Superfast Coloring in CONGEST via Efficient Color Sampling
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We present a procedure for efficiently sampling colors in the model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to Θ(log n) semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with O(log^* Δ) complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an O(log^* Δ)-round CONGEST algorithm for (1+ϵ)Δ-edge coloring when Δ≥log^1+1/log^*n n, and a poly(loglog n)-round algorithm for (2Δ-1)-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.
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