The wrong direction of Jensen's inequality is algorithmically right

TL;DR

This paper introduces a method to convert algorithms with random exponential running times into ones with expected running times exponential in the expected logarithm of the original time, improving efficiency without distribution knowledge.

Contribution

It presents a novel algorithmic transformation that guarantees expected running time bounds based on the expectation of the logarithm of the original running time.

Findings

Expected running time is improved to O(e^{E[ln T]})

Transformation does not require distribution knowledge

Applicable to any algorithm with random running time

Abstract

Let $\mathcal{A}$ be an algorithm with expected running time $e^X$, conditioned on the value of some random variable $X$. We construct an algorithm $\mathcal{A'}$ with expected running time $O(e^{E[X]})$, that fully executes $\mathcal{A}$. In particular, an algorithm whose running time is a random variable $T$ can be converted to one with expected running time $O(e^{E[\ln T]})$, which is never worse than $O(E[T])$. No information about the distribution of $X$ is required for the construction of $\mathcal{A}'$.