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^b for any arbitrary constant b ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the n^b 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(n) = poly(log log 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(log log log n) rounds matches the bound of log(T(n)), 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
