Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces

2026-07-04

AI Summary

This work presents the first constant-round algorithm in the Massively Parallel Computation (MPC) model for efficiently constructing a $(1/\varepsilon)$-well-separated pair decomposition (WSPD) in both doubling metric spaces and Euclidean spaces. By integrating doubling dimension theory, grid-based partitioning, and distributed randomized sampling with local aggregation, the algorithm overcomes the previous $O(\log n)$-round barrier. The resulting WSPD has size $(1/\varepsilon)^{O(\mathrm{ddim})} \cdot \tilde{O}(n)$ in general doubling spaces and improves to $(1/\varepsilon)^{O(d)} n$ in $d$-dimensional Euclidean space. This decomposition enables constant-round MPC solutions to fundamental geometric problems, including diameter approximation, closest pair, $(1+\varepsilon)$-spanner construction, and $k$-nearest neighbors.

Abstract

In this paper, we study the problem of constructing a $(1/\varepsilon)$-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 $O(1/\varepsilon)$-WSPD of size $(1/\varepsilon)^{O(ddim)}\cdot \tilde O(n)$ for point sets in a metric space of a constant doubling dimension $ddim$, with high probability, using $(1/\varepsilon)^{O(ddim)} \cdot \tilde O(n)$ total space and $O(n^δ)$ space per machine for a constant $δ\in (0,1)$. In the $d$-dimensional Euclidean space, we can improve the size of the WSPD and the total space to $(1/\varepsilon)^{O(d)} n$. 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 $(1+\varepsilon)$-spanner, a $(1-\varepsilon)$-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.

Problem

• WSPD
• Massively Parallel Computation
• doubling dimension
• Euclidean space
• well-separated pair decomposition

Innovation

• Massively Parallel Computation
• Well-Separated Pair Decomposition
• Doubling Dimension
• Constant-Round Algorithm
• Euclidean Space