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 +logε ^-1)) , and a lower bound of (log n / loglog n) for deterministic algorithms. We obtain a deterministic4 algorithm with quasi-polylogarithmic cost ( ε ^-1log n) ^O( loglog n) . Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost O(ε ^-1log ^2 n) .
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