单纯形范围报告的最优下界

P. Afshani, P. Cheng
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引用次数: 2

摘要

给出了单纯形值域报告问题的简化改进下界。我们展示了在$\mathbb{R}^d$中给定一组$P$的$n$点,任何使用$S(n)$空间来回答此类查询的数据结构都必须具有$Q(n)=\Omega((n^2/S(n))^{(d-1)/d}+k)$查询时间,其中$k$是输出大小。对于近线性空间数据结构,即$S(n)=O(n\log^{O(1)}n)$,这改进了Chazelle和Rosenberg [CR96]和afshai [A12]先前的下界,但也许更重要的是,它是有史以来第一个对$d\ge 3$维的单纯形范围搜索的任何变体的紧下界。我们通过与入射几何中研究得很好的问题建立一个简单的联系来获得下界,这使得我们可以使用该区域的已知结构。我们观察到,对一个简单的已经存在的结构进行一个小的修改就可以导致我们的下界。我们相信我们的证明可以被更广泛的受众所接受,至少与之前由Chazelle和Rosenberg [CR96]以及Afshani [A12]基于度量参数的复杂概率证明相比。缺乏近线性空间数据结构的紧或几乎紧(多对数因子)下界是在证明多层数据结构下界等问题上取得进展的主要瓶颈。我们希望,这种基于入射几何的新攻线能够在这一领域取得进一步进展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An Optimal Lower Bound for Simplex Range Reporting
We give a simplified and improved lower bound for the simplex range reporting problem. We show that given a set $P$ of $n$ points in $\mathbb{R}^d$, any data structure that uses $S(n)$ space to answer such queries must have $Q(n)=\Omega((n^2/S(n))^{(d-1)/d}+k)$ query time, where $k$ is the output size. For near-linear space data structures, i.e., $S(n)=O(n\log^{O(1)}n)$, this improves the previous lower bounds by Chazelle and Rosenberg [CR96] and Afshani [A12] but perhaps more importantly, it is the first ever tight lower bound for any variant of simplex range searching for $d\ge 3$ dimensions. We obtain our lower bound by making a simple connection to well-studied problems in incident geometry which allows us to use known constructions in the area. We observe that a small modification of a simple already existing construction can lead to our lower bound. We believe that our proof is accessible to a much wider audience, at least compared to the previous intricate probabilistic proofs based on measure arguments by Chazelle and Rosenberg [CR96] and Afshani [A12]. The lack of tight or almost-tight (up to polylogarithmic factor) lower bounds for near-linear space data structures is a major bottleneck in making progress on problems such as proving lower bounds for multilevel data structures. It is our hope that this new line of attack based on incidence geometry can lead to further progress in this area.
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