We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the $(1+1)$ evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the $(1+1)$ evolutionary algorithm on a new benchmark consisting of $n/\ell$ plateaus of effective size $2^\ell-1$ which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For $\ell = o(n)$, the optimal static mutation rate is approximately $1.59/n$. The optimal fitness dependent mutation rate, when the first $k$ fitness-relevant bits have been found, is asymptotically $1/(k+1)$. These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.

翻译：我们提出了一种基于离散傅里叶分析的新方法，用于分析算法在平顶上所花费的时间。这立即给出了Garnier，Kallel和Schoenauer（1999）所提出的关于针问题 $(1 + 1)$ 进化算法期望运行时间的精确证明估计。我们还使用这种方法分析了由 $n / \ell$ 个有效大小为 $2^\ell -1$ 的平顶组成的新基准上 $(1 + 1)$ 进化算法的运行时间，这些运行时间必须按LeadingOnes方式依次进行优化。使用我们的新方法，我们确定了静态和与适应度相关的突变速率的精确期望运行时间。当 $\ell = o(n)$ 时，最优静态突变速率约为 $1.59 / n$。在找到前 $k$ 个与适应度相关的位时，最优的与适应度相关的变异率在渐近意义下为 $1 / (k + 1)$。因此，这些结果，目前仅针对单个实例LeadingOnes问题而证明，适用于更广泛的问题类别。我们期望类似的扩展方法也适用于LeadingOnes的其他重要结果。我们也对我们的傅里叶分析方法能够应用于其他平顶问题持乐观态度。