A generalization of Amdahl's law and relative conditions of parallelism

TL;DR

This paper extends Amdahl's law to better understand the limits of parallel computing, establishing mathematical relations for speedup, classes of parallelism, and conditions for superlinear speedup using differential calculus.

Contribution

It introduces a generalized form of Amdahl's law, providing new mathematical relations and conditions for superlinear speedup in parallel implementations.

Findings

Derived conditions for superlinear speedup.

Defined classes of parallelism based on speedup.

Established mathematical relations involving processors and problem size.

Abstract

In this work I present a generalization of Amdahl's law on the limits of a parallel implementation with many processors. In particular I establish some mathematical relations involving the number of processors and the dimension of the treated problem, and with these conditions I define, on the ground of the reachable speedup, some classes of parallelism for the implementations. I also derive a condition for obtaining superlinear speedup. The used mathematical technics are those of differential calculus. I describe some examples from classical problems offered by the specialized literature on the subject.