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

Abstract

We present a deterministic (log log log )-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ + 1)-coloring on -vertex graphs. In this model, every machine has sublinear local space of size for any arbitrary constant ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the 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 = (log log ) rounds, which sets the state-ofthe-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 (log log log ) rounds matches the bound of log(), 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. CCS CONCEPTS• Computing methodologies → Distributed algorithms; • Mathematics of computing → Graph algorithms; • Theory of computation → Pseudorandomness and derandomization.

Deterministic Massively Parallel Connectivity

“…One interesting feature of our work is that the derandomization techniques incorporated in our work are highly non-component-stable. As the result, our work is another example (see also [CDP20a,CDP21b,CDP21c]) showing that the powerful framework of conditional MPC lower bounds due to Ghaffari et al [GKU19] (which as for now, is arguably the most general framework of lower bounds known for MPC algorithms) is not always suitable, and a lower bound (even if only conditioned on the 1-vs-2 cycles Conjecture 1) for component-stable MPC algorithms may not preclude the existence of more efficient non-component-stable MPC algorithms.…”

“…We will incorporate the approach used recently in [CPS20] and [CDP20a] (see also recent results in [CDP20b,CDP21b,CDP21c]; though a similar approach has been used in earlier, classic works on derandomization, see, e.g., [MNN94] or [Lub93,Rag88], with the only difference being that now we have to process by blocks of O(log n/δ) bits, whereas in earlier works one was processing individual bits, one bit at a time 2 ). We will search for function h * by splitting the O(log n)-bit seed defining it into smaller parts of χ < log S = O(δ log n) = O(log n) bits each 3 , and then processing one part at time, iteratively extending a fixed prefix of the seed until we have fixed the entire seed.…”

Deterministic massively parallel connectivity

“…While there are some deterministic MPC algorithms with low round complexity and linear global space, such results are still rare; some recent examples include [8,18,24,25,27]. The occasional use of extra global space is largely because the derandomization has some overhead which may lead to extra 𝑂 (𝑛 𝛿 ) (or even larger) factor in the global space bounds.…”

Brooks' Theorem in Graph Streams: A Single-Pass Semi-Streaming Algorithm for $Δ$-Coloring

“…Many of these work on streaming algorithms for graph coloring also extend to other models such as sublinear-time and massively parallel computation (MPC) algorithms. For instance, for (∆ + 1) coloring, there are randomized sublinear-time algorithms in O(n 3/2 ) time [ACK19] or deterministic MPC algorithms with O(1) rounds and O(n) per-machine memory [CDP20] (see also [CDP21]).…”