Improved Online Sorting

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