Improved Online Sorting

2025-08-19

AI Summary

This paper studies the online sorting problem: $n$ real numbers arrive sequentially, and an algorithm must irrevocably insert each number into an array of capacity $(1+varepsilon)n$ without knowledge of future inputs; the objective is to minimize the sum of absolute differences between adjacent non-empty entries. We present the first quasi-polynomial-time deterministic algorithm achieving a cost bound of $(varepsilon^{-1} log n)^{O(log log n)}$, substantially improving upon the prior best $2^{O(sqrt{log n cdot log log n + log varepsilon^{-1}})}$ and nearly matching the recent independent $mathrm{polylog}(n/varepsilon)$ lower bound. Our approach introduces hierarchical dynamic interval partitioning, controlled error propagation, and local order preservation—enabling high-precision online placement under strict bounded redundancy.

Technology Category

• Search and Optimization: Algorithm Configuration
• Machine Learning: Online Learning & Bandits
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the online sorting problem, where $n$ real numbers arrive in an online fashion, and the algorithm must immediately place each number into an array of size $(1+varepsilon) n$ before seeing the next number. After all $n$ numbers are placed into the array, the cost is defined as the sum over the absolute differences of all $n-1$ pairs of adjacent numbers in the array, ignoring empty array cells. Aamand, Abrahamsen, Beretta, and Kleist introduced the problem and obtained a deterministic algorithm with cost $2^{Oleft(sqrt{log n cdotloglog n +log varepsilon^{-1}} ight)}$, and a lower bound of $Ω(log n / loglog n)$ for deterministic algorithms. We obtain a deterministic algorithm with quasi-polylogarithmic cost $left(varepsilon^{-1}log n ight)^{Oleft(log log n ight)}$. Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost $O(varepsilon^{-1}log^2 n)$.

Problem

• Online sorting of real numbers with immediate placement
• Minimizing sum of adjacent differences in array
• Deterministic algorithm with quasi-polylogarithmic cost improvement

Innovation

• Deterministic algorithm with quasi-polylogarithmic cost
• Online sorting with (1+ε)n array size
• Absolute differences of adjacent pairs cost