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Fully Scalable MPC Algorithms for Euclidean k-Center

Authors Artur Czumaj , Guichen Gao , Mohsen Ghaffari , Shaofeng H.-C. Jiang

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ACM Subject Classification
  • Theory of computation → Massively parallel algorithms
  • Theory of computation → Facility location and clustering
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Massively Parallel Computing
  • Euclidean Spaces
  • k-Center Clustering

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Abstract

The k-center problem is a fundamental optimization problem with numerous applications in machine learning, data analysis, data mining, and communication networks. The k-center problem has been extensively studied in the classical sequential setting for several decades, and more recently there have been some efforts in understanding the problem in parallel computing, on the Massively Parallel Computation (MPC) model. For now, we have a good understanding of k-center in the case where each local MPC machine has sufficient local memory to store some representatives from each cluster, that is, when one has Ω(k) local memory per machine. While this setting covers the case of small values of k, for a large number of clusters these algorithms require undesirably large local memory, making them poorly scalable. The case of large k has been considered only recently for the fully scalable low-local-memory MPC model for the Euclidean instances of the k-center problem. However, the earlier works have been considering only the constant dimensional Euclidean space, required a super-constant number of rounds, and produced only k(1+o(1)) centers whose cost is a super-constant approximation of k-center.
In this work, we significantly improve upon the earlier results for the k-center problem for the fully scalable low-local-memory MPC model. In the low dimensional Euclidean case in ℝ^d, we present the first constant-round fully scalable MPC algorithm for (2+ε)-approximation. We push the ratio further to (1 + ε)-approximation albeit using slightly more (1 + ε)k centers. All these results naturally extends to slightly super-constant values of d. In the high-dimensional regime, we provide the first fully scalable MPC algorithm that in a constant number of rounds achieves an O(log n/ log log n)-approximation for k-center.

Cite As Get BibTex

Artur Czumaj, Guichen Gao, Mohsen Ghaffari, and Shaofeng H.-C. Jiang. Fully Scalable MPC Algorithms for Euclidean k-Center. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 64:1-64:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.64

Author Details

Artur Czumaj
  • Department of Computer Science, University of Warwick, Coventry, UK
Guichen Gao
  • School of Computer Science, Peking University, Beijing, China
Mohsen Ghaffari
  • Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA
Shaofeng H.-C. Jiang
  • School of Computer Science, Peking University, Beijing, China

Funding

  • Czumaj, Artur: Research supported in part by the Centre for Discrete Mathematics and its Applications (DIMAP), by EPSRC award EP/V01305X/1, by a Weizmann-UK Making Connections Grant, by an IBM Award.
  • Jiang, Shaofeng H.-C.: Research partially supported by a national key R&D program of China No. 2021YFA1000900.

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