Sorting and Selection with Imprecise Comparisons

Abstract

In experimental psychology, the method of paired comparisons was proposed as a means for ranking preferences amongst n elements of a human subject. The method requires performing all $\binom{n}{2}$ comparisons then sorting elements according to the number of wins. The large number of comparisons is performed to counter the potentially faulty decision-making of the human subject, who acts as an imprecise comparator.

We consider a simple model of the imprecise comparisons: there exists some δ> 0 such that when a subject is given two elements to compare, if the values of those elements (as perceived by the subject) differ by at least δ, then the comparison will be made correctly; when the two elements have values that are within δ, the outcome of the comparison is unpredictable. This δ corresponds to the just noticeable difference unit (JND) or difference threshold in the psychophysics literature, but does not require the statistical assumptions used to define this value.

In this model, the standard method of paired comparisons minimizes the errors introduced by the imprecise comparisons at the cost of $\binom{n}{2}$ comparisons. We show that the same optimal guarantees can be achieved using 4 n ^3/2 comparisons, and we prove the optimality of our method. We then explore the general tradeoff between the guarantees on the error that can be made and number of comparisons for the problems of sorting, max-finding, and selection. Our results provide close-to-optimal solutions for each of these problems.