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

TL;DR

This paper introduces an adaptive online policy for selecting a monotone increasing subsequence from a sequence of independent random variables, achieving near-optimal performance within an $O( ext{log} n)$ gap.

Contribution

The paper presents a simple, adaptive policy that attains $O( ext{log} n)$-optimality for the online monotone subsequence selection problem, with a direct proof method.

Findings

Expected selections are within $O( ext{log} n)$ of the optimal.

The policy is simple and adaptive, suitable for online implementation.

Provides a new proof technique avoiding complex inequalities.

Abstract

Given a sequence of $n$ independent random variables with common continuous distribution, we propose a simple adaptive online policy that selects a monotone increasing subsequence. We show that the expected number of monotone increasing selections made by such a policy is within $O(\log n)$ of optimal. Our construction provides a direct and natural way for proving the $O(\log n)$-optimality gap. An earlier proof of the same result made crucial use of a key inequality of Bruss and Delbaen (2001) and of de-Poissonization.