Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

Abstract

In the semialgebraic range searching problem, we are given a set of n points in ℝ^d and we want to preprocess the points such that for any range belonging to a family of constant complexity semialgebraic sets (Tarski cells), all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, then the problem is well-understood: it can be solved using S(n) and with Q(n) time with S(n)Q^d(n) = O(n^d) where the O(⋅) notation hides polylogarithmic factors and this trade-off is tight (up to n^o(1) factors). Consequently, there exists low space structures that use O(n) with O(n^{1-1/d}) time and fast query structures that use O(n^d) with O(log^{d+1} n) 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, with O(n) and O(n^{1-1/d}) time. 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 other related problems. More precisely, we show that any data structure for reporting the points between two concentric circles, a problem that we call 2D annulus reporting problem, with Q(n) time must use S(n) = Ω^o(n³/Q(n)⁵) where the Ω^o(⋅) notation hides n^o(1) factors, meaning, for Q(n) = O(log^{O(1)}n), Ω^o(n³) must be used. In addition, we study the problem of reporting the subset of input points between two polynomials of the form Y = ∑_{i=0}^Δ a_i Xⁱ where values a_0,⋯,a_Δ are given at the time, a problem that we call polynomial slab reporting. For this, we show a lower bound of Ω^o(n^{Δ+1}/Q(n)^{Δ²+Δ}), which shows for Q(n) = O(log^{O(1)}n), we must use Ω^o(n^{Δ+1}) space. We also consider the dual problems of semialgebraic range searching, semialgebraic stabbing problems, and present lower bounds for them. In particular, we show that in linear space, any data structure that solves 2D annulus stabbing problems must use Ω(n^{2/3}) time. Note that this almost matches the upper bound obtained by lifting 2D annuli to 3D. Like semialgebraic range searching, we also present lower bounds for general semialgebraic slab stabbing problems. Again, our lower bounds are almost tight for linear size data structures in this case.