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

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.