We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.

翻译：我们研究了并行Moser-Tardos算法的一个变体。我们证明，如果关注一类依赖图具有亚指数增长的问题，则该算法使用的随机比特总数的期望值为常数；特别地，该期望值与变量数量无关。这一结果是通过在依赖图中距离足够远的变量上复用相同的随机比特来实现的。该结论有两个推论。首先，我们获得了一个用于寻找满足赋值的确定性算法，该算法对于上一段中所述的任何问题类别，其运行时间为O(n)，其中n为变量数量。其次，我们给出了Lovász局部引理的一个Borel版本。