Transformations that preserve malignness of universal distributions

Abstract

A function μ(x) that assigns a nonnegative real number μ(x) to each bit string x is said to be malign if, for any algorithm, the worst-case computation time and the average computation time of the algorithm are functions, of the same order when each bit string x is given to the algorithm as an input with the probability that is proportional to the value μ(x). M. Li and P. M. B. Vitányi found that functions that are known as “universal distributions” are malign. We show that if μ(x) is a universal distribution and t is a positive real number, then the function μ(x) ^t is malign or not according as t≥1 or t<1. For t>1, μ(x) ^t is an example of malign functions that are not universal distributions.