Online Sorting with Our Eyes Wide Shut

Abstract

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 $\Omega(\log n)$ 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.