Sorting and Selection with Imprecise Comparisons

Reviewer: M. Rahman

Sorting and selection with imprecise comparisons has long been the focus of extensive research attention among theoreticians. There have been a number of models and frameworks of imprecision considered in the literature. In this paper, the authors study the following simple model: when comparing two elements, the comparison is made correctly if the values of those elements differ by at least some constant δ>0; otherwise, the outcome of the comparison is unpredictable. The authors here present a number of interesting algorithmic results, as well as some useful lower bounds. They first present a deterministic max-finding algorithm with error 2 using 2 n 3/2 comparisons. Using this algorithm recursively, they present deterministic algorithms with error k that require comparisons. They further prove a lower bound of comparisons for the problem. They also present a linear-time randomized algorithm that achieves error 3 with probability at least 1 - 1/ n 2, establishing that randomization greatly changes the complexity of the problem. Subsequently, the authors prove that the selection of an element of any order i can be achieved using comparisons, and sorting with error k can be done using comparisons. They also prove a lower bound of comparisons for sorting with error k . All of the algorithms have the same time order as the number of comparisons they make. Notably, to obtain the algorithms for larger error k , the authors use several different ways to partition elements, recursively apply algorithms for smaller error, and then finally combine results. As noted by the authors, a similar approach to finding a small set of "good" elements was used by Borgstrom and Kosaraju [1] in the context of noisy binary search. The model studied by the authors is inspired by both imprecision in human judgment of values and also by bounded but potentially adversarial errors in sporting tournaments, which make it all the more interesting and useful. In summary, the results achieved by the authors basically establish that there exist algorithms that are robust to imprecision in comparisons while using substantially fewer comparisons than the naive methods. Additionally, the deterministic algorithms presented here can be used in another interesting well-studied problem of finding a k -king in any tournament graph while minimizing the number of edges checked. Finally, the authors list a number of interesting and natural open problems. Online Computing Reviews Service