单纯形范围报告的最优下界
An Optimal Lower Bound for Simplex Range Reporting
摘要：
我们给出了一个简化和改进的下界的单纯形范围报告问题。我们表明，给定一组$P$的$n$点$\mathbb{R}^d$，任何数据结构，使用$S(n)$空间来回答这样的查询必须有$Q(n)=Ω((n^2/S(n))^{(d-1)/d}+k)$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Afshani [A12]以前的下界，但也许更重要的是，它是第一个严格的下界，用于搜索$d\ge 3$维的单纯形范围的任何变体。我们得到我们的下界，使一个简单的连接，充分研究的问题，在入射几何，使我们能够使用已知的建设在该地区。我们观察到，一个简单的已经存在的结构的一个小的修改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Chazelle和Rosenberg [CR96]和Afshani [A12]基于测度论证的复杂概率证明相比。近线性空间数据结构的紧或几乎紧（直到多对数因子）下界的缺乏是在诸如证明多级数据结构的下界之类的问题上取得进展的主要瓶颈。我们希望这种基于关联几何的新攻击路线能够在该领域取得进一步进展。
英文摘要：
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 $Q(n)=Ω((n^2/S(n))^{(d-1)/d}+k)$ 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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