We present ${\rm poly\log\log n}$-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into $k$ parts such that a node of degree $d(u)$ has $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lov\'asz Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+\epsilon)\Delta$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $\Delta$, for any $\epsilon>0$. Further, our results improve the computation of defective colorings and certain tight list coloring problems. All the results improve the state-of-the-art round complexity exponentially, even in the LOCAL model.

翻译：我们提出了 $ rm 多元\ log\ log n} 圆形随机分布算法, 用于计算顶点分割, 将图表的顶部分割成 $k$ 部分, 这样每个部分的节点 $d( u) 都有 $ approx d( u) / k$ 邻居 。 我们的技术可以被视为 任何 $\ perm ollog\ log n} 通用的 $ pool logn $ roup 运算法的第一个进步。 作为我们结果的主要应用, 我们获得了一个随机化的 $ rm pool\ log\ log n} 。 整个 CONEST 算法, $ ( 1 ⁇ ipslon)\ Delta $- edge $n- node 图形, 足够恒定最大 $\ delta$@ 0 $。 此外, 我们的计算结果可以改进对有缺陷的颜色和某些严格列表问题的计算方法。 所有结果都会模型都改善了 。 。