$O(1)$-Round MPC Algorithms for Multi-dimensional Grid Graph Connectivity, EMST and DBSCAN

TL;DR

This paper presents new constant-round MPC algorithms for grid graph connectivity, approximate Euclidean Minimum Spanning Tree, and DBSCAN clustering, improving efficiency and determinism over prior methods in massively parallel computation.

Contribution

It introduces the first $O(1)$-round MPC algorithms for grid graph connectivity, deterministic approximate EMST, and approximate DBSCAN clustering, specifically tailored for grid graphs and low-dimensional spaces.

Findings

Achieves $O(1)$-round MPC algorithms for grid graph connectivity.

Provides deterministic edge-wise weight approximation for EMST.

Develops $O(1)$-round MPC algorithm for approximate DBSCAN clustering.

Abstract

In this paper, we investigate three fundamental problems in the Massively Parallel Computation (MPC) model: (i) grid graph connectivity, (ii) approximate Euclidean Minimum Spanning Tree (EMST), and (iii) approximate DBSCAN. Our first result is a $O(1)$-round Las Vegas (i.e., succeeding with high probability) MPC algorithm for computing the connected components on a $d$-dimensional $c$-penetration grid graph ($(d,c)$-grid graph), where both $d$ and $c$ are positive integer constants. In such a grid graph, each vertex is a point with integer coordinates in $\mathbb{N}^d$, and an edge can only exist between two distinct vertices with $\ell_\infty$-norm at most $c$. To our knowledge, the current best existing result for computing the connected components (CC's) on $(d,c)$-grid graphs in the MPC model is to run the state-of-the-art MPC CC algorithms that are designed for general graphs: they achieve $O(\log \log n + \log D)$[FOCS19] and $O(\log \log n + \log \frac{1}{\lambda})$[PODC19] rounds, respectively, where $D$ is the {\em diameter} and $\lambda$ is the {\em spectral gap} of the graph. With our grid graph connectivity technique, our second main result is a $O(1)$-round Las Vegas MPC algorithm for computing approximate Euclidean MST. The existing state-of-the-art result on this problem is the $O(1)$-round MPC algorithm proposed by Andoni et al.[STOC14], which only guarantees an approximation on the overall weight in expectation. In contrast, our algorithm not only guarantees a deterministic overall weight approximation, but also achieves a deterministic edge-wise weight approximation.The latter property is crucial to many applications, such as finding the Bichromatic Closest Pair and DBSCAN clustering. Last but not the least, our third main result is a $O(1)$-round Las Vegas MPC algorithm for computing an approximate DBSCAN clustering in $O(1)$-dimensional space.