AI 中文总结
该研究针对Stone Game III，证明即使在无平局、输入有胜者或值无界的情况下，任何确定性算法都需$\u03a9(n)$次输入检查，从而确立标准$O(n)$时间、$O(1)$空间方案的渐近最优性。
AI 中文摘要
Stone Game III存在一种标准的反向动态规划算法，时间复杂度为$O(n)$，辅助空间复杂度为$O(1)$。该上界是显而易见的，但它的最优性引出了一个看似简单的问题：正确算法是否真的需要检查线性数量的输入值？对于原问题，全零实例提供了一个简短的不可区分性证明，表明每个位置都必须被检查。这一论点似乎高度依赖于平局的可能性。我们证明并非如此：即使在每个输入都有一个胜者的前提下，敌手也可以通过组合模块化移动控制与不可区分的输入补全，迫使任何确定性算法进行$\u03a9(n)$次检查。我们还将该论点扩展到正且无界的值，在无零或平局的情况下得到相同的线性下界。这些结果共同确立了标准$O(n)$时间、$O(1)$空间解决方案在多个愈发受限变体中的渐近最优性。
英文摘要
Stone Game III admits a standard backward dynamic program using $O(n)$ time and $O(1)$ auxiliary space. The upper bound is immediate, but its optimality raises a deceptively simple question: must a correct algorithm really inspect a linear number of input values? For the original problem, an all-zero instance gives a short indistinguishability proof that every position must be inspected. This argument appears to depend strongly on the possibility of a tie. We show that it does not. Even under the promise that every input has a winner, an adversary can force any deterministic algorithm to make $Ω(n)$ inspections by combining modular move control with indistinguishable input completions. We also extend the argument to positive but unbounded values, obtaining the same linear lower bound without zeros or ties. Together these results establish the asymptotic optimality of the standard $O(n)$-time, $O(1)$-space solution in several increasingly restrictive variants.
