In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius $k$ on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter $k$ on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.

翻译：在最近的工作中，C. Doerr 与 Krejca（《进化计算汇刊》，2023年）证明了随机局部搜索启发式算法在广义针函数上期望运行时间的上界。基于这些上界，他们以不严格的方式推断出针半径 $k$ 对运行时间具有显著影响。本文补充了确定参数 $k$ 对运行时间影响所缺失的下界。为此，我们推导出期望运行时间的精确表达式，并显著改进了 C. Doerr 与 Krejca 给出的上界。此外，我们还给出了期望运行时间的渐近估计。