SAT and Polytime Reductions If an algorithm for $SAT$ runs in $O(n^{\log n})$ time, and if $L$ belongs to $\mathsf{NP}$, is there an algorithm for $L$ that runs in $O(n^{\log n})$ time?
Since any input x for L can be reduced to SAT in O(|x|^c) time, the created SAT instance will have size at most n=O(|x|^c).  So the n^(lg n)-time algorithm for SAT will be an |x|^O(lg |x|)-time algorithm for L.