We present a poly $\log \log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized complexity between $\log n$ and poly $\log \log n$. Furthermore, we provide extensions to the network decomposition algorithms given in the recent breakthrough by Rozhon and Ghaffari [STOC2020] and the follow up by Ghaffari, Grunau, and Rozhon [SODA2021]. In particular, we show how to obtain a large distance separated weak network decomposition with a negligible dependency on the range of unique identifiers.
翻译：我们用恒定度图为Lovasz Lemma (LLLL) 自然等级的Lovasz 地方 Lemma (LLLL) 实例展示了 $\ log\ log n 时间随机的 CONEST 算法。 这意味着,除其他外,在 $\ log n$ 和 poli $\ log\ log n$ 之间不存在随机复杂的 LLLCL 问题。 此外,我们扩展了Rozhon 和 Ghaffari 最近的突破[STOC/2020] 和 Ghaffari 、 Grunau 和 Rozhon 的跟踪所给出的网络分解算法[SODA-2021], 特别是我们展示了如何获得一个大距离分离的薄弱网络分解,对独特识别特征范围的依赖微不足道的网络分解。
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