AI 中文总结
本文针对奶牛路径问题，详细证明了Kao等人提出的循环顺序随机算法在所有路径数下的最优性，即无算法能优于最佳循环算法。
AI 中文摘要
在奶牛路径问题中，一头奶牛必须找到位于仅由原点连接的$w$条路径之一上、距离未知的目标，性能通过竞争比来衡量。Kao、Reif和Tate设计了一种高效的随机算法，其中奶牛按固定的循环顺序访问各路径。他们证明了该算法在$w=2$时是最优的，随后Kao、Ma、Sipser和Yin证明了其对所有$w$的最优性，并声称没有算法能优于最佳循环算法。本文提供了该声称的详细证明。
英文摘要
In the cow-path problem, a cow must find a goal lying at an unknown distance on one of $w$ paths connected only at the origin, and performance is measured by competitive ratio. Kao, Reif and Tate designed an efficient randomized algorithm in which the cow visits the paths in a fixed cyclic order. They proved the algorithm is optimal for $w=2$, and subsequently Kao, Ma, Sipser and Yin proved its optimality for all $w$, with a claim that no algorithm does better than the best cyclic one. This note provides a detailed proof of that claim.
