Spin-the-bottle sort and annealing sort: Oblivious sorting via round-robin random comparisons

We study sorting algorithms based on randomized round-robin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require Ω(n2 log n) expected time in order to succeed, and that in O(n2 log n) time this algorithm succeeds with high probability for any input. We also show there is an implementation of Annealing sort that runs in O(n log n) time and succeeds with very high probability.