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

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.