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Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces

Authors Eunjin Oh , Hyeonjun Shin

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LIPIcs.ESA.2026.88.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • MPC model
  • parallel algorithms
  • Computational Geometry
  • well-separated pair decomposition
  • doubling metric spaces

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Abstract

In this paper, we study the problem of constructing a (1/ε)-well-separated pair decomposition (WSPD) for a point set of size n in the Massively Parallel Computation (MPC) model, where multiple machines work in parallel and communicate in synchronous rounds. We present an O(1)-round MPC algorithm that constructs a O(1/ε)-WSPD of size (1/ε)^O(ddim) ⋅ Õ(n) for point sets in a metric space of a constant doubling dimension ddim, with high probability, using (1/ε)^O(ddim) ⋅ Õ(n) total space and O(n^δ) space per machine for a constant δ ∈ (0,1). In the d-dimensional Euclidean space, we can improve the size of the WSPD and the total space to (1/ε)^O(d) n. This improves the best-known algorithm [FOCS'93] for computing a WSPD which requires O(log n) rounds and works only in Euclidean spaces. As a consequence, the following problems can be solved in O(1) rounds in the MPC model: computing a (1+ε)-spanner, a (1-ε)-approximation of the diameter, the closest pair, and the k-nearest neighbors (k-NN). While our k-NN algorithm is specific to Euclidean space, the other three problems can be solved in both Euclidean and doubling metric spaces.

Cite As Get BibTex

Eunjin Oh and Hyeonjun Shin. Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 88:1-88:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.88

Author Details

Eunjin Oh
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Hyeonjun Shin
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea

Funding

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).

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