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\lhead{HW Chap. 5\\\ \\\ }
\rhead{Josh Holtrop\\2008-11-05\\CS 677}
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\noindent
\begin{enumerate}
\item[1.]{
    The best known sequential sorting algorithms have a complexity of $O (n \log n)$.
    So, the speedup factor is given by
    $$ s = \frac{T_s}{T_p} = \frac{n \log n}{cn} = \frac{\log n}{c} $$
}

\item[2.]{
    The total processing time when the program is run on $p$ processors
        will be given by the initialization phase plus the compute phase
        divided by $p$ processors.
    So, the speedup is given by
    $$ s = \frac{T_s}{T_p} = \frac{n + n^3}{n + \frac{n^3}{p}} $$
}

\item[3.]{
    Using Amdahl's law, the maximum speedup is $1/f$, where $f$ is the
        serial fraction of execution time.
    So, the maximum fraction of execution time a program can spend on
        serial code if the parallel version must achieve a speedup
        factor of 10 is 10\%.
}

\vskip 1em
\item[4.]{
    Using Gustafson's law, the scaled speedup factor is given by
    $$ S_G = p + (1 - p) T_s = 8 + (1 - 8) \frac{1}{24} = 7.708 $$
}

\end{enumerate}

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