Locally-iterative (Δ+1)-Coloring in Sublinear (in Δ) Rounds
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Distributed graph coloring is one of the most extensively studied problems in distributed computing. There is a canonical family of distributed graph coloring algorithms known as the locally-iterative coloring algorithms, first formalized in the seminal work of [Szegedy and Vishwanathan, STOC'93]. In such algorithms, every vertex iteratively updates its own color according to a predetermined function of the current coloring of its local neighborhood. Due to the simplicity and naturalness of its framework, locally-iterative coloring algorithms are of great significance both in theory and practice. In this paper, we give a locally-iterative (Δ+1)-coloring algorithm with O(Δ^3/4logΔ)+log^*n running time. This is the first locally-iterative (Δ+1)-coloring algorithm with sublinear-in-Δ running time, and answers the main open question raised in a recent breakthrough [Barenboim, Elkin, and Goldberg, JACM'21]. A key component of our algorithm is a locally-iterative procedure that transforms an O(Δ^2)-coloring to a (Δ+O(Δ^3/4logΔ))-coloring in o(Δ) time. Inside this procedure we work on special proper colorings that encode (arb)defective colorings, and reduce the number of used colors quadratically in a locally-iterative fashion. As a main application of our result, we also give a self-stabilizing distributed algorithm for (Δ+1)-coloring with O(Δ^3/4logΔ)+log^*n stabilization time. To the best of our knowledge, this is the first self-stabilizing algorithm for (Δ+1)-coloring with sublinear-in-Δ stabilization time.
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