Fast Parallel Degree+1 List Coloring
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Graph coloring problems are arguably among the most fundamental graph problems in parallel and distributed computing with numerous applications. In particular, in recent years the classical (Δ+1)-coloring problem became a benchmark problem to study the impact of local computation for parallel and distributed algorithms. In this work, we study the parallel complexity of a generalization of the (Δ+1)-coloring problem: the problem of (degree+1)-list coloring (𝖣1𝖫𝖢), where each node has an input palette of acceptable colors, of size one more than its degree, and the objective is to find a proper coloring using these palettes. In a recent work, Halldórsson et al. (STOC'22) presented a randomized O(log^3log n)-rounds distributed algorithm for 𝖣1𝖫𝖢 in the 𝖫𝖮𝖢𝖠𝖫 model, matching for the first time the state-of-the art complexity for (Δ+1)-coloring due to Chang et al. (SICOMP'20). In this paper, we obtain a similar connection for 𝖣1𝖫𝖢 in the Massively Parallel Computation (𝖬𝖯𝖢) model with sublinear local space: we present a randomized O(logloglog n)-round 𝖬𝖯𝖢 algorithm for 𝖣1𝖫𝖢, matching the state-of-the art 𝖬𝖯𝖢 algorithm for the (Δ+1)-coloring problem. We also show that our algorithm can be efficiently derandomized.
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