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：C. Doerr 与 Krejca 近期在《IEEE Transactions on Evolutionary Computation》（2023）中证明了随机局部搜索启发式算法在广义针函数上的预期运行时间上界。基于这些上界，他们以不完全严谨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 $k$ 影响程度所必需的下界。为此，我们推导出预期运行时间的精确表达式，该结果同时显著改进了 C. Doerr 与 Krejca 给出的上界。我们还描述了预期运行时间的渐近估计。