Finding a Mediocre Player

Abstract

Consider a totally ordered set $S$ of $n$ elements; as an example, a set of tennis players and their rankings. Further assume that their ranking is a total order and thus satisfies transitivity and anti-symmetry. Following Frances Yao (1974), an element (player) is said to be $(i,j)$-\emph{mediocre} if it is neither among the top $i$ nor among the bottom $j$ elements of $S$. Finding a mediocre element is closely related to finding the median element. More than $40$ years ago, Yao suggested a very simple and elegant algorithm for finding an $(i,j)$-mediocre element: Pick $i+j+1$ elements arbitrarily and select the $(i+1)$-th largest among them. She also asked: "Is this the best algorithm?" No one seems to have found a better algorithm ever since. We first provide a deterministic algorithm that beats the worst-case comparison bound in Yao's algorithm for a large range of values of $i$ (and corresponding suitable $j=j(i)$) even if the current best selection algorithm is used. We then repeat the exercise for randomized algorithms; the average number of comparisons of our algorithm beats the average comparison bound in Yao's algorithm for another large range of values of $i$ (and corresponding suitable $j=j(i)$) even if the best selection algorithm is used; the improvement is most notable in the symmetric case $i=j$. Moreover, the tight bound obtained in the analysis of Yao's algorithm allows us to give a definite answer for this class of algorithms. In summary, we answer Yao's question as follows: (i)~"Presently not" for deterministic algorithms and (ii)~"Definitely not" for randomized algorithms. (In fairness, it should be said however that Yao posed the question in the context of deterministic algorithms.)