Improved Deterministic (Δ+1)-Coloring in Low-Space MPC
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We present a deterministic O(logloglog 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 a sublinear local memory of size n^ϕ for any arbitrary constant ϕ∈ (0,1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential (in n^ϕ) local computation, 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, most 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 were previously known.
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