Derandomization that is rarely wrong from short advice that is typically good

For every ε < 0, we present a deterministic log-space algorithm that correctly decides undirected graph connectivity on all but at most 2^nε of the nvertex graphs. The same holds for every problem in Symmetric Log-space (i.e., SL). Using a plausible complexity assumption (i.e., that P cannot be approximated by SIZE(p)^SAT, for every polynomial p) we showthat, for every ε > 0, each problem in BPP has a deterministic polynomial-time algorithm that errs on at most 2n of the n-bit long inputs. (The complexity assumption that we use is not known to imply BPP = P.) All results are obtained as special cases of a general methodology that explores which probabilistic algorithms can be derandomized by generating their coin tosses deterministically from the input itself.We show that this is possible (for all but extremely few inputs) for algorithms which take advice (in the usual Karp- Lipton sense), in which the advice string is short, and most choices of the advice string are good for the algorithm. To get the applications above and others, we show that algorithms with short and typically-good advice strings do exist, unconditionally for SL, and under reasonable assumptions for BPP and AM.