Linear time bounds for median computations

Abstract

New upper and lower bounds are presented for the maximum number of comparisons, f(i,n), required to select the i-th largest of n numbers. An upper bound is found, by an analysis of a new selection algorithm, to be a linear function of n: f(i,n) ≤ 103n/18 < 5.73n, for 1 ≤ i ≤ n. A lower bound is shown deductively to be: f(i,n) ≥ n+min(i,n−i+l) + [log2(n)] − 4, for 2 ≤ i ≤ n−1, or, for the case of computing medians: f([n/2],n) ≥ 3n/2 − 3