Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces

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 $O(n^\delta)$ space per machine for a constant $\delta\in (0,1)$. 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.