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. 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.
Highlights
The online increasing subsequence problems are stochastic optimisation problems concerned with non-anticipating policies aimed to select an increasing subsequence from a sequence of random items X1, X2, . . . with known continuous distribution F, which, without loss of generality, will be assumed uniform on [0, 1]
An online policy for selecting an increasing subsequence is a collection of stopping times τ = (τ1, τ2, . . .) adapted to the sequence of sigma-fields Fi = σ {X1, X2, . . . , Xi}, 1 ≤ i < ∞, and satisfying (i) τ1 < τ2 < · · ·, (ii) Xτ1 < Xτ2 < · · ·
We studied two classical sequential selection problems initiated by Samuels and Steele [20] and Arlotto et al [3]
Summary
The online increasing subsequence problems are stochastic optimisation problems concerned with non-anticipating policies aimed to select an increasing subsequence from a sequence of random items X1, X2, . . . with known continuous distribution F , which, without loss of generality, will be assumed uniform on [0, 1]. The online increasing subsequence problems are stochastic optimisation problems concerned with non-anticipating policies aimed to select an increasing subsequence from a sequence of random items X1, X2, . With known continuous distribution F , which, without loss of generality, will be assumed uniform on [0, 1]. The online constraint requires to accept or reject Xi at time i, when the item is observed, with the decision becoming immediately terminal. An online policy for selecting an increasing subsequence is a collection of stopping times τ = . .) adapted to the sequence of sigma-fields Fi = σ {X1, X2, . Xi}, 1 ≤ i < ∞, and satisfying (i) τ1 < τ2 < · · · , (ii) Xτ1 < Xτ2 < · · ·
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