A Polylogarithmic Algorithm for Stochastic Online Sorting

In the \emph{Online Sorting Problem}, an array of $n$ initially empty cells is given. At each time step $t$, a real number $x_t \in [0,1]$ arrives and must be placed immediately and irrevocably into an empty cell. The objective is to minimize the sum of absolute differences between consecutive entries. The problem was introduced by Aamand, Abrahamsen, Beretta, and Kleist (SODA 2023) as a technical tool for proving lower bounds in online geometric packing problems. In follow-up work, Abrahamsen, Bercea, Beretta, Klausen, and Kozma (ESA 2024) studied the \emph{Stochastic Online Sorting Problem}, where each $x_t$ is drawn i.i.d.\ from $\mathcal{U}(0,1)$, and presented a $\widetilde{O}(n^{1/4})$-competitive algorithm, showing that stochastic input enables much stronger guarantees than in the adversarial setting. They also introduced the \emph{Online Travelling Salesperson Problem (TSP)} as a multidimensional generalization. More recently, Hu, independently and in parallel, obtained a $\log n \cdot 2^{O(\log^* n)}$-competitive algorithm together with a logarithmic lower bound for the \emph{Stochastic Online Sorting Problem}. We give an $O(\log^{2} n)$-competitive algorithm for the \emph{Stochastic Online Sorting Problem} that succeeds w.h.p., achieving an exponential improvement over the $\widetilde{O}(n^{1/4})$ bound of Abrahamsen et al.(ESA 2024). Our approach further extends to the \emph{Stochastic Online TSP} in fixed dimension $d$, where it achieves an $O(\log^2 n)$-competitive ratio.