Sorting and Selection with Imprecise Comparisons

TL;DR

This paper studies algorithms for sorting and selection under imprecise comparisons, establishing optimal methods that balance accuracy and comparison complexity, inspired by human judgment and adversarial errors.

Contribution

It introduces a model of imprecise comparisons, proves optimality of the paired comparison method, and develops near-optimal algorithms for sorting and selection with fewer comparisons.

Findings

Paired comparison method is optimal but comparison-intensive.

Achieves the same accuracy with only 4 n^{3/2} comparisons.

Provides lower bounds and near-optimal algorithms for sorting and selection.

Abstract

We consider a simple model of imprecise comparisons: there exists some $\delta>0$ 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 $\delta$, then the comparison will be made correctly; when the two elements have values that are within $\delta$, the outcome of the comparison is unpredictable. This model is inspired by both imprecision in human judgment of values and also by bounded but potentially adversarial errors in the outcomes of sporting tournaments. Our model is closely related to a number of models commonly considered in the psychophysics literature where $\delta$ corresponds to the {\em just noticeable difference unit (JND)} or {\em difference threshold}. 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 show that in our model the method of paired comparisons has optimal accuracy, minimizing the errors introduced by the imprecise comparisons. However, it is also wasteful, as it requires all $\binom{n}{2}$. 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 strong lower bounds and close-to-optimal solutions for each of these problems.