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Range Counting Oracles for Geometric Problems

Authors Anne Driemel , Morteza Monemizadeh , Eunjin Oh , Frank Staals , David P. Woodruff

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Range counting oracles
  • minimum spanning trees
  • Earth Mover’s Distance

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Abstract

In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size n in a discrete space [Δ]^d, where queries can be made to an oracle that responds to orthogonal range counting requests. The query complexity of an optimization problem is measured by the number of oracle queries required to compute an estimator for the problem. We investigate two problems in this framework, the Euclidean Minimum Spanning Tree (MST) and Earth Mover Distance (EMD). For EMD, we show the existence of an estimator that approximates the cost of EMD with O(log Δ)-relative error and O(nΔ/(s^{1+1/d}))-additive error using O(s polylog Δ) range counting queries for any parameter s with 1 ≤ s ≤ n. Moreover, we prove that this bound is tight. For MST, we demonstrate that the weight of MST can be estimated within a factor of (1 ± ε) using Õ(√n) range counting queries.

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Anne Driemel, Morteza Monemizadeh, Eunjin Oh, Frank Staals, and David P. Woodruff. Range Counting Oracles for Geometric Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.42

Author Details

Anne Driemel
  • University of Bonn, Germany
Morteza Monemizadeh
  • Department of Mathematics and Computer Science, TU Eindhoven, the Netherlands
Eunjin Oh
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Oh, Eunjin: Supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.RS-2024-00440239, Sublinear Scalable Algorithms for Large-Scale Data Analysis) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2024-00358505).
  • Woodruff, David P.: supported in part by Office of Naval Research award number N000142112647 and a Simons Investigator Award.

Acknowledgements

This research was initiated during the Workshop "Massive Data Models and Computational Geometry" held at the University of Bonn in September 2024 and funded by DFG, German Research Foundation, through EXC 2047 Hausdorff Center for Mathematics and FOR 5361 KI-FOR Algorithmic Data Analytics for Geodesy (AlgoForGe). We thank the organizers and participants for the stimulating environment that inspired this research.

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