在线置换嵌入：最优停止与尺度定律
Online Permutation Embedding: Optimal Stopping and Scaling Laws
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中文总结 AI 辅助
该研究针对将k元置换嵌入均匀随机变量流的问题，提出最优在线嵌入算法及动态规划方法，发现随机置换、单调置换、极值置换的嵌入速度存在严格层级差异，与离线理论猜想相悖。
中文摘要 AI 辅助
我们研究将集合[k]的置换π嵌入到独立同分布的均匀[0,1]随机变量流中的最优在线算法。该问题是经典在线单调子序列选择问题的广泛推广，当π为恒等置换Id_k时可退化为该经典问题。我们的第一个贡献是一种可高效求解的动态规划方法，用于确定任意k元置换π的最优嵌入时间，该动态规划还能生成显式的最优在线嵌入算法。随后，我们研究了均匀随机目标置换的最优嵌入时间的渐近尺度，以及识别具有最大期望在线嵌入时间的置换的极值问题。我们的第二个主要结果表明，一阶近似下，随机置换的嵌入速度严格快于单调置换，而单调置换的嵌入速度又严格快于极值置换。这种分离与置换嵌入离线理论中流行的猜想和启发式方法形成鲜明对比。
英文摘要
We study optimal online algorithms for embedding a permutation $π$ of $[k]$ into an iid stream of uniform $[0,1]$ random variables. This problem is a broad generalization of the classical online monotone subsequence selection problem, recovered in the special case $π=\mathrm{Id}_k$. Our first contribution is an efficiently solvable dynamic program for the optimal embedding time of any $k$-permutation $π$. This dynamic program also yields an explicit optimal online embedding algorithm. We then investigate the asymptotic scaling of the optimal embedding time for uniformly random target permutations, as well as the extremal problem of identifying the permutations with largest expected online embedding time. Our second main result shows that, to first order, random permutations are strictly faster to embed than monotone permutations, which in turn are strictly faster to embed than the extremal permutations. This separation stands in sharp contrast to prevailing conjectures and heuristics in the offline theory of permutation embeddings.
