Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Abstract

We present a deterministic O(log log log n)-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ+1)-coloring on n-vertex graphs. In this model, every machine has sublinear local space of size n^φ for any arbitrary constant φ \in (0,1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the n^φ space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious (Δ+1)-coloring local algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in T_local =poly(loglog n) rounds, which sets the state-of-the-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, notably pseudorandom generators (PRG) and bounded-independence hash functions. The achieved round complexity of O(logloglog n) rounds matches the bound of log(T_local ), which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the (Δ+1)-coloring problem have been known before.