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

2025-01-21

AI Summary

This paper addresses three fundamental problems—connectivity computation on multidimensional grid graphs, Euclidean minimum spanning tree (EMST) construction, and DBSCAN clustering—within the Massively Parallel Computation (MPC) model. We propose novel MPC protocols based on local probing and hierarchical aggregation, geometric hashing, and randomized sampling with deterministic verification. Our contributions are: (1) an $O(1)$-round Las Vegas algorithm for grid graph connectivity, improving upon the prior $O(log log n + log D)$-round bound; (2) the first EMST algorithm in MPC that simultaneously achieves deterministic approximation guarantees on both total weight and individual edge weights; and (3) the first $O(1)$-round *exact* DBSCAN approximation algorithm in constant-dimensional Euclidean space, achieving theoretically optimal round complexity. All results hold under standard MPC constraints (sublinear memory per machine, near-linear total memory), and significantly advance the state of parallel geometric and graph processing.

Technology Category

• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• MPC Model
• Multi-dimensional Grid Graphs
• Approximate Algorithms

Innovation

• MPC Model
• O(1) Approximation Algorithms
• Multi-dimensional Grid Connectivity