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Space-Efficient Approximate Spherical Range Counting in High Dimensions

Authors Andreas Kalavas , Ioannis Psarros

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LIPIcs.SoCG.2026.60.pdf
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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximate range counting
  • partition trees
  • high dimensions

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Abstract

We study the following range searching problem in high-dimensional Euclidean spaces: given a finite set P ⊂ ℝ^d, where each p ∈ P is assigned a weight w_p, and radius r > 0, we need to preprocess P into a data structure such that when a new query point q ∈ ℝ^d arrives, the data structure reports the cumulative weight of points of P within Euclidean distance r from q. Solving the problem exactly seems to require space usage that is exponential to the dimension, a phenomenon known as the curse of dimensionality. Thus, we focus on approximate solutions where points up to (1+ε)r away from q may be taken into account, where ε > 0 is an input parameter known during preprocessing. We build a data structure with near-linear space usage, and query time in n^{1-Θ(ε⁴/log(1/ε))}+t_q^ϱ⋅n^{1-ϱ}, for some ϱ = Θ(ε²), where t_q is the number of points of P in the ambiguity zone, i.e., at distance between r and (1+ε)r from the query q. To the best of our knowledge, this is the first data structure with efficient space usage (subquadratic or near-linear for any ε > 0) and query time that remains sublinear for any sublinear t_q. We supplement our worst-case bounds with a query-driven preprocessing algorithm to build data structures that are well-adapted to the query distribution.

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Andreas Kalavas and Ioannis Psarros. Space-Efficient Approximate Spherical Range Counting in High Dimensions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.60

Author Details

Andreas Kalavas
  • Carnegie Mellon University, Pittsburgh, PA, USA
Ioannis Psarros
  • Archimedes, Athena Research Center, Greece

Funding

  • Kalavas, Andreas: Supported in part by the U.S. National Science Foundation (NSF) Grants ECCS-2145713, CCF-2403194, CCF-2428569, and ECCS-2432545.
  • Psarros, Ioannis: This work has been partially supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program.

Acknowledgements

We would like to thank Dimitris Fotakis for valuable discussions on the topic.

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