Improved Online Sorting

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 $(1+\epsilon )n$ 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 $2^{O\left(\sqrt{\log n\cdot \log \log n+\log {\epsilon }^{-1}}\right)}$, and a lower bound of $\hat{I}\text{c◯}(\log n/\log \log n)$ for deterministic algorithms. We obtain a deterministic algorithm with quasi-polylogarithmic cost ${\left({\epsilon }^{-1}\log n\right)}^{O\left(\log \log n\right)}$. Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost $O({\epsilon }^{-1}{\log }^{2}n)$.

10 pages