Density-Dependent Graph Orientation and Coloring in Scalable MPC

TL;DR

This paper introduces scalable MPC algorithms for graph orientation and coloring that adapt to subgraph density, running in poly(log log n) rounds and surpassing previous round complexity barriers.

Contribution

It presents novel density-dependent algorithms for graph orientation and coloring in scalable MPC, achieving faster round complexity than prior methods.

Findings

Algorithms run in poly(log log n) rounds.

Orientation with maximum outdegree O(α log log n).

Coloring with O(α log log n) colors.

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.