2025/08/20 by Nirjhor, Jubayer, Wein, Nicole
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2508.14361
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+ε) 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(√log n ⋅loglog n +log ε-1), and a lower bound of Ω(log n / loglog n) for deterministic algorithms. We obtain a deterministic algorithm with quasi-polylogarithmic cost (ε-1log n)O(log log n). Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost O(ε-1log2 n).