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Lower bounds for semialgebraic range searching and stabbing problems

Research output: Contribution to book/anthology/report/proceedingArticle in proceedingsResearchpeer-review

In the semialgebraic range searching problem, we are given a set of n points in ℝd 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)Qd(n) = Õ(nd) where the Õ(·) notation hides polylogarithmic factors and this trade-off is tight (up to no(1) factors). Consequently, there exists “low space” structures that use O(n) space with O(n1−1/d) query time and “fast query” structures that use O(nd) space with O(logd+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(n1−1/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 o problem, with Q(n) query time must use S(n) = Ω(n3/Q(n)5) space where the Ω(·) notation hides o o no(1) factors, meaning, for Q(n) = O(logO(1) n), Ω(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 aiXi where values a0,···,a∆ 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∆+1/Q(n)∆2+∆), which shows for Q(n) = O(logO(1) n), o we must use Ω(n∆+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(n2/3) 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.

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
Number of pages15
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Publication yearJun 2021
Article number8
ISBN (Electronic)9783959771849
Publication statusPublished - Jun 2021
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021


Conference37th International Symposium on Computational Geometry, SoCG 2021
LandUnited States
ByVirtual, Buffalo
SeriesLeibniz International Proceedings in Informatics, LIPIcs

Bibliographical note

Publisher Copyright:
© Peyman Afshani and Pingan Cheng; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

    Research areas

  • Algorithms, Computational geometry, Data structures

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