An Optimal Lower Bound for Simplex Range Reporting

TL;DR

This paper establishes a tight lower bound for simplex range reporting in higher dimensions, improving previous bounds and connecting the problem to incidence geometry for broader applicability.

Contribution

It provides the first tight lower bound for simplex range searching in dimensions three and above, using a simplified approach linked to incidence geometry.

Findings

Improved lower bounds for near-linear space data structures.

First tight bounds for $d extgreater 2$ in simplex range searching.

Accessible proof technique based on incidence geometry.

Abstract

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.