Document Open Access Logo

Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights

Authors Shangqi Lu, Yufei Tao

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2022.57.pdf
  • Filesize: 0.9 MB
  • 16 pages

Document Identifiers

Author Details

Shangqi Lu
  • Chinese University of Hong Kong, New Territories, Hong Kong
Yufei Tao
  • Chinese University of Hong Kong, New Territories, Hong Kong

Funding

This research was supported in part by GRF Projects 14207820 and 14203421 from HKRGC.

Cite AsGet BibTex

Shangqi Lu and Yufei Tao. Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.57

Abstract

Let P be a set of n points in ℝ^d where each point p ∈ P carries a weight drawn from a commutative monoid (ℳ, +, 0). Given a d-rectangle r_upd (i.e., an orthogonal rectangle in ℝ^d) and a value Δ ∈ ℳ, a range update adds Δ to the weight of every point p ∈ P∩ r_upd; given a d-rectangle r_qry, a range sum query returns the total weight of the points in P ∩ r_qry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of Õ(n) space that handles an update in Õ(T_upd) time and a query in Õ(T_qry) time for arbitrary functions T_upd(n) and T_qry(n) satisfying T_upd ⋅ T_qry = n. The result holds for any fixed dimensionality d ≥ 2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Range Updates
  • Range Sum Queries
  • Data Structures
  • Lower Bounds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Amir Abboud and Søren Dahlgaard. Popular conjectures as a barrier for dynamic planar graph algorithms. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 477-486, 2016. Google Scholar
  2. Jon Louis Bentley. Decomposable searching problems. Information Processing Letters (IPL), 8(5):244-251, 1979. Google Scholar
  3. Thiago Bergamaschi, Monika Henzinger, Maximilian Probst Gutenberg, Virginia Vassilevska Williams, and Nicole Wein. New techniques and fine-grained hardness for dynamic near-additive spanners. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1836-1855, 2021. Google Scholar
  4. Christoph Berkholz, Jens Keppeler, and Nicole Schweikardt. Answering conjunctive queries under updates. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 303-318, 2017. Google Scholar
  5. Christoph Berkholz, Jens Keppeler, and Nicole Schweikardt. Answering ucqs under updates and in the presence of integrity constraints. In Proceedings of International Conference on Database Theory (ICDT), pages 8:1-8:19, 2018. Google Scholar
  6. Christoph Berkholz and Maximilian Merz. Probabilistic databases under updates: Boolean query evaluation and ranked enumeration. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 402-415, 2021. Google Scholar
  7. Katrin Casel and Markus L. Schmid. Fine-grained complexity of regular path queries. In Proceedings of International Conference on Database Theory (ICDT), pages 19:1-19:20, 2021. Google Scholar
  8. Raphaël Clifford, Allan Grønlund, Kasper Green Larsen, and Tatiana Starikovskaya. Upper and lower bounds for dynamic data structures on strings. In Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS), pages 22:1-22:14, 2018. Google Scholar
  9. Soren Dahlgaard. On the hardness of partially dynamic graph problems and connections to diameter. In Proceedings of International Colloquium on Automata, Languages and Programming (ICALP), pages 48:1-48:14, 2016. Google Scholar
  10. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008. Google Scholar
  11. Maximilian Probst Gutenberg, Virginia Vassilevska Williams, and Nicole Wein. New algorithms and hardness for incremental single-source shortest paths in directed graphs. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 153-166, 2020. Google Scholar
  12. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 21-30, 2015. Google Scholar
  13. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. CoRR, abs/1511.06773, 2015. URL: http://arxiv.org/abs/1511.06773.
  14. Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional hardness for sensitivity problems. In Innovations in Theoretical Computer Science (ITCS), pages 26:1-26:31, 2017. Google Scholar
  15. Monika Henzinger, Andrea Lincoln, and Barna Saha. The complexity of average-case dynamic subgraph counting. Electronic Colloquium on Computational Complexity, page 157, 2021. Google Scholar
  16. Nabil Ibtehaz, M. Kaykobad, and M. Sohel Rahman. Multidimensional segment trees can do range updates in poly-logarithmic time. Theoretical Computer Science, 854:30-43, 2021. Google Scholar
  17. Ahmet Kara, Hung Q. Ngo, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Maintaining triangle queries under updates. ACM Transactions on Database Systems (TODS), 45(3):11:1-11:46, 2020. Google Scholar
  18. Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Trade-offs in static and dynamic evaluation of hierarchical queries. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 375-392, 2020. Google Scholar
  19. Joshua Lau and Angus Ritossa. Algorithms and hardness for multidimensional range updates and queries. In Innovations in Theoretical Computer Science (ITCS), pages 35:1-35:20, 2021. Google Scholar
  20. Hung Le, Lazar Milenkovic, Shay Solomon, and Virginia Vassilevska Williams. Dynamic matching algorithms under vertex updates. In Innovations in Theoretical Computer Science (ITCS), pages 96:1-96:24, 2022. Google Scholar
  21. Shangqi Lu and Yufei Tao. Towards optimal dynamic indexes for approximate (and exact) triangle counting. In Proceedings of International Conference on Database Theory (ICDT), pages 6:1-6:23, 2021. Google Scholar
  22. Pushkar Mishra. On updating and querying sub-arrays of multidimensional arrays. CoRR, abs/1311.6093, 2013. URL: http://arxiv.org/abs/1311.6093.
  23. Yufei Tao and Ke Yi. Intersection joins under updates. Journal of Computer and System Sciences (JCSS), 124:41-64, 2022. Google Scholar
  24. Jason Yang and Jun Wan. On updating and querying submatrices. CoRR, abs/2010.13180, 2020. URL: http://arxiv.org/abs/2010.13180.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact