Spin-the-bottle Sort and Annealing Sort: Oblivious Sorting via Round-robin Random Comparisons

TL;DR

This paper compares two randomized, data-oblivious sorting algorithms, Spin-the-bottle sort and Annealing sort, demonstrating Annealing sort's superior efficiency with an implementation achieving near-optimal sorting time.

Contribution

The paper introduces and analyzes Annealing sort as a more efficient data-oblivious sorting algorithm compared to Spin-the-bottle sort, with proven high-probability success and optimal runtime.

Findings

Spin-the-bottle sort can take $oldsymbol{ ext{Omega}(n^2 ext{log} n)}$ expected time on some inputs.

Spin-the-bottle sort succeeds in $oldsymbol{O(n^2 ext{log} n)}$ time with high probability.

Annealing sort can be implemented to run in $oldsymbol{O(n ext{log} n)}$ time with very high probability.

Abstract

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 \emph{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 $\Omega(n^2\log n)$ expected time in order to succeed, and that in $O(n^2\log n)$ 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.