In \emph{Online Sorting}, an array of $n$ initially empty cells is given. At each time step $t$, an element $x_t \in [0,1]$ arrives and must be placed irrevocably into an empty cell without any knowledge of future arrivals. We aim to minimize the sum of absolute differences between pairs of elements placed in consecutive array cells, seeking an online placement strategy that results in a final array close to a sorted one. An interesting multidimensional generalization, a.k.a. the \emph{Online Travelling Salesperson Problem}, arises when the request sequence consists of points in the $d$-dimensional unit cube and the objective is to minimize the sum of euclidean distances between points in consecutive cells. Motivated by the recent work of (Abrahamsen, Bercea, Beretta, Klausen and Kozma; ESA 2024), we consider the \emph{stochastic version} of Online Sorting (\textit{resp.} Online TSP), where each element (\textit{resp.} point) $x_t$ is an i.i.d. sample from the uniform distribution on $[0, 1]$ (\textit{resp.} $[0,1]^d$). By carefully decomposing the request sequence into a hierarchy of balls-into-bins instances, where the balls to bins ratio is large enough so that bin occupancy is sharply concentrated around its mean and small enough so that we can efficiently deal with the elements placed in the same bin, we obtain an online algorithm that approximates the optimal cost within a factor of $O(\log^2 n)$ with high probability. Our result comprises an exponential improvement on the previously best known competitive ratio of $\tilde{O}(n^{1/4})$ for Stochastic Online Sorting due to (Abrahamsen et al.; ESA 2024) and $O(\sqrt{n})$ for (adversarial) Online TSP due to (Bertram, ESA 2025).

翻译：在\emph{在线排序}问题中，给定一个包含$n$个初始为空单元的数组。在每个时间步$t$，元素$x_t \in [0,1]$到达且必须被不可撤销地放置到一个空单元中，且无法预知未来到达的元素。我们的目标是最小化放置在连续数组单元中的元素对之间的绝对差值之和，从而寻求一种在线放置策略，使得最终数组接近排序状态。一个有趣的多维推广（即\emph{在线旅行商问题}）出现在请求序列由$d$维单位立方体中的点构成时，其目标是最小化连续单元中点之间的欧几里得距离之和。受（Abrahamsen, Bercea, Beretta, Klausen and Kozma; ESA 2024）近期工作的启发，我们考虑在线排序（\textit{相应地}在线TSP）的\emph{随机版本}，其中每个元素（\textit{相应地}点）$x_t$是从$[0, 1]$（\textit{相应地}$[0,1]^d$）上的均匀分布中独立同分布抽取的样本。通过将请求序列仔细分解为一系列球入箱实例的层次结构（其中球与箱的比例足够大使箱的占用率高度集中于其均值附近，同时又足够小使得我们能高效处理放置在同一箱中的元素），我们得到一种在线算法，能以高概率在$O(\log^2 n)$因子内逼近最优成本。我们的结果相较于先前已知的最佳竞争比实现了指数级改进：针对随机在线排序的$\tilde{O}(n^{1/4})$（Abrahamsen等人; ESA 2024）以及针对（对抗性）在线TSP的$O(\sqrt{n})$（Bertram, ESA 2025）。