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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML>
<HEAD>
<META NAME="generator" CONTENT="http://txt2tags.org">
<LINK REL="stylesheet" TYPE="text/css" HREF="DOC.css">
<TITLE>Comparison Between BMZ And CHM Algorithms</TITLE>
</HEAD><BODY BGCOLOR="white" TEXT="black">
<CENTER>
<H1>Comparison Between BMZ And CHM Algorithms</H1>
</CENTER>
<HR NOSHADE SIZE=1>
<H2>Characteristics</H2>
<P>
Table 1 presents the main characteristics of the two algorithms.
The number of edges in the graph <IMG ALIGN="middle" SRC="figs/img27.png" BORDER="0" ALT=""> is <IMG ALIGN="middle" SRC="figs/img236.png" BORDER="0" ALT="">,
the number of keys in the input set <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
The number of vertices of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> is equal
to <IMG ALIGN="bottom" SRC="figs/img12.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img237.png" BORDER="0" ALT=""> for BMZ algorithm and the CHM algorithm, respectively.
This measure is related to the amount of space to store the array <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">.
This improves the space required to store a function in BMZ algorithm to <IMG ALIGN="middle" SRC="figs/img238.png" BORDER="0" ALT=""> of the space required by the CHM algorithm.
The number of critical edges is <IMG ALIGN="middle" SRC="figs/img76.png" BORDER="0" ALT=""> and 0, for BMZ algorithm and the CHM algorithm,
respectively.
BMZ algorithm generates random graphs that necessarily contains cycles and the
CHM algorithm
generates
acyclic random graphs.
Finally, the CHM algorithm generates <A HREF="concepts.html">order preserving functions</A>
while BMZ algorithm does not preserve order.
</P>
<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Characteristics </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=2><SMALL CLASS="FOOTNOTESIZE"> <SPAN>Algorithms</SPAN></SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> BMZ </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> CHM </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img1.png"
ALT="$c$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1.15 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.09 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img239.png"
ALT="$\vert E(G)\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="89" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img240.png"
ALT="$\vert V(G)\vert=\vert g\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="20" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img241.png"
ALT="$cn$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="20" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img241.png"
ALT="$cn$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<!-- MATH
$|E(G_{\rm crit})|$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img111.png"
ALT="$\vert E(G_{\rm crit})\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img242.png"
ALT="$0.5\vert E(G)\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 0</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img32.png"
ALT="$G$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> cyclic </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> acyclic </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Order preserving </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> no </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> yes </SMALL></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 1:</B> Main characteristics of the algorithms.</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<H2>Memory Consumption</H2>
<UL>
<LI>Memory consumption to generate the minimal perfect hash function (MPHF):
</UL>
<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
<TR>
<TH>Algorithm</TH>
<TH><I>c</I></TH>
<TH>Memory consumption to generate a MPHF</TH>
</TR>
<TR>
<TD ALIGN="center">BMZ</TD>
<TD>0.93</TD>
<TD ALIGN="center"><I>24.80n + O(1)</I></TD>
</TR>
<TR>
<TD ALIGN="center">BMZ</TD>
<TD>1.15</TD>
<TD ALIGN="center"><I>26.42n + O(1)</I></TD>
</TR>
<TR>
<TD ALIGN="center">CHM</TD>
<TD>2.09</TD>
<TD ALIGN="center"><I>33.00n + O(1)</I></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 2:</B> Memory consumption to generate a MPHF using the algorithms BMZ and CHM.</TD>
</TR>
</TABLE>
<UL>
<LI>Memory consumption to store the resulting minimal perfect hash function (MPHF):
</UL>
<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
<TR>
<TH>Algorithm</TH>
<TH><I>c</I></TH>
<TH>Memory consumption to store a MPHF</TH>
</TR>
<TR>
<TD ALIGN="center">BMZ</TD>
<TD>0.93</TD>
<TD ALIGN="center"><I>3.72n</I></TD>
</TR>
<TR>
<TD ALIGN="center">BMZ</TD>
<TD>1.15</TD>
<TD ALIGN="center"><I>4.60n</I></TD>
</TR>
<TR>
<TD ALIGN="center">CHM</TD>
<TD>2.09</TD>
<TD ALIGN="center"><I>8.36n</I></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 3:</B> Memory consumption to store a MPHF generated by the algorithms BMZ and CHM.</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<H2>Run times</H2>
<P>
We now present some experimental results to compare the BMZ and CHM algorithms.
The data consists of a collection of 100 million universe resource locations
(URLs) collected from the Web.
The average length of a URL in the collection is 63 bytes.
All experiments were carried on
a computer running the Linux operating system, version 2.6.7,
with a 2.4 gigahertz processor and
4 gigabytes of main memory.
</P>
<P>
Table 4 presents time measurements.
All times are in seconds.
The table entries represent averages over 50 trials.
The column labelled as <IMG ALIGN="middle" SRC="figs/img243.png" BORDER="0" ALT=""> represents
the number of iterations to generate the random graph <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> in the
mapping step of the algorithms.
The next columns represent the run times
for the mapping plus ordering steps together and the searching
step for each algorithm.
The last column represents the percent gain of our algorithm
over the CHM algorithm.
</P>
<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE"> <SPAN> BMZ </SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE">
<SPAN>CHM algorithm</SPAN></SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> Gain</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> (%)</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,562,500 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.28 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 8.54 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.37 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 10.91 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.70 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 14.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1.57 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 16.13 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 48 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3,125,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.16 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 15.92 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 4.88 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 20.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.85 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 30.36 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3.20 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 61 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 6,250,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.20 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.09 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 10.48 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 43.57 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.90 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 62.26 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 6.76 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 69.02 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 58 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 12,500,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.00 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 63.26 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 23.04 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 86.30 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.60 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 117.99 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 14.94 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 132.92 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 54 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.00 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 130.79 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 51.55 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 182.34 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 262.05 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.68 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 295.73 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 62 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 50,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.07 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 273.75 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 114.12 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 387.87 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.90 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 577.59 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 73.97 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 651.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 68 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 100,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.07 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 567.47 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 243.13 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 810.60 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,131.06 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 157.23 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,288.29 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 59 </SMALL></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 4:</B> Time measurements for BMZ and the CHM algorithm.</TD>
</TR>
</TABLE>
<P>
The mapping step of the BMZ algorithm is faster because
the expected number of iterations in the mapping step to generate <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> are
2.13 and 2.92 for BMZ algorithm and the CHM algorithm, respectively
(see <A HREF="bmz.html#papers">[2</A>] for details).
The graph <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> generated by BMZ algorithm
has <IMG ALIGN="bottom" SRC="figs/img12.png" BORDER="0" ALT=""> vertices, against <IMG ALIGN="bottom" SRC="figs/img237.png" BORDER="0" ALT=""> for the CHM algorithm.
These two facts make BMZ algorithm faster in the mapping step.
The ordering step of BMZ algorithm is approximately equal to
the time to check if <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> is acyclic for the CHM algorithm.
The searching step of the CHM algorithm is faster, but the total
time of BMZ algorithm is, on average, approximately 59 % faster
than the CHM algorithm.
It is important to notice the times for the searching step:
for both algorithms they are not the dominant times,
and the experimental results clearly show
a linear behavior for the searching step.
</P>
<P>
We now present run times for BMZ algorithm using a <A HREF="bmz.html#heuristic">heuristic</A> that
reduces the space requirement
to any given value between <IMG ALIGN="bottom" SRC="figs/img12.png" BORDER="0" ALT=""> words and <IMG ALIGN="bottom" SRC="figs/img13.png" BORDER="0" ALT=""> words.
For example, for <IMG ALIGN="bottom" SRC="figs/img244.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img6.png" BORDER="0" ALT="">, the analytical expected number
of iterations are <IMG ALIGN="bottom" SRC="figs/img245.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img246.png" BORDER="0" ALT="">, respectively
(for <IMG ALIGN="middle" SRC="figs/img247.png" BORDER="0" ALT="">, the number of iterations are 2.78 for <IMG ALIGN="bottom" SRC="figs/img244.png" BORDER="0" ALT=""> and 3.04
for <IMG ALIGN="bottom" SRC="figs/img6.png" BORDER="0" ALT="">).
Table 5 presents the total times to construct a
function for <IMG ALIGN="middle" SRC="figs/img247.png" BORDER="0" ALT="">, with an increase from <IMG ALIGN="bottom" SRC="figs/img237.png" BORDER="0" ALT=""> seconds
for <IMG ALIGN="bottom" SRC="figs/img128.png" BORDER="0" ALT=""> (see Table 4) to <IMG ALIGN="bottom" SRC="figs/img249.png" BORDER="0" ALT=""> seconds for <IMG ALIGN="bottom" SRC="figs/img244.png" BORDER="0" ALT=""> and
to <IMG ALIGN="bottom" SRC="figs/img250.png" BORDER="0" ALT=""> seconds for <IMG ALIGN="bottom" SRC="figs/img6.png" BORDER="0" ALT="">.
</P>
<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE"> <SPAN> BMZ <SPAN CLASS="MATH"><IMG
WIDTH="60" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img5.png"
ALT="$c=1.00$"></SPAN></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE">
<SPAN> BMZ <SPAN CLASS="MATH"><IMG
WIDTH="60" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img6.png"
ALT="$c=0.93$"></SPAN></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Total </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 12,500,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.78 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 76.68 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25.06 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 101.74 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3.04 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 76.39 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 102.19 </SMALL></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 5:</B> Time measurements for BMZ tuned algorithm with <IMG ALIGN="bottom" SRC="figs/img5.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img6.png" BORDER="0" ALT="">.</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><A HREF="index.html">Home</A></TD>
<TD><A HREF="chd.html">CHD</A></TD>
<TD><A HREF="bdz.html">BDZ</A></TD>
<TD><A HREF="bmz.html">BMZ</A></TD>
<TD><A HREF="chm.html">CHM</A></TD>
<TD><A HREF="brz.html">BRZ</A></TD>
<TD><A HREF="fch.html">FCH</A></TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<P>
Enjoy!
</P>
<P>
<A HREF="mailto:davi@users.sourceforge.net">Davi de Castro Reis</A>
</P>
<P>
<A HREF="mailto:db8192@users.sourceforge.net">Djamel Belazzougui</A>
</P>
<P>
<A HREF="mailto:fc_botelho@users.sourceforge.net">Fabiano Cupertino Botelho</A>
</P>
<P>
<A HREF="mailto:nivio@dcc.ufmg.br">Nivio Ziviani</A>
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