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arXiv · 2603.10639

Density-Dependent Graph Orientation and Coloring in Scalable MPC

Mohsen Ghaffari·Christoph Grunau
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Abstract

This paper presents massively parallel computation (MPC) algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density. Our algorithms run in $poly(\log\log n)$ rounds and compute an orientation of the edges with maximum outdegree $O(\alpha \log\log n)$ as well as a coloring of the vertices with $O(\alpha \log\log n)$ colors. Here, $\alpha$ denotes the density of the densest subgraph. Our algorithm's round complexity is notable because it breaks the $\tilde{\Theta}(\sqrt{\log n})$ barrier, which applied to the previously best known density-dependent orientation algorithm [Ghaffari, Lattanzi, and Mitrovic ICML'19] and is common to many other scalable MPC algorithms.

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Mohsen Ghaffari, Christoph Grunau. 2026-03-11. Density-Dependent Graph Orientation and Coloring in Scalable MPC. https://arxiv.org/abs/2603.10639

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