research-article

Improved pointer machine and I/O lower bounds for simplex range reporting and related problems

Author:

Peyman AfshaniAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 339 - 346

https://doi.org/10.1145/2261250.2261301

Published: 17 June 2012 Publication History

Abstract

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems.

Simplex range reporting is the problem of storing n points in a data structure such that the $k$ points that lie inside a query simplex can be outputted efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using ~O( (n / Q(n))^d) space in which the ~O(.) notation either hides a polylogarithmic or an n^ε factor for any constant ε > 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who proved a space lower bound of Ω(n^d-ε-dγ) for pointer machine data structures that can answer queries in O(n^γ + k) time.

For data structures with Q(n) + O(k) query time, we improve the space lower bound to Ω( (n/Q(n))^d / 2^{O(√log Q(n))}). Not only this reduces the overhead from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our lower bound is within a o(log n) factor of the conjectured trade-off curve.

By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we also study the external memory model and offer a new framework for proving lower bound in this model. For the first time we show that answering simplex range reporting queries with Q(n) + k/B I/Os requires Ω(B (n/(BQ(n)))^d / 2^{O(√log Q(n))} space in which B is the block size.

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P. Afshani, L. Arge, and K. G. Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. submitted to SoCG'12.

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Cited By

Afshani PCheng P(2023)Lower Bounds for Semialgebraic Range Searching and Stabbing ProblemsJournal of the ACM10.1145/357857470:2(1-26)Online publication date: 18-Apr-2023
https://dl.acm.org/doi/10.1145/3578574
Afshani PKillmann R(2023)Rectangle stabbing and orthogonal range reporting lower bounds in moderate dimensionsComputational Geometry10.1016/j.comgeo.2022.101959111(101959)Online publication date: Apr-2023
https://doi.org/10.1016/j.comgeo.2022.101959
Afshani PCheng P(2023)On Semialgebraic Range ReportingDiscrete & Computational Geometry10.1007/s00454-023-00574-171:1(4-39)Online publication date: 24-Oct-2023
https://doi.org/10.1007/s00454-023-00574-1
Show More Cited By

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Improved pointer machine and I/O lower bounds for simplex range reporting and related problems

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cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright © 2012 ACM.

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Published: 17 June 2012

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SoCG '12

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Afshani PCheng P(2023)Lower Bounds for Semialgebraic Range Searching and Stabbing ProblemsJournal of the ACM10.1145/357857470:2(1-26)Online publication date: 18-Apr-2023
https://dl.acm.org/doi/10.1145/3578574
Afshani PKillmann R(2023)Rectangle stabbing and orthogonal range reporting lower bounds in moderate dimensionsComputational Geometry10.1016/j.comgeo.2022.101959111(101959)Online publication date: Apr-2023
https://doi.org/10.1016/j.comgeo.2022.101959
Afshani PCheng P(2023)On Semialgebraic Range ReportingDiscrete & Computational Geometry10.1007/s00454-023-00574-171:1(4-39)Online publication date: 24-Oct-2023
https://doi.org/10.1007/s00454-023-00574-1
Afshani PMarx D(2021)A lower bound for dynamic fractional cascadingProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458197(2229-2248)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458197
Goswami MJacob RPagh RSuciu DTao YWei Z(2020)On the I/O Complexity of the k-Nearest Neighbors ProblemProceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3375395.3387649(205-212)Online publication date: 14-Jun-2020
https://dl.acm.org/doi/10.1145/3375395.3387649
Afshani PCheng P(2020)2D Generalization of Fractional Cascading on Axis-aligned Planar Subdivisions2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00072(716-727)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00072
Afshani PDriemel ACzumaj A(2018)On the complexity of range searching among curvesProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175328(898-917)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175328
Afshani PSheng CTao YWilkinson BChekuri C(2014)Concurrent range reporting in two-dimensional spaceProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634147(983-994)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634147
Afshani PArge LLarsen KDey TWhitesides S(2012)Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine modelProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261299(323-332)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261299
Larsen KNguyen HDey TWhitesides S(2012)Improved range searching lower boundsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261275(171-178)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261275

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