Maximum Selection and Sorting with Adversarial Comparators and an Application to Density Estimation

We study maximum selection and sorting of $n$ numbers using pairwise comparators that output the larger of their two inputs if the inputs are more than a given threshold apart, and output an adversarially-chosen input otherwise. We consider two adversarial models. A non-adaptive adversary that decides on the outcomes in advance based solely on the inputs, and an adaptive adversary that can decide on the outcome of each query depending on previous queries and outcomes. Against the non-adaptive adversary, we derive a maximum-selection algorithm that uses at most $2n$ comparisons in expectation, and a sorting algorithm that uses at most $2n \ln n$ comparisons in expectation. These numbers are within small constant factors from the best possible. Against the adaptive adversary, we propose a maximum-selection algorithm that uses $\Theta(n\log (1/{\epsilon}))$ comparisons to output a correct answer with probability at least $1-\epsilon$. The existence of this algorithm affirmatively resolves an open problem of Ajtai, Feldman, Hassadim, and Nelson. Our study was motivated by a density-estimation problem where, given samples from an unknown underlying distribution, we would like to find a distribution in a known class of $n$ candidate distributions that is close to underlying distribution in $\ell_1$ distance. Scheffe's algorithm outputs a distribution at an $\ell_1$ distance at most 9 times the minimum and runs in time $\Theta(n^2\log n)$. Using maximum selection, we propose an algorithm with the same approximation guarantee but run time of $\Theta(n\log n)$.