Optimal Parallel Sorting with Comparison Errors

We present comparison-based parallel algorithms for sorting n comparable items subject to comparison errors. We consider errors to occur according to a well-studied framework, where the comparison of two elements returns the wrong answer with a fixed probability. In the persistent model, the result of the comparison of two given elements, x and y, always has the same result, and is independent of all other pairs of elements. In the non-persistent model, the result of the comparison of each pair of elements, x and y, is independent of all prior comparisons, including for x and y. It is not possible to always correctly sort a given input set in the persistent model, so we study algorithms that achieve a small maximum dislocation and small total dislocation of the elements in the output permutation. In this paper, we provide parallel algorithms for sorting with comparison errors in the persistent and non-persistent models. Our algorithms are asymptotically optimal in terms of their span, work, and, in the case of persistent errors, maximum and total dislocation. The main results are algorithms for the binary-forking parallel model with atomics, but we also provide algorithms for the CREW PRAM model. Our algorithms include a number of novel techniques and analysis tools, including a PRAM-to-binary-forking-model simulation result, and are the first optimal parallel algorithms for the persistent model and the non-persistent model in the binary-forking parallel model with atomics. In particular, our algorithms have O(log n) span, O(n log n) work, and, in the case of the persistent model, O(log n) maximum dislocation and O(n) total dislocation, with high probability. We achieve similar results for the CREW PRAM model, which are the first optimal methods for the persistent model and the first optimal results for the non-persistent model with reasonable constant factors in the performance bounds.