Quickest Online Selection of an Increasing Subsequence of Specified Size

TL;DR

This paper studies an online algorithm for selecting an increasing subsequence of fixed length from a sequence of random variables, aiming to minimize the expected selection time, and provides asymptotic formulas for its mean and variance.

Contribution

It introduces a dual approach to the online subsequence selection problem, simplifying the derivation of asymptotic formulas for optimal expected time.

Findings

Derived asymptotic formulas for mean and variance of minimal selection time.

Presented a dual formulation that simplifies the analysis of online increasing subsequence selection.

Provided insights into the optimal online decision strategy for subsequence selection.

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.