On malign input distributions for algorithms

Abstract

By a measure we mean a function Μ from {0,1}^* (the set of all binary sequences) to real numbers such that Μ(x)≥0 and Μ({0,1}*)<∞. A malign measure is a measure such that if an input x in {0,1}ⁿ (the set of all binary sequences of length n) is selected with the probability Μ(i)/Μ({0,1}ⁿ) then the worst-case computation time t ^wo _A (n) and the average-case computation time t ^{av, μ} _A (n) of an algorithm A for inputs of length n are functions of n of the same order for any algorithm A. Li and Vitányi found that a priori measures are malign. We show that “a priori”-ness and malignness are different in one strong sense.