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 $c= 6/\pi^2$). 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.

翻译：我们研究构建给定随机搜索算法 \texttt{alg} 的模拟器问题，其期望运行时间为 $O( \mathcal{O} \log \mathcal{O})$，其中 $\mathcal{O}$ 是任何此类模拟器的最优期望运行时间。反直觉的是，这些模拟器在使 alg 执行单次成功运行方面，可能比原算法快得多，且这一过程无需任何关于 alg、其运行时间分布等先验知识。例如，考虑一个算法：它根据整数上的某种分布随机选取某个整数 $t$，并运行 $t$ 秒；随后以 $1/2$ 的概率停止，否则永远运行（即发生灾难）。针对此情形，本文描述的模拟器均以常数期望时间终止，且模拟运行时间服从指数衰减分布。Luby 等人此前已研究过该问题——我们的主要贡献在于，除了他们描述的方案外，提出了多种额外的模拟策略。特别地，我们的一种（最优）模拟策略异常简洁：以概率 $c/t^2$（其中 $c= 6/\pi^2$）随机选取整数 $t>0$，运行算法 $t$ 秒。若 alg 的运行在此阈值前终止，则模拟成功并退出；否则，模拟器重复此过程直至成功。