The refutation of amdahl’s law and its variants

Abstract

Amdahl’s law, imposing a restriction on the speedup achievable by a multiple number of processors, based on the concept of sequential and parallelizable fractions of computations, has been used to justify, among others, asymmetric chip multiprocessor architectures and concerns of “dark silicon”. This paper demonstrates flaws in Amdahl’s law that (i) in theory no inherently sequential fractions of computations exists (ii) sequential fractions appearing in practice are inherently different from parallelizable fractions and therefore usually have different growth rates and that (iii) the time requirement of sequential fractions can be proportional to the number of processors. However, mathematical analyses are also provided to demonstrate that sequential fractions have negligible effect on speedup if the growth rate of the parallelizable fraction is higher than that of the sequential fraction. Examples from computational geometry are given that Amdahl’s law and its variants fail to represent limits to parallel computation. In particular, Gustafson’s law, claimed to be a refutation of Amdahl’s law by some authors, is shown to contradict established theoretical results. We can conclude that no simple formula or law governing concurrency exists.