179 lines
7.9 KiB
TeX
Executable File
179 lines
7.9 KiB
TeX
Executable File
\section{Experimental Results}
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% We now present some experimental results.
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% The same experiments were run with our algorithm and
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% the algorithm due to Czech, Havas and Majewski~\cite{chm92}, referred to as
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% the CHM algorithm.
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% The two algorithms were implemented in the C language and
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% are available at \texttt{http://cmph.sf.net}.
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% Our data consists
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% of a collection of 100 million
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% universe resource locations (URLs) collected from the Web.
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% The average length of a URL in the collection is 63 bytes.
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% All experiments were carried out on
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% a computer running the Linux operating system, version 2.6.7,
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% with a 2.4 gigahertz processor and
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% 4 gigabytes of main memory.
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%
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% Table~\ref{tab:characteristics} presents the main characteristics
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% of the two algorithms.
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% The number of edges in the graph $G=(V,E)$ is~$|S|=n$,
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% the number of keys in the input set~$S$.
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% The number of vertices of $G$ is equal to $1.15n$ and $2.09n$
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% for our algorithm and the CHM algorithm, respectively.
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% This measure is related to the amount of space to store the array $g$.
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% This improves the space required to store a function in our algorithm to
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% $55\%$ of the space required by the CHM algorithm.
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% The number of critical edges
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% is $\frac{1}{2}|E(G)|$ and 0 for our algorithm and the CHM algorithm,
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% respectively.
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% Our algorithm generates random graphs that contain cycles with high
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% probability and the
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% CHM algorithm
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% generates
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% acyclic random graphs.
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% Finally, the CHM algorithm generates order preserving functions
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% while our algorithm does not preserve order.
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%
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% \vspace{-10pt}
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% \begin{table}[htb]
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% {\footnotesize
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% \begin{center}
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% \begin{tabular}{|c|c|c|c|c|c|c|}
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% \hline
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% & $c$ & $|E(G)|$ & $|V(G)|=|g|$ & $|E(G_\crit)|$ & $G$ & Order preserving \\
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% \hline
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% Our algorithm & 1.15 & $n$ & $cn$ & $0.5|E(G)|$ & cyclic & no \\
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% \hline
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% CHM algorithm & 2.09 & $n$ & $cn$ & 0 & acyclic & yes \\
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% \hline
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% \end{tabular}
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% \end{center}
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% }
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% \caption{Main characteristics of the algorithms}
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% \vspace{-25pt}
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% \label{tab:characteristics}
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% \end{table}
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%
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% Table~\ref{tab:timeresults} presents time measurements.
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% All times are in seconds.
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% The table entries are averages over 50 trials.
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% The column labelled $N_i$ gives
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% the number of iterations to generate the random graph $G$
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% in the mapping step of the algorithms.
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% The next columns give the running times
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% for the mapping plus ordering steps together and the searching
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% step for each algorithm.
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% The last column gives the percentage gain of our algorithm
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% over the CHM algorithm.
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%
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% \begin{table*}
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% {\footnotesize
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% \begin{center}
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% \begin{tabular}{|c|cccc|cccc|c|}
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% \hline
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% \raisebox{-0.7em}{$n$} & \multicolumn{4}{c|}{\raisebox{-1mm}{Our algorithm}} &
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% \multicolumn{4}{c|}{\raisebox{-1mm}{CHM algorithm}}& \raisebox{-0.2em}{Gain}\\
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% \cline{2-5} \cline{6-9}
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% & \raisebox{-1mm}{$N_i$} &\raisebox{-1mm}{Map+Ord} &
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% \raisebox{-1mm}{Search} &\raisebox{-1mm}{Total} &
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% \raisebox{-1mm}{$N_i$} &\raisebox{-1mm}{Map+Ord} &\raisebox{-1mm}{Search} &
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% \raisebox{-1mm}{Total} & \raisebox{0.2em}{(\%)}\\
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% \hline
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% %1,562,500 & 2.28 & 8.54 & 2.37 & 10.91 & 2.70 & 14.56 & 1.57 & 16.13 & 48 \\ %[1mm]
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% %3,125,000 & 2.16 & 15.92 & 4.88 & 20.80 & 2.85 & 30.36 & 3.20 & 33.56 & 61 \\ %[1mm]
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% 6,250,000 & 2.20 & 33.09 & 10.48 & 43.57 & 2.90 & 62.26 & 6.76 & 69.02 & 58 \\ %[1mm]
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% 12,500,000 & 2.00 & 63.26 & 23.04 & 86.30 & 2.60 & 117.99 & 14.94 & 132.92 & 54 \\ %[1mm]
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% 25,000,000 & 2.00 & 130.79 & 51.55 & 182.34 & 2.80 & 262.05 & 33.68 & 295.73 & 62 \\ %[1mm]
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% %50,000,000 & 2.07 & 273.75 & 114.12 & 387.87 & 2.90 & 577.59 & 73.97 & 651.56 & 68 \\ %[1mm]
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% 100,000,000 & 2.07 & 567.47 & 243.13 & 810.60 & 2.80 & 1,131.06 & 157.23 & 1,288.29 & 59 \\ %[1mm]
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% \hline
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% \end{tabular}
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% \end{center}
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% \caption{Time measurements
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% for our algorithm and the CHM algorithm}
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% \vspace{-25pt}
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% \label{tab:timeresults}
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% }\end{table*}
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%
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% \enlargethispage{\baselineskip}
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% The mapping step of the new algorithm is faster because
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% the expected number of iterations in the mapping step to generate
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% $G$ are 2.13 and 2.92 for our algorithm and the CHM algorithm, respectively.
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% The graph $G$ generated by our algorithm
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% has $1.15n$ vertices, against $2.09n$ for the CHM algorithm.
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% These two facts make our algorithm faster in the mapping step.
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% The ordering step of our algorithm is approximately equal to
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% the time to check if $G$ is acyclic for the CHM algorithm.
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% The searching step of the CHM algorithm is faster, but the total
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% time of our algorithm is, on average, approximately 58\% faster
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% than the CHM algorithm.
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%
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% The experimental results fully backs the theoretical results.
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% It is important to notice the times for the searching step:
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% for both algorithms they are not the dominant times,
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% and the experimental results clearly show
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% a linear behavior for the searching step.
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%
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% We now present a heuristic that reduces the space requirement
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% to any given value between $1.15n$ words and $0.93n$ words.
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% The heuristic reuses, when possible, the set
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% of $x$ values that caused reassignments, just before trying $x+1$
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% (see Section~\ref{sec:searching}).
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% The lower limit $c=0.93$ was obtained experimentally.
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% We generate $10{,}000$ random graphs for
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% each size $n$ ($n=10^5$, $5 \times 10^5$, $10^6$, $2\times 10^6$).
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% With $c=0.93$ we were always able to generate~$h$, but with $c=0.92$ we never
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% succeeded.
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% Decreasing the value of $c$ leads to an increase in the number of
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% iterations to generate $G$.
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% For example, for $c=1$ and $c=0.93$, the analytical expected number
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% of iterations are $2.72$ and $3.17$, respectively
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% (for $n=12{,}500{,}000$, the number of iterations are 2.78 for $c=1$ and 3.04
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% for $c=0.93$).
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% Table~\ref{tab:timeresults2} presents the total times to construct a
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% function for $n=12{,}500{,}000$, with an increase from $86.31$ seconds
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% for $c=1.15$ (see Table~\ref{tab:timeresults}) to
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% $101.74$ seconds for $c=1$ and to $102.19$ seconds for $c=0.93$.
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%
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% \vspace{-5pt}
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% \begin{table*}
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% {\footnotesize
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% \begin{center}
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% \begin{tabular}{|c|cccc|cccc|}
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% \hline
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% \raisebox{-0.7em}{$n$} & \multicolumn{4}{c|}{\raisebox{-1mm}{Our algorithm $c=1.00$}} &
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% \multicolumn{4}{c|}{\raisebox{-1mm}{Our algorithm $c=0.93$}} \\
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% \cline{2-5} \cline{6-9}
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% & \raisebox{-1mm}{$N_i$} &\raisebox{-1mm}{Map+Ord} &
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% \raisebox{-1mm}{Search} &\raisebox{-1mm}{Total} &
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% \raisebox{-1mm}{$N_i$} &\raisebox{-1mm}{Map+Ord} &\raisebox{-1mm}{Search} &
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% \raisebox{-1mm}{Total} \\%[0.3mm]
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% \hline%\\[-2mm]
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% 12,500,000 & 2.78 & 76.68 & 25.06 & 101.74 & 3.04 & 76.39 & 25.80 & 102.19 \\ %[1mm]
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% \hline
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% \end{tabular}
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% \end{center}
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% \caption{Time measurements
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% for our tuned algorithm with $c=1.00$ and $c=0.93$}
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% \vspace{-25pt}
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% \label{tab:timeresults2}
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% }
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% \end{table*}
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%
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% We compared our algorithm with the ones proposed by Pagh~\cite{p99} and
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% Dietzfelbinger and Hagerup~\cite{dh01}, respectively. The authors sent to us their
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% source code. In their implementation the set of keys is a set of random integers.
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% We modified our implementation to generate our~$h$ from a set of random
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% integers in order to make a fair comparison. For a set of $10^6$ random integers,
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% the times to generate a minimal perfect hash function were $2.7 s$, $4 s$ and $4.5 s$ for
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% our algorithm, Pagh's algorithm and Dietzfelbinger and Hagerup's algorithm, respectively.
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% Thus, our algorithm was 48\% faster than Pagh's algorithm and 67\% faster than
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% Dietzfelbinger and Hagerup's algorithm, on average. This gain was maintained for sets with different
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% sizes.
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% Our algorithm needs $kn$ ($k \in [0.93, 1.15]$) words to store
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% the resulting function, while Pagh's algorithm needs $kn$ ($k > 2$) words and
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% Dietzfelbinger and Hagerup's algorithm needs $kn$ ($k \in [1.13, 1.15]$) words.
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% The time to generate the functions is inversely proportional to the value of $k$.
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% \enlargethispage{\baselineskip}
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