2026-07-23T13:53:55Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16011 2024-03-06T10:56:43Z ddc:004 open_access

On Semialgebraic Range Reporting Afshani, Peyman Cheng, Pingan Computational Geometry Range Searching Data Structures and Algorithms Lower Bounds Semialgebraic range searching, arguably the most general version of range searching, is a fundamental problem in computational geometry. In the problem, we are to preprocess a set of points in ℝ^D such that the subset of points inside a semialgebraic region described by a constant number of polynomial inequalities of degree Δ can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" data structures [Agarwal et al., 2013; Matoušek and Patáková, 2015] with almost optimal query time of Q(n) = O(n^{1-1/D+o(1)}) were obtained. For "fast query" data structures (i.e., when Q(n) = n^{o(1)}), it was conjectured that a similar improvement is possible, i.e., it is possible to achieve space S(n) = O(n^{D+o(1)}). The conjecture was refuted very recently by Afshani and Cheng [Afshani and Cheng, 2021]. In the plane, i.e., D = 2, they proved that S(n) = Ω(n^{Δ+1 - o(1)}/Q(n)^{(Δ+3)Δ/2}) which shows Ω(n^{Δ+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 best known upper bounds have S(n) = O(n^{m+o(1)}/Q(n)^{(m-1)D/(D-1)}) where m = binom(D+Δ,D)-1 = Ω(Δ^D) is the maximum number of parameters to define a monic degree-Δ D-variate polynomial, for any constant dimension D and degree Δ. In this paper, we resolve two of the issues: we prove a lower bound in D-dimensions, for constant D, and show that when the query time is n^{o(1)}+O(k), the space usage is Ω(n^{m-o(1)}), which almost matches the Õ(n^{m}) upper bound and essentially closes the problem for the fast-query case, 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 [Afshani and Cheng, 2021] 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 to obtain better lower bounds a new fundamentally different input set needs to be constructed. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Peyman Afshani and Pingan Cheng 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.3 urn:nbn:de:0030-drops-160117 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.3 eng https://creativecommons.org/licenses/by/4.0/legalcode