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

TL;DR

The paper develops $O(1)$-round Las Vegas MPC algorithms for three core problems: grid-graph connectivity (and MSF), approximate Euclidean MST (EMST), and approximate DBSCAN, in the setting of constants-dimensional space with sublinear per-machine memory. Central to the approach is the pseudo $s$-separator technique, an efficient, $O(1)$-round construct that partitions implicit $(d,c)$-grid graphs into subproblems that fit in local memory, enabling constant-round CC and MSF. The EMST algorithm combines MSF on progressively coarsened implicit grid graphs with a deterministic edge-weight guarantee, while the DBSCAN method identifies core points via grid-based counting and forms primitive clusters that yield a $\rho$-approximate clustering in $O(1)$ rounds. Together, these results advance the practical and theoretical understanding of large-scale geometric computation in the MPC model and offer derandomized options with strong guarantees for downstream tasks. The techniques promise broader applicability to other MPC-enabled geometric and clustering computations on sparse, grid-like structures.

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 $O(\log \log n + \log \frac{1}λ)$[PODC19] rounds, respectively, where $D$ is the {\em diameter} and $λ$ 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.

Figures (3)

• Figure 1: A $(d,c)$-grid graph $G$ with $d =2$ and $c=2$ and a separator. The blue regions are the target $c$-dividers: the vertices in them are separator vertices. Removing all the separator vertices disconnects $G$ into $5$ sub-graphs.
• Figure 2: A running example of our approximate EMST algorithm
• Figure 3: A running example of Algorithm \ref{['algo:coredete']} with $\mathit{minPts}=3$. The input points are in six non-empty grid cells stored across five machines as shown in the figure. Specifically, $g_2,g_4,g_5$ are dense grid cells and all the points in them are core points, while $g_1,g_3,g_6$ are sparse grid cells. To label the points in these sparse grid cells, Algorithm \ref{['algo:coredete']} performs in two MPC rounds. Take grid cell $g_3$ which is stored in $M_1$ as an example; the neighbour grid cell set of $g_3$ is $\text{Nei}_{g_3}=\{g_1,g_2,g_4\}$, and the storage locations of them include: $M_1$ for both $g_1$ and $g_2$, and $M_2, M_3, M_4$ for $g_4$. The communications between $M_1$ and $M_1$ (i.e., local computation) and $M_2,M_3,M_4$ are incurred, which are shown by the blue arrows in the figure.

Theorems & Definitions (6)

• Definition 1: Implicit Graph
• Definition 2: Pseudo $s$-Separator
• Lemma 3: Binary Partition Lemma
• Theorem 4
• Theorem 5
• Lemma 8: Core Point Identification