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oai:drops-oai.dagstuhl.de:20193 2024-07-02T07:52:50Z ddc:004 open_access

Fully-Scalable MPC Algorithms for Clustering in High Dimension Czumaj, Artur Gao, Guichen Jiang, Shaofeng H.-C. Krauthgamer, Robert Veselý, Pavel Massively parallel computing high dimension facility location k-median k-means We design new parallel algorithms for clustering in high-dimensional Euclidean spaces. These algorithms run in the Massively Parallel Computation (MPC) model, and are fully scalable, meaning that the local memory in each machine may be n^σ for arbitrarily small fixed σ > 0. Importantly, the local memory may be substantially smaller than the number of clusters k, yet all our algorithms are fast, i.e., run in O(1) rounds. We first devise a fast MPC algorithm for O(1)-approximation of uniform Facility Location. This is the first fully-scalable MPC algorithm that achieves O(1)-approximation for any clustering problem in general geometric setting; previous algorithms only provide poly(log n)-approximation or apply to restricted inputs, like low dimension or small number of clusters k; e.g. [Bhaskara and Wijewardena, ICML'18; Cohen-Addad et al., NeurIPS'21; Cohen-Addad et al., ICML'22]. We then build on this Facility Location result and devise a fast MPC algorithm that achieves O(1)-bicriteria approximation for k-Median and for k-Means, namely, it computes (1+ε)k clusters of cost within O(1/ε²)-factor of the optimum for k clusters. A primary technical tool that we introduce, and may be of independent interest, is a new MPC primitive for geometric aggregation, namely, computing for every data point a statistic of its approximate neighborhood, for statistics like range counting and nearest-neighbor search. Our implementation of this primitive works in high dimension, and is based on consistent hashing (aka sparse partition), a technique that was recently used for streaming algorithms [Czumaj et al., FOCS'22]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Artur Czumaj and Guichen Gao and Shaofeng H.-C. Jiang and Robert Krauthgamer and Pavel Veselý 2024 Is Part Of LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2024.50 urn:nbn:de:0030-drops-201938 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.50 eng https://creativecommons.org/licenses/by/4.0/legalcode