Abstract
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 a small probability that the wrong comparison outcome is observed. We design a randomized algorithm that solves this problem with a success probability of at least 1-q for q in (0, frac{n-k}{n}) and any k in [1, n-1] using Obig ( frac{n}{k} big lceil log frac{1}{q} big rceil big ) comparisons in expectation (if k ge n or q ge frac{n-k}{n} the problem becomes trivial). Then, we prove that the expected number of comparisons needed by any algorithm that succeeds with probability at least 1-q must be {varOmega }(frac{n}{k}log frac{1}{q}) whenever q is bounded away from frac{n-k}{n}, thus implying that the expected number of comparisons performed by our algorithm is asymptotically optimal in this range. Moreover, we show that the approximate minimum selection problem can be solved using O( (frac{n}{k} + log log frac{1}{q}) log frac{1}{q}) comparisons in the worst case, which is optimal when q is bounded away from frac{n-k}{n} and k = Obig ( frac{n}{log log frac{1}{q}}big ).
Highlights
In an ideal world, computational tasks are always carried out reliably, i.e., every operation performed by an algorithm behaves exactly as intended
An alternative approach deliberately allows errors to interfere with the execution of an algorithm, in the hope that the computed solution will still be good, at least in an approximate sense. This begs the question: is it possible to devise algorithms that cope with faults by design and return solutions that are demonstrably good?. We investigate this question by considering a generalization of the fundamental problem of finding the minimum element in a totally-ordered set: in the fault-tolerant approximate minimum selection problem (FT-Min(k) for short) we wish to return one of the smallest k elements in a collection of size n > k using only unreliable pairwise comparisons, i.e., comparisons in which the result can sometimes be incorrect due to errors
This provides the sought lower bound on ν and completes the proof
Summary
Computational tasks are always carried out reliably, i.e., every operation performed by an algorithm behaves exactly as intended. We investigate this question by considering a generalization of the fundamental problem of finding the minimum element in a totally-ordered set: in the fault-tolerant approximate minimum selection problem (FT-Min(k) for short) we wish to return one of the smallest k elements in a collection of size n > k using only unreliable pairwise comparisons, i.e., comparisons in which the result can sometimes be incorrect due to errors. This allows, for example, to find a representative in the top percentile of the input set, or to obtain a good estimate of the minimum from a set of noisy observations. Before presenting our results in more detail, we briefly discuss the considered error model
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