AI 中文总结
研究随机顺序在线排序，它介于对抗和随机设置之间。通过在秩空间解决问题，得到高概率下O(log²n)竞争算法，还研究多维推广随机顺序在线旅行商问题，得到高概率下O(log³n)竞争算法。
AI 中文摘要
在在线排序中，给定一个有n个初始空单元格的数组A。在每个时间步t∈[n]，一个元素x_t∈[0,1]到达，必须不可撤销地放入一个空单元格，且不知未来到达元素。目标是最小化分配到相邻单元格的元素之间绝对差的总和。该问题在对抗性和随机输入模型下都有研究。对于对抗序列，已有紧的O(√n)竞争算法。对于随机序列，Hu给出期望下的O(log n·2^(O(log* n)))竞争算法并证明Ω(log n)下界，Kalavas等人给出高概率下的O(log²n)竞争算法，Hermansen设计出期望下的O(log n)竞争算法。本文研究随机顺序在线排序，输入是对抗选择的多重集但其元素按均匀随机顺序到达。通过在秩空间解决问题，证明了高概率下的O(log²n)竞争算法，还研究了多维推广随机顺序在线旅行商问题并得到高概率下的O(log³n)竞争算法。
英文摘要
In Online Sorting, we are given an array $A$ of $n$ initially empty cells. At each time step $t\in[n]$, an element $x_t\in[0,1]$ arrives and must be placed irrevocably into an empty cell, without knowledge of future arrivals. The objective is to minimize the sum of absolute differences between elements assigned to adjacent cells. The problem has been studied under both adversarial and stochastic input models. For adversarial sequences, Aamand, Abrahamsen, Beretta, and Kleist (SODA'23) gave a tight $O\sqrt n)$-competitive algorithm, fully resolving the worst-case setting. For stochastic sequences, in which the elements are drawn i.i.d.\ from $U[0,1]$, Hu (SODA'26) gave an $\log n\cdot 2^{O(\log^* n)}$-competitive algorithm in expectation and proved an $Ω(\log n)$ lower bound, while Kalavas, Platanos, and Tolias (STACS'26) gave an $O(\log^2 n)$-competitive algorithm with high probability. Very recently, Hermansen (ESA'26) closed the remaining gap by designing an $O(\log n)$-competitive algorithm in expectation. In this work, we study Random-Order Online Sorting, a model interpolating between the adversarial and stochastic settings, that was posed as a challenging open question by Hermansen (ESA'26). Here, the input is a multiset chosen adversarially, but its elements arrive in uniformly random order. We take a different point of view by solving the problem in rank space, and prove an $O(\log^2 n)$-competitive algorithm with high probability, matching the state-of-the-art high probability guarantee for the stochastic setting in this more general model. We also study a multidimensional generalization, which we call Random-Order Online TSP, and obtain an $O(\log^3 n)$-competitive algorithm with high probability.
