A Polylogarithmic Algorithm for Stochastic Online Sorting

2025-08-17

AI Summary

This paper studies the stochastic online sorting problem: a sequence of real numbers drawn i.i.d. from the uniform distribution arrives sequentially and must be irrevocably placed into an initially empty array of length $n$; the objective is to minimize the sum of absolute differences between adjacent elements in the final array. We propose a probabilistic analysis framework based on a hierarchical bucketing structure and dynamic position assignment. Under the stochastic input assumption, our algorithm achieves an $O(log^2 n)$ competitive ratio—exponentially improving upon the previous best $ ilde{O}(n^{1/4})$, and this bound holds with high probability. Furthermore, we extend our framework to the fixed-dimensional stochastic online TSP, obtaining for the first time an $O(log^2 n)$ competitive ratio in this setting, thereby breaking the performance barriers inherent in classical geometric online algorithms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

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.

Problem

• Develops polylogarithmic competitive algorithm for stochastic online sorting
• Improves exponential bound over prior work for sorting problem
• Extends approach to multidimensional online TSP with fixed dimension

Innovation

• Polylogarithmic competitive algorithm for stochastic sorting
• Exponential improvement over previous n^1/4 bound
• Extends to Online TSP with logarithmic competitive ratio