On Adaptive Integer Sorting

This paper considers integer sorting on a RAM. We show that adaptive sorting of a sequence with qn inversions is asymptotically equivalent to multisorting groups of at most q keys, and a total of n keys. Using the recent \(O(n\sqrt{\log\log n})\) expected time sorting of Han and Thorup on each set, we immediately get an adaptive expected sorting time of \(O(n\sqrt{\log\log q})\). Interestingly, for any positive constant e, we show that multisorting and adaptive inversion sorting can be performed in linear time if \(q\leq 2^{(\log n)^{1-\varepsilon}}\). We also show how to asymptotically improve the running time of any traditional sorting algorithm on a class of inputs much broader than those with few inversions.