A new formula for speedup and its characterization

Abstract.

We address the task of measuring the relative speed (speedup) of two systems $A$ and $B$ for solving the same problem. For example, $B$ may be a parallel algorithm, parametrized by the number of processors used, whose running time has to be related to a serial standard algorithm $A$. If $A$ and/or $B$ are randomized or if we are interested in their performance on a (discrete) probability distribution of problem instances, the running times are described by random variables $T^A$ and $T^B$. The speedup of $B$ over $A$ is usually defined as $E(T^A)/E(T^B)$ where $E$ denotes the expected value. In many cases this definition is not appropriate for the user of $A$ or $B$, because the summation in $E(T^A)$ and $E(T^B)$ hides information about the speedup of individual runs. We propose an alternative speedup definition of the form $M(T^A/ T^B)$ and present a set of intuitive functional equations, which any such function $M(T^A/T^B)$ should fulfill. Finally, we prove that the weighted geometric mean is the only solution of these equations.