The Lov\'{a}sz Local Lemma (LLL) is a cornerstone principle in the
probabilistic method of combinatorics, and a seminal algorithm of Moser &
Tardos (2010) provides an efficient randomized algorithm to implement it. This
can be parallelized to give an algorithm that uses polynomially many processors
and runs in O(log3n) time on an EREW PRAM, stemming from O(logn)
adaptive computations of a maximal independent set (MIS). Chung et al. (2014)
developed faster local and parallel algorithms, potentially running in time
O(log2n), but these algorithms require more stringent conditions than the
LLL.
We give a new parallel algorithm that works under essentially the same
conditions as the original algorithm of Moser & Tardos but uses only a single
MIS computation, thus running in O(log2n) time on an EREW PRAM. This can
be derandomized to give an NC algorithm running in time O(log2n) as well,
speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013).
We also provide improved and tighter bounds on the run-times of the
sequential and parallel resampling-based algorithms originally developed by
Moser & Tardos. These apply to any problem instance in which the tighter
Shearer LLL criterion is satisfied