Quickest online selection of an increasing subsequence of specified size

Abstract

Abstract. Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length n. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time.Mathematics Subject Classification (2010): Primary: 60C05, 90C40; Secondary: 60G40, 90C27, 90C39Key Words: Increasing subsequence problem, online selection, sequential selection, time-focused decision problem, dynamic programming, Markov decision problem. Increasing Subsequences and Time-Focused SelectionIf X 1 , X 2 , . . . is a sequence of independent random variables with a common continuous distribution F , then Here we consider a new kind of decision problem where one seeks to select as quickly as possible an increasing subsequence of a prespecified length n. More precisely, at time i, when the decision maker is first presented with X i , a decision

An adaptive O(log n)‐optimal policy for the online selection of a monotone subsequence from a random sample

“…Many and it would be worthwhile to understand how the right-hand side changes with the initial state value s. When s = 0 we have from Theorem 1 that E[L n (π * n )] ≤ E[L n ( π n )] + 2{log(n) + 1} for all n ≥ 1, but the actual expected performance of policy π n seems to be much tighter. Based on an extensive numerical analysis, 1 we conjecture that there is a constant 0 < c < ∞ such that E[L n (π * n )] ≤ E[L n ( π n )] + c for all n ≥ 1.…”

“…. with common continuous distribution F and seeks to construct a monotone subsequence X τ 1 ≤ X τ 2 ≤ · · · ≤ X τ j (1) where the indices 1 ≤ τ 1 < τ 2 < · · · < τ j are stopping times with respect to the σ -fields F i = σ {X 1 , X 2 , . .…”

“…In the first, the decision maker seeks to maximize the expected number of selected elements when n are sequentially revealed . In contrast, in the second approach the decision maker's objective is to minimize the expected time it takes to construct a monotone subsequence with n elements . Here, we confine our attention to the first—more classical—approach.…”

On sequential selection and a first passage problem for the Poisson process

“…In the special case of uniform [0, 1] marks and the bin of unit capacity, the equivalence in terms of the optimal policies has been commonly argued by comparing the dynamic programming equations for the value function [1,9]. It was also noticed in [9] (p. 455) that the greedy policy which selects every item that fits in the remaining capacity outputs the same length as the policy selecting every consequitive record (a mark bigger than all marks seen so far).…”

Sequential selection of a monotone subsequence from a random permutation

“…Gnedin (1999) later gave a much different proof of (23) and generalized the bound in a way that accommodate random samples with random sizes. More recently, Arlotto, Mossel and Steele (2015) obtained yet another proof (23) as a corollary to bounds on the quickest selection problem, which is an informal dual to the traditional selection problem.…”