The online monotone array completion problem

2026-06-30

AI Summary

This study addresses the online problem of filling an initially empty array by irrevocably placing each arriving sample from the uniform distribution on $[0,1]$ into an empty slot such that the filled entries remain non-decreasing from left to right, with the goal of minimizing the expected time to fill the entire array. The work establishes the first tight characterization of the optimal expected time as $(\frac{1}{2} + o(1))n \log n$ and presents an explicit deterministic strategy achieving this bound. In a variant allowing overwriting of already placed elements, the authors design a novel strategy that reduces the expected filling time to $O(n\sqrt{\log n})$, highlighting a fundamental gap between the two models. Combining probabilistic analysis, online algorithm design, and adversarial lower-bound techniques, they also prove a universal lower bound of $(\frac{1}{2} - o(1))n \log n$.

Abstract

Consider the following online filling game. An array of length $n$ is initially empty. At each time step one observes an independent sample from $\mathrm{Unif}[0,1]$ and must either discard it or place it irrevocably into an empty position of the array, while preserving the constraint that the occupied entries are non-decreasing from left to right. Among all possible strategies, what is the optimal expected time required to fill the array? Let $v_n$ denote this optimal expected completion time. Our main result determines $v_n$ up to lower-order terms: \[ v_n=\left(\frac12+o(1)\right)n\log n. \] More precisely, no strategy, even if randomized and adaptive, can have expected completion time below $\left(\frac12-o(1)\right)n\log n$, while we provide an explicit deterministic strategy whose expected completion time is at most $\left(\frac12+o(1)\right)n\log n$. For comparison, the natural coupon-collector strategy, which partitions $[0,1]$ into $n$ equal intervals and reserves one array position for each interval, has expected completion time $(1+o(1))n\log n$. We also consider a with-replacement version of the game, in which previously placed entries may be overwritten. For this variant, we give a deterministic strategy with expected completion time $O(n\sqrt{\log n})$, thereby establishing a separation between the two models.

Problem

• online monotone array completion
• irrevocable placement
• non-decreasing constraint
• expected completion time
• uniform random samples

Innovation

• online monotone array completion
• optimal expected time
• deterministic strategy
• coupon collector
• with-replacement model