Superfast Coloring in CONGEST via Efficient Color Sampling

We present a procedure for efficiently sampling colors in the {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to $\Theta(\log n)$ semi-random colors unused by their neighbors in $O(1)$ rounds, even in the distance-2 setting. This yields algorithms with $O(\log^* \Delta)$ complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an $O(\log^* \Delta)$-round CONGEST algorithm for $(1+\epsilon)\Delta$-edge coloring when $\Delta \ge \log^{1+1/\log^*n} n$, and a poly($\log\log n$)-round algorithm for $(2\Delta-1)$-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.