Parallel Range, Segment and Rectangle Queries with Augmented Maps

Abstract

The range, segment and rectangle query problems are fundamental problems in computational geometry, and have extensive applications in many domains. Despite the significant theoretical work on these problems, efficient implementations can be complicated. We know of very few practical implementations of the algorithms in parallel, and most implementations do not have tight theoretical bounds. In this paper, we focus on simple and efficient parallel algorithms and implementations for range, segment and rectangle queries, which have tight worst-case bound in theory and good parallel performance in practice. We propose to use 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, segment and rectangle 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 (O(n log n) sequential work) and achieve a low parallel depth (polylogarithmic for the multi-level tree structures, and O(n ) for sweepline algorithms). The query time is almost linear to the output size.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 10 8 elements) and a machine with 72-cores (144 hyperthreads). The parallel construction achieves 32-68x speedup. Speedup numbers on queries are up to 126-fold. Our sequential implementation outperforms the CGAL library by at least 2x in both construction and queries. Our sequential implementation can be slightly slower than the R-tree in the Boost library in some cases (0.6-2.5x), but has significantly better query performance (1.6-1400x) than Boost.Many data structures are designed for solving range, segment and rectangle queries such as range trees [17], segment trees [19], kd-trees [16], R-trees [14,50,52], priority trees [44], and many others [58,30,42,48], which are then applied to later research in various areas [2,38,4,20,35,25,54,5,8,7,18,49,21]. Many of them are augmented tree structures. The standard range tree has construction time O(n log n) and query time O(k + log 2 n) for input size n and output size k. Using fractional cascading [41,57], the query time can be reduced to O(k + log n). We did not employ such optimizations, but instead show that the our version using parallel augmented maps achieve good parallelism in practice, and is simple and easy for engineering. The terminology "segment tree" refers to different data structures in the literature. Our version is similar to some previous works [27,12,3]. Previous solutions for rectangle queries usually use combinations of range trees, segment trees, interval trees, and priority

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