Finding a Mediocre Player

01/25/2019
by   Adrian Dumitrescu, et al.
0

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)-mediocre if it is neither among the top i nor among the bottom j elements of S. More than 40 years ago, Yao suggested a stunningly simple 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 such an 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)). 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)); 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.)

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/09/2019

Fragile Complexity of Comparison-Based Algorithms

We initiate a study of algorithms with a focus on the computational comp...
research
05/05/2018

A Nearly Optimal Algorithm for Approximate Minimum Selection with Unreliable Comparisons

We consider the approximate minimum selection problem in presence of ind...
research
03/02/2020

Concurrent Disjoint Set Union

We develop and analyze concurrent algorithms for the disjoint set union ...
research
07/18/2018

Deterministic oblivious distribution (and tight compaction) in linear time

In an array of N elements, M positions and M elements are "marked". We s...
research
11/15/2018

Secretary Ranking with Minimal Inversions

We study a twist on the classic secretary problem, which we term the sec...
research
11/26/2021

An Optimal Algorithm for Finding Champions in Tournament Graphs

A tournament graph T = (V, E ) is an oriented complete graph, which can ...
research
04/12/2020

Multiparty Selection

Given a sequence A of n numbers and an integer (target) parameter 1≤ i≤ ...

Please sign up or login with your details

Forgot password? Click here to reset