Quickly Avoiding a Random Catastrophe

We study the problem of constructing simulations of a given randomized search algorithm \texttt{alg} with expected running time $O( \mathcal{O} \log \mathcal{O})$, where $\mathcal{O}$ is the optimal expected running time of any such simulation. Counterintuitively, these simulators can be dramatically faster than the original algorithm in getting alg to perform a single successful run, and this is done without any knowledge about alg, its running time distribution, etc. For example, consider an algorithm that randomly picks some integer $t$ according to some distribution over the integers, and runs for $t$ seconds. then with probability $1/2$ it stops, or else runs forever (i.e., a catastrophe). The simulators described here, for this case, all terminate in constant expected time, with exponentially decaying distribution on the running time of the simulation. Luby et al. studied this problem before -- and our main contribution is in offering several additional simulation strategies to the one they describe. In particular, one of our (optimal) simulation strategies is strikingly simple: Randomly pick an integer $t>0$ with probability $c/t^2$ (with $c= 6/π^2$). Run the algorithm for $t$ seconds. If the run of alg terminates before this threshold is met, the simulation succeeded and it exits. Otherwise, the simulator repeat the process till success.