Approximate Selection with Unreliable Comparisons in Optimal Expected Time

Abstract

Given n elements, an integer k ≤ n/2 and a parameter ε ≥ 1/n, we study the problem of selecting an element with rank in (k-nε, k+nε] using unreliable comparisons where the outcome of each comparison is incorrect independently with a constant error probability, and multiple comparisons between the same pair of elements are independent. In this fault model, the fundamental problems of finding the minimum, selecting the k-th smallest element and sorting have been shown to require Θ(n log 1/Q), Θ(n log k/Q) and Θ(n log n/Q) comparisons, respectively, to achieve success probability 1-Q [Uriel Feige et al., 1994]. Considering the increasing complexity of modern computing, it is of great interest to develop approximation algorithms that enable a trade-off between the solution quality and the number of comparisons. In particular, approximation algorithms would even be able to attain a sublinear number of comparisons. Very recently, Leucci and Liu [Stefano Leucci and Chih-Hung Liu, 2022] proved that the approximate minimum selection problem, which covers the case that k ≤ nε, requires expected Θ(ε^{-1} log 1/Q) comparisons, but the general case, i.e., for nε < k ≤ n/2, is still open.
We develop a randomized algorithm that performs expected O(k/n ε^{-2} log 1/Q) comparisons to achieve success probability at least 1-Q. For k = n ε, the number of comparisons is O(ε^{-1} log 1/Q), matching Leucci and Liu’s result [Stefano Leucci and Chih-Hung Liu, 2022], whereas for k = n/2 (i.e., approximating the median), the number of comparisons is O(ε^{-2} log 1/Q). We also prove that even in the absence of comparison faults, any randomized algorithm with success probability at least 1-Q performs expected Ω(min{n, k/n ε^{-2} log 1/Q}) comparisons. As long as n is large enough, i.e., when n = Ω(k/n ε^{-2} log 1/Q), our lower bound demonstrates the optimality of our algorithm, which covers the possible range of attaining a sublinear number of comparisons. Surprisingly, for constant Q, our algorithm performs expected O(k/n ε^{-2}) comparisons, matching the best possible approximation algorithm in the absence of computation faults. In contrast, for the exact selection problem, the expected number of comparisons is Θ(n log k) with faults versus Θ(n) without faults. Our results also indicate a clear distinction between approximating the minimum and approximating the k-th smallest element, which holds even for the high probability guarantee, e.g., if k = n/2, Q = 1/n and ε = n^{-α} for α ∈ (0, 1/2), the asymptotic difference is almost quadratic, i.e., Θ̃(n^α) versus Θ̃(n^{2α}).