A Nearly Optimal Algorithm for Approximate Minimum Selection with Unreliable Comparisons
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We consider the approximate minimum selection problem in presence of independent random comparison faults. This problem asks to select one of the smallest k elements in a linearly-ordered collection of n elements by only performing unreliable pairwise comparisons: whenever two elements are compared, there is constant probability that the wrong answer is returned. We design a randomized algorithm that solves this problem with high probability (w.h.p.) for the whole range of values of k using O( n · ( n/k + n )) expected time. Then, we prove that the expected running time of any algorithm that succeeds w.h.p. must be Ω(n/k n), thus implying that our algorithm is optimal, in expectation, for almost all values of k (and it is optimal up to triple- factors for k = ω(n/ n)). These results are quite surprising in the sense that for k between Ω( n) and c · n, for any constant c<1, the expected running time must still be Ω(n/k n) even in absence of comparison faults. Informally speaking, we show how to deal with comparison errors without any substantial complexity penalty w.r.t. the fault-free case. Moreover, we prove that as soon as k = O( n/ n), it is possible to achieve a worst-case running time of O(n/k n).
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