Quasi-optimal upper bounds for simplex range searching and new zone theorems
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This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ℜd so that, given any query simplexq, the points inP ∩q can be counted or reported efficiently. Ifm units of storage are available (n <m <n d ), then we show that it is possible to answer any query inO(n 1+ɛ/m 1/d ) query time afterO(m 1+ɛ) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieveO(logn) query time at the expense ofO(n d+ɛ) storage for any fixed ɛ > 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.
Key wordsComputational geometry Range searching Space-time tradeoff
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