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原文摘要

We study Euclidean $k$-Means and $k$-Median under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully scalable} setting. Previously, fully scalable algorithms for $k$-Means and/or $k$-Median either required $O(\log\log n\cdot\log\log\log n)$ rounds [Cohen-Addad, Kuhn, and Parsaeian; SODA'26] or guaranteed only an $O(\log n)$-approximation [Goranci, Jiang, Kiss, Kong, Qian, and Szilagyi; SODA'26]. In this paper, we close this gap by giving the first fully scalable algorithms for both problems, which work even when $k$ is polynomially larger than the local memory, and \emph{simultaneously} achieve an $O(1)$-approximation and $O(1)$ round complexity. Technically, we devise a new variant of the MP algorithm [Mettu and Plaxton, SICOMP'03] that computes a \emph{fractional solution satisfying the Lagrangian Multiplier Preserving (LMP) property}. Importantly, this LMP guarantee holds even under arbitrary distance distortions, a feature that is crucial for efficient MPC implementations. We then give a novel MPC rounding algorithm that first rounds to a bicriteria solution with $k+O(n^{1-ε})$ centers and then uses an \emph{iterative kernelization} idea to convert it into one with exactly $k$ centers.