Abstract
We present an improved space/query-time tradeoff for the general simplex range searching problem, matching known lower bounds up to small polylogarithmic factors. In particular, we construct a linear-space simplex range searching data structure withO(n1−1/d) query time, which is optimal ford=2 and probably also ford>2. Further, we show that multilevel range searching data structures can be built with only a polylogarithmic overhead in space and query time per level (the previous solutions require at least a small fixed power ofn). We show that Hopcroft's problem (detecting an incidence amongn lines andn points) can be solved in time\(n^{4/3} 2^{O(\log ^4 n)}\). In all these algorithms we apply Chazelle's results on computing optimal cuttings.
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Part of this research was done during the First Utrecht Computational Geometry Workshop, supported by the Dutch Organization for Scientific Research (NWO).
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Matoušek, J. Range searching with efficient hierarchical cuttings. Discrete Comput Geom 10, 157–182 (1993). https://doi.org/10.1007/BF02573972
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DOI: https://doi.org/10.1007/BF02573972