Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

TL;DR

This paper studies the optimal method for sequentially selecting a unimodal subsequence from a random sequence, showing near-optimal performance compared to an all-knowing prophet, and extends to more complex monotone block subsequences.

Contribution

It introduces an optimal sequential selection strategy for unimodal and d+1 monotone block subsequences, providing performance bounds relative to the prophet.

Findings

Achieves within a factor of sqrt(2) of the prophet's performance.

Extends analysis to subsequences with multiple monotone blocks.

Covers the special case of monotone subsequences.

Abstract

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.