Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

TL;DR

This paper establishes the first nontrivial lower bounds for semialgebraic range searching, disproving a conjecture and showing that fast query solutions require significantly more space than previously known.

Contribution

It provides the first nontrivial lower bounds for semialgebraic range searching, challenging existing conjectures and clarifying the space-query trade-offs.

Findings

Any data structure for concentric circle range reporting with polylogarithmic query time requires near cubic space.

Reporting points between two polynomial curves requires near polynomial space for polylogarithmic query time.

Linear space data structures for 2D ring stabbing have near-linear query time, nearly matching upper bounds.

Abstract

In the semialgebraic range searching problem, we are to preprocess $n$ points in $\mathbb{R}^d$ s.t. for any query range from a family of constant complexity semialgebraic sets, all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem can be solved using $S(n)$ space and with $Q(n)$ query time with $S(n)Q^d(n) = \tilde{O}(n^d)$ and this trade-off is almost tight. Consequently, there exists low space structures that use $\tilde{O}(n)$ space with $O(n^{1-1/d})$ query time and fast query structures that use $O(n^d)$ space with $O(\log^{d} n)$ query time. However, for the general semialgebraic ranges, only low space solutions are known, but the best solutions match the same trade-off curve as the simplex queries. It has been conjectured that the same could be done for the fast query case but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and related problems. We show that any data structure for reporting the points between two concentric circles with $Q(n)$ query time must use $S(n)=\Omega(n^{3-o(1)}/Q(n)^5)$ space, meaning, for $Q(n)=O(\log^{O(1)}n)$, $\Omega(n^{3-o(1)})$ space must be used. We also study the problem of reporting the points between two polynomials of form $Y=\sum_{i=0}^\Delta a_i X^i$ where $a_0, \cdots, a_\Delta$ are given at the query time. We show $S(n)=\Omega(n^{\Delta+1-o(1)}/Q(n)^{\Delta^2+\Delta})$. So for $Q(n)=O(\log^{O(1)}n)$, we must use $\Omega(n^{\Delta+1-o(1)})$ space. For the dual semialgebraic stabbing problems, we show that in linear space, any data structure that solves 2D ring stabbing must use $\Omega(n^{2/3})$ query time. This almost matches the linearization upper bound. For general semialgebraic slab stabbing problems, again, we show an almost tight lower bounds.