On Semialgebraic Range Reporting

TL;DR

This paper advances understanding of semialgebraic range reporting by establishing new lower bounds in higher dimensions and matching upper bounds in 2D, clarifying the complexity of the problem.

Contribution

It proves a lower bound in D-dimensions and demonstrates tight bounds for certain query times, resolving key open issues in semialgebraic range searching.

Findings

Lower bound in D-dimensions for semialgebraic range reporting.

Matching upper bounds for uniform random point sets in 2D.

Clarification of the tightness of existing bounds for specific query times.

Abstract

In the problem of semialgebraic range searching, we are to preprocess a set of points in $\mathbb{R}^D$ such that the subset of points inside a semialgebraic region described by $O(1)$ polynomial inequalities of degree $\Delta$ can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" structures [AMS13,MP15] with almost optimal query time of $Q(n)=O(n^{1-1/D+o(1)})$ were obtained. For "fast query" structures (i.e., when $Q(n)=n^{o(1)}$), it was conjectured that a structure with space $S(n) = O(n^{D+o(1)})$ is possible. The conjecture was refuted recently by Afshani and Cheng [AC21]. In the plane, they proved that $S(n) = \Omega(n^{\Delta+1 - o(1)}/Q(n)^{(\Delta+3)\Delta/2})$ which shows $\Omega(n^{\Delta+1-o(1)})$ space is needed for $Q(n) = n^{o(1)}$. While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of $n$ or $Q(n)$ seem to be tight even for $D=2$, as the current upper bound is $S(n) = O(n^{\boldsymbol{m}+o(1)}/Q(n)^{(\boldsymbol{m}-1)D/(D-1)})$ where $\boldsymbol{m}=\binom{D+\Delta}{D}-1 = \Omega(\Delta^D)$ is the maximum number of parameters to define a monic degree-$\Delta$ $D$-variate polynomial, for any $D,\Delta=O(1)$. In this paper, we resolve two of the issues: we prove a lower bound in $D$-dimensions and show that when $Q(n)=n^{o(1)}+O(k)$, $S(n)=\Omega(n^{\boldsymbol{m}-o(1)})$, which is almost tight as far as the exponent of $n$ is considered in the pointer machine model. When considering the exponent of $Q(n)$, we show that the analysis in [AC21] is tight for $D=2$, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or a new fundamentally different input set is needed to get a better lower bound.