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 incor...
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(纸本)9783959772662
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 [9]. 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 [23] 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 −2 log 1/Q) comparisons to achieve success probability at least 1 − Q. For k = nΕ, the number of comparisons is O(Ε−1 log 1/Q), matching Leucci and Liu’s result [23], 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 expe
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