Finding Mode Using Equality Comparisons

Abstract

We consider the problem of finding the mode (an element that appears the maximum number of times) in a list of elements that are not necessarily from a totally ordered set. Here, the relation between elements is determined by ‘equality’ comparisons whose outcome is $=$ when the two elements being compared are equal and $\ne$ otherwise. In sharp contrast to the $\varTheta (\frac{n\lg n}{m})$ bound known in the classical three way comparison model where elements are from a totally ordered set, a recent paper gave an $O(\frac{n^2}{m})$ upper bound and $\varOmega (\frac{n^2}{m})$ lower bound for the number of comparisons required to find the mode, where m is the frequency of the mode. While the number of comparisons made by the algorithm is roughly $\frac{n^2}{m}$, it is not clear how the necessary bookkeeping required can be done to make the rest of the operations take $\varTheta (\frac{n^2}{m})$ time.

In this paper, we give two mode finding algorithms, one taking at most $\frac{2 n^2}{m}$ comparisons and another taking at most $\frac{3n^2}{2m} + O(\frac{n^2}{m^2})$ comparisons. The bookkeeping required for both the algorithms are simple enough to be implemented in $O(\frac{n^2}{m})$ time. The second algorithm generalizes a classical majority finding algorithm due to Fischer and Salzberg.