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 adversariallychosen 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 Θ(n log(1/ǫ)) comparisons to output a correct answer with probability at least 1 − ǫ. The existence of this algorithm affirmatively resolves an open problem of Ajtai, Feldman, Hassadim, and Nelson [AFHN15].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 ℓ1 distance. Scheffe's algorithm [DL01] outputs a distribution at an ℓ1 distance at most 9 times the minimum and runs in time Θ(n 2 log n). Using maximum selection, we propose an algorithm with the same approximation guarantee but run time of Θ(n log n). * Part of this paper appeared in [AJOS14].However, in many applications, the pairwise comparisons may be imprecise. For example, when comparing two random numbers, such as stock performances, or team strengths, the output of the comparison may vary due to chance. Consequently, a number of researchers have considered maximum selection and sorting with imperfect, or noisy, comparators.The comparators in these models mostly function correctly, but occasionally may produce an inaccurate comparison result, where the form of inaccuracy is dictated by the application. The Bradley-Terry-Luce [BT52] model assumes that if two values x and y are compared, then x is selected as the larger with probability x/(x + y). Observe that the comparison is correct with probability max{x, y}/(x + y) ≥ 1/2. Algorithms for ranking and estimating weights under this model were proposed, e.g., in [NOS12]. Another model assumes that the output of any comparator gets reversed with probability less than 1/2. Algorithms applying this model for maximum selection were proposed in [AGHB + 94] and for ranking in [KK07, BM08].We consider a third model where, unlike the previous models, the comparison outcome can be adversarial. If the numbers compared are more than a threshold ∆ apart, the comparison is correct, while if they differ by at most ∆, the comparison is arbitrary, and possibly even adversarial.This model can be partially motivated by physical observations. Measurements are regularly quantized and often adulterated with some measurement...

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Abstract-The problem of finding optimal querying policy, for expected query complexity of symmetric boolean threshold functions was solved in [1] in the context of collocated networks. In this paper, instead of considering the optimal policy to compute the functions, we define the problem of verification of the function value. We use this idea to provide a simpler proof of the optimal querying policy for threshold functions. The method is more generic and is extended to delta and some other symmetric functions. We also provide some partial results for interval functions and finally address a question posed in [1]. Recently we have extended these results to any symmetric function of boolean inputs, which we mention at the end.

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