Document

On Semialgebraic Range Reporting

File

LIPIcs.SoCG.2022.3.pdf
• Filesize: 0.75 MB
• 14 pages

Cite As

Peyman Afshani and Pingan Cheng. On Semialgebraic Range Reporting. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.3

Abstract

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.

Subject Classification

ACM Subject Classification
• Theory of computation → Computational geometry
Keywords
• Computational Geometry
• Range Searching
• Data Structures and Algorithms
• Lower Bounds

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. Peyman Afshani. Improved pointer machine and I/O lower bounds for simplex range reporting and related problems. In Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry, SoCG '12, pages 339-346, New York, NY, USA, 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2261250.2261301.
2. Peyman Afshani. A new lower bound for semigroup orthogonal range searching. In 35th International Symposium on Computational Geometry, volume 129 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 3, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019.
3. Peyman Afshani and Pingan Cheng. Lower Bounds for Semialgebraic Range Searching and Stabbing Problems. In Kevin Buchin and Éric Colin de Verdière, editors, 37th International Symposium on Computational Geometry (SoCG 2021), volume 189 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:15, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.8.
4. Pankaj K. Agarwal. Simplex range searching and its variants: a review. In A journey through discrete mathematics, pages 1-30. Springer, Cham, 2017.
5. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. Efficient algorithm for generalized polynomial partitioning and its applications. SIAM J. Comput., 50(2):760-787, 2021. URL: https://doi.org/10.1137/19M1268550.
6. Pankaj K. Agarwal, Jiří Matoušek, and Micha Sharir. On range searching with semialgebraic sets. II. SIAM J. Comput., 42(6):2039-2062, 2013. URL: https://doi.org/10.1137/120890855.
7. Sunil Arya, Theocharis Malamatos, and David M. Mount. On the importance of idempotence. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 564-573. ACM, New York, 2006. URL: https://doi.org/10.1145/1132516.1132598.
8. Sunil Arya, David M. Mount, and Jian Xia. Tight lower bounds for halfspace range searching. Discrete Comput. Geom., 47(4):711-730, 2012. URL: https://doi.org/10.1007/s00454-012-9412-x.
9. Hervé Brönnimann, Bernard Chazelle, and János Pach. How hard is half-space range searching? Discrete Comput. Geom., 10(2):143-155, 1993. URL: https://doi.org/10.1007/BF02573971.
10. Bernard Chazelle. Lower bounds on the complexity of polytope range searching. J. Amer. Math. Soc., 2(4):637-666, 1989. URL: https://doi.org/10.2307/1990891.
11. Bernard Chazelle. Lower bounds for orthogonal range searching. I. The reporting case. J. Assoc. Comput. Mach., 37(2):200-212, 1990. URL: https://doi.org/10.1145/77600.77614.
12. Bernard Chazelle. Lower bounds for orthogonal range searching. II. The arithmetic model. J. Assoc. Comput. Mach., 37(3):439-463, 1990. URL: https://doi.org/10.1145/79147.79149.
13. Bernard Chazelle and Burton Rosenberg. Simplex range reporting on a pointer machine. Comput. Geom., 5(5):237-247, 1996. URL: https://doi.org/10.1016/0925-7721(95)00002-X.
14. Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors. Handbook of discrete and computational geometry. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2018. Third edition of [ MR1730156].
15. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Ann. of Math. (2), 181(1):155-190, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.2.
16. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete Comput. Geom., 10(2):157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
17. Jiří Matoušek. Geometric range searching. ACM Comput. Surv., 26(4):421-461, 1994. URL: https://doi.org/10.1145/197405.197408.
18. Jiří Matoušek and Zuzana Patáková. Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom., 54(1):22-41, 2015. URL: https://doi.org/10.1007/s00454-015-9701-2.
19. Andrew Chi-Chih Yao and F. Frances Yao. A general approach to d-dimensional geometric queries (extended abstract). In Robert Sedgewick, editor, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, May 6-8, 1985, Providence, Rhode Island, USA, pages 163-168. ACM, 1985. URL: https://doi.org/10.1145/22145.22163.
20. A. Young. On Quantitative Substitutional Analysis. Proc. Lond. Math. Soc., 33:97-146, 1901. URL: https://doi.org/10.1112/plms/s1-33.1.97.
X

Feedback for Dagstuhl Publishing