Sorting with adversarial comparators and application to density estimation

We consider the problems of sorting and maximum-selection of n elements using adversarial comparators. We derive a maximum-selection algorithm that uses 8n comparisons in expectation, and a sorting algorithm that uses 4n log₂ n comparisons in expectation. Both are tight up to a constant factor. Our adversarial-comparator model was motivated by the practically important problem of density-estimation, where we observe samples from an unknown distribution, and try to determine which of n known distributions is closest to it. Existing algorithms run in Ω(n²) time. Applying the adversarial comparator results, we derive a density-estimation algorithm that runs in only O(n) time.