We study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size \(n\). In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length \(k\) from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with \(O(1)\) remainder in the first problem and \(O(k)\) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

我们研究了递增子序列在线选择中两个密切相关的问题。在第一个问题，由塞缪尔和Steele引入（安Probab 9（6）：937-947，1981），目的是最大限度地从给定尺寸的随机样本由nonanticipating策略中选择的子序列的长度\ (n\)。在对偶问题中，最近由 Arlotto 等人研究。（Random Struct. Algorithms 49:235–252, 2016），目标是最小化从无限长度序列中选择给定长度\(k\)递增子序列所需的预期时间。开发一种基于动态规划方程单调性的方法，我们推导出最优值的两项渐近展开式，在第一个问题中具有\(O(1)\)余数，并且\(O(k)\)在第二个。解决 Arlotto 等人的猜想。（Random Struct. Algorithms 52:41–53, 2018），我们还设计了选择策略以在这些范围内实现最优，从某种意义上说，这是最好的。