research-article

Tight Space Bounds for Two-Dimensional Approximate Range Counting

Authors:

Ke YiAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 2

Article No.: 23, Pages 1 - 17

https://doi.org/10.1145/3205454

Published: 04 June 2018 Publication History

Abstract

We study the problem of two-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n× n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, it can return the number of points in P ∩ R with additive error ε n. A well-known solution for this problem is obtained by using ε-approximation, which is a subset A⊆ P that can estimate the number of points in P ∩ R with the number of points in A ∩ R. It is known that an ε-approximation of size O(1/ε log ^2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε).

The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points. In this article, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses O(1/ε(log ²1/ε + log n)) bits and answers queries with error ε n. We then prove a lower bound that any data structure that answers queries with error ε n will have to use Ω(1/ε (log ²1/ε + log n)) bits. Our lower bound is information-theoretic: We show that there is a collection of 2^Ω(nlog n) point sets with large union combinatorial discrepancy and thus are hard to distinguish unless we use Ω(nlog n) bits.

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Cited By

Friedman RShahout R(2022)Box queries over multi-dimensional streamsInformation Systems10.1016/j.is.2022.102086109(102086)Online publication date: Nov-2022
https://doi.org/10.1016/j.is.2022.102086
Cormode GGarofalakis MShekelyan MLibkin LPichler RGuagliardo P(2021)Data-Independent Space Partitionings for SummariesProceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3452021.3458316(285-298)Online publication date: 20-Jun-2021
https://dl.acm.org/doi/10.1145/3452021.3458316
Shekelyan MDignos AGamper JGarofalakis M(2021)Approximating Multidimensional Range Counts with Maximum Error Guarantees2021 IEEE 37th International Conference on Data Engineering (ICDE)10.1109/ICDE51399.2021.00141(1595-1606)Online publication date: Apr-2021
https://doi.org/10.1109/ICDE51399.2021.00141

Index Terms

Tight Space Bounds for Two-Dimensional Approximate Range Counting
1. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
      1. Cell probe models and lower bounds
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 2

April 2018

339 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3196491

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 June 2018

Accepted: 01 April 2018

Revised: 01 March 2018

Received: 01 December 2015

Published in TALG Volume 14, Issue 2

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HKRGC
Fundamental Research Funds for the Central Universities
Research Funds of Renmin University of China
National Natural Science Foundation of China

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Cited By

Friedman RShahout R(2022)Box queries over multi-dimensional streamsInformation Systems10.1016/j.is.2022.102086109(102086)Online publication date: Nov-2022
https://doi.org/10.1016/j.is.2022.102086
Cormode GGarofalakis MShekelyan MLibkin LPichler RGuagliardo P(2021)Data-Independent Space Partitionings for SummariesProceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3452021.3458316(285-298)Online publication date: 20-Jun-2021
https://dl.acm.org/doi/10.1145/3452021.3458316
Shekelyan MDignos AGamper JGarofalakis M(2021)Approximating Multidimensional Range Counts with Maximum Error Guarantees2021 IEEE 37th International Conference on Data Engineering (ICDE)10.1109/ICDE51399.2021.00141(1595-1606)Online publication date: Apr-2021
https://doi.org/10.1109/ICDE51399.2021.00141

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