论文标题
LCTR的Sprague-Grundy值和复杂性
Sprague-Grundy values and complexity for LCTR
论文作者
论文摘要
鉴于整数分区为$ n $，我们考虑了公正的组合游戏LCTR，其中移动包括删除其年轻图的左列或顶行。我们表明，对于正常和误会的比赛，最佳策略都可以主要包括反映对手的举动。我们还确定，LCTR和彻头彻尾的既是国内又是可退货，另一方面都不是驯服也不被迫。对于这两种游戏，这些结构观察都允许计算$ O（\ log（n））$时间中任何位置的sprague-grundy值，假设时间单元允许读取整数或执行基本的算术操作。由于Ilić（2019），这改善了先前已知的$ O（N）$。我们还涵盖了两种游戏的其他复杂度度量，例如状态空间复杂性以及相应的游戏树中的叶子和节点的数量。
Given an integer partition of $n$, we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of its Young diagram. We show that for both normal and misère play, the optimal strategy can consist mostly of mirroring the opponent's moves. We also establish that both LCTR and Downright are domestic as well as returnable, and on the other hand neither tame nor forced. For both games, those structural observations allow for computing the Sprague-Grundy value any position in $O(\log(n))$ time, assuming that the time unit allows for reading an integer, or performing a basic arithmetic operation. This improves on the previously known bound of $O(n)$ due to Ilić (2019). We also cover some other complexity measures of both games, such as state-space complexity, and number of leaves and nodes in the corresponding game tree.
