Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

Abstract

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 $n$ 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 $k$ 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 $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ 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.

Asymptotics and Renewal Approximation in the Online Selection of Increasing Subsequence

“…Arlotto et al [2] employed (1) and a de-Poissonisation argument to show that, indeed, the strategy given by ( 33) is within O(log n) from the optimum for n large. Extending the methods of the present paper, the latter result has been strengthened recently in [17] : the expected length of subsequence selected with acceptance window (33) is √ 2n − (log n)/12 + O(1), and this is within O(1) from the optimum.…”

“…Since θ * (z) = z for small z this has a simple pole at 0, but the singularity is compensated in (17), so Lemma 3 and ( 13) ensure that…”

“…Proof. Using the integral representation (17) The lemma applied to the right-hand side of (19) gives u ′′ (z) = O(z −2 ). In ( 9) we replace θ * by 1/2, expand u(z − y) − u(z) = −yu ′ (z) + O(z −2 ) and integrate to obtain with some algebra…”