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# Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

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## Acknowledgements

The authors would like to thank Esther Ezra for sparking the initial ideas behind the proof.

## Cite As

Peyman Afshani and Pingan Cheng. Lower Bounds for Semialgebraic Range Searching and Stabbing Problems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.8

## 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 query 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) space and with Q(n) query time with S(n)Q^d(n) = Õ(n^d) where the Õ(⋅) 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) space with O(n^{1-1/d}) query time and "fast query" structures that use O(n^d) space with O(log^{d+1} 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, with O(n) space and Õ(n^{1-1/d}) query 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) query time must use S(n) = Ω^o(n³/Q(n)⁵) space where the Ω^o(⋅) notation hides n^o(1) factors, meaning, for Q(n) = O(log^{O(1)}n), Ω^o(n³) space 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 query time, a problem that we call polynomial slab reporting. For this, we show a space 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}) query 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Computational Geometry
• Data Structures and Algorithms

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