A Randomised Approach to Distributed Sorting

We introduce and analyse a new, extremely simple, randomised sorting algorithm: - choose a pair of indices $\{i,j\}$ according to some distribution $q$; - sort the elements in positions $i$ and $j$ of the array in ascending order. Choosing $q_{\{i,j\}}\propto 1\mathrm{/}\mathrm{\mid }j-i\mathrm{\mid }$ yields an order-$n(\log n{)}^2$ sorting time. We call it the harmonic sorter. The sorter trivially parallelises in the asynchronous setting, yielding a linear speed-up. We also exhibit a low-communication, synchronous version with a linear speed-up. We compare and contrast this algorithm with other sorters, and discuss some of its benefits, particularly its robustness and amenability to parallelisation and distributed computing.