving 33-to-63-fold speedup. Speedups on queries are up to 108-fold. Our sequential range query implementations outperforms the CGAL implementation by more than a factor of 2, and gets an additional 63-fold speedup on all cores.
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