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PPSZ, for long time the fastest known algorithm for $k$-SAT, works by goingthrough the variables of the input formula in random order; each variable isthen set randomly to $0$ or $1$, unless the correct value can be inferred by anefficiently implementable rule (like small-width resolution; or being impliedby a small set of clauses).  We show that PPSZ performs exponentially better than previously known, forall $k \geq 3$. For Unique-$3$-SAT we bound its running time by$O(1.306973^{n})$, which is somewhat better than the algorithm of Hansen,Kaplan, Zamir, and Zwick, which runs in time $O(1.306995^n)$. Before that, thebest known upper bound for Unique-$3$-SAT was $O(1.3070319^n)$.  All improvements are achieved without changing the original PPSZ. The coreidea is to pretend that PPSZ does not process the variables in uniformly randomorder, but according to a carefully designed distribution. We write "pretend"since this can be done without any actual change to the algorithm.

PPSZ is better than you think