68M20-ProofOfAmdahlsLaw.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ProofOfAmdahlsLaw}
\pmcreated{2013-03-22 13:10:36}
\pmmodified{2013-03-22 13:10:36}
\pmowner{lieven}{1075}
\pmmodifier{lieven}{1075}
\pmtitle{proof of Amdahl's Law}
\pmrecord{5}{33622}
\pmprivacy{1}
\pmauthor{lieven}{1075}
\pmtype{Proof}
\pmcomment{trigger rebuild}
\pmclassification{msc}{68M20}

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Suppose an algorithm needs $n$ operations to compute the result. With 1 processor, the algorithm will take $n$ time units. With $N$ processors, the $(1-f)n$ parallelizable operations will take $\frac{(1-f)n}{N}$ time units and the remaining $fn$ non parallelizable operations will take $fn$ time units for a total running time of $fn+\frac{(1-f)n}{N}$ time units. So the speedup $S$ is $\frac{n}{fn+\frac{(1-f)n}{N}}=\frac{1}{f+\frac{1-f}{N}}$.
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