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external memory based algorithm documentation updated

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fc_botelho 2006-04-25 19:34:06 +00:00
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@ -16,7 +16,7 @@ a daily task.
For instance, the simple assignment of number identifiers to web pages of a collection
can be a challenging task.
While traditional databases simply cannot handle more traffic once the working
set of URLs does not fit in main memory anymorei[[4 #papers]], the algorithm we propose here to
set of URLs does not fit in main memory anymore[[4 #papers]], the algorithm we propose here to
construct MPHFs can easily scale to billions of entries.
As there are many applications for MPHFs, it is
@ -301,6 +301,121 @@ as shown in [[3 #papers]].
%!include(html): ''TABLEBRZ1.t2t''
| **Table 1:** Internal memory based algorithm: average time in seconds for constructing a MPHF, the standard deviation (SD), and the confidence intervals considering a confidence level of 95 %.
Figure 6 presents the runtime for each trial. In addition,
the solid line corresponds to a linear regression model
obtained from the experimental measurements.
As we can see, the runtime for a given //n// has a considerable
fluctuation. However, the fluctuation also grows linearly with //n//.
| [figs/brz/bmz_temporegressao.png]
| **Figure 6:** Time versus number of keys in //S// for the internal memory based algorithm. The solid line corresponds to a linear regression model.
The observed fluctuation in the runtimes is as expected; recall that this
runtime has the form [figs/brz/img159.png] with //Z// a geometric random variable with
mean //1/p=e//. Thus, the runtime has mean [figs/brz/img181.png] and standard
deviation [figs/brz/img182.png].
Therefore, the standard deviation also grows
linearly with //n//, as experimentally verified
in Table 1 and in Figure 6.
----------------------------------------
===Performance of the External Memory Based Algorithm===
The runtime of the external memory based algorithm is also a random variable,
but now it follows a (highly concentrated) normal distribution, as we discuss at the end of this
section. Again, we are interested in verifying the linearity claim made in
[[2, Section 5.1 #papers]]. Therefore, we ran the algorithm for
several numbers //n// of keys in //S//.
The values chosen for //n// were 1, 2, 4, 8, 16, 32, 64, 128, 512 and 1000
million.
We limited the main memory in 500 megabytes for the experiments.
The size [figs/brz/img8.png] of the a priori reserved internal memory area
was set to 250 megabytes, the parameter //b// was set to //175// and
the building block algorithm parameter //c// was again set to //1//.
We show later on how [figs/brz/img8.png] affects the runtime of the algorithm. The other two parameters
have insignificant influence on the runtime.
We again use a statistical method for determining a suitable sample size
to estimate the number of trials to be run for each value of //n//. We got that
just one trial for each //n// would be enough with a confidence level of 95 %.
However, we made 10 trials. This number of trials seems rather small, but, as
shown below, the behavior of our algorithm is very stable and its runtime is
almost deterministic (i.e., the standard deviation is very small).
Table 2 presents the runtime average for each //n//,
the respective standard deviations, and
the respective confidence intervals given by
the average time [figs/brz/img167.png] the distance from average time
considering a confidence level of 95 %.
Observing the runtime averages we noticed that
the algorithm runs in expected linear time,
as shown in [[2, Section 5.1 #papers]]. Better still,
it is only approximately 60 % slower than the BMZ algorithm.
To get that value we used the linear regression model obtained for the runtime of
the internal memory based algorithm to estimate how much time it would require
for constructing a MPHF for a set of 1 billion keys.
We got 2.3 hours for the internal memory based algorithm and we measured
3.67 hours on average for the external memory based algorithm.
Increasing the size of the internal memory area
from 250 to 600 megabytes,
we have brought the time to 3.09 hours. In this case, the external memory based algorithm is
just 34 % slower in this setup.
%!include(html): ''TABLEBRZ2.t2t''
| **Table 2:**The external memory based algorithm: average time in seconds for constructing a MPHF, the standard deviation (SD), and the confidence intervals considering a confidence level of 95 %.
Figure 7 presents the runtime for each trial. In addition,
the solid line corresponds to a linear regression model
obtained from the experimental measurements.
As we were expecting the runtime for a given //n// has almost no
variation.
| [figs/brz/brz_temporegressao.png]
| **Figure 7:** Time versus number of keys in //S// for our algorithm. The solid line corresponds to a linear regression model.
An intriguing observation is that the runtime of the algorithm is almost
deterministic, in spite of the fact that it uses as building block an
algorithm with a considerable fluctuation in its runtime. A given bucket
//i//, [figs/brz/img47.png], is a small set of keys (at most 256 keys) and,
as argued in last Section, the runtime of the
building block algorithm is a random variable [figs/brz/img207.png] with high fluctuation.
However, the runtime //Y// of the searching step of the external memory based algorithm is given
by [figs/brz/img209.png]. Under the hypothesis that
the [figs/brz/img207.png] are independent and bounded, the {\it law of large numbers} (see,
e.g., [[6 #papers]]) implies that the random variable [figs/brz/img210.png] converges
to a constant as [figs/brz/img83.png]. This explains why the runtime of our
algorithm is almost deterministic.
----------------------------------------
=== Controlling disk accesses ===
In order to bring down the number of seek operations on disk
we benefit from the fact that our algorithm leaves almost all main
memory available to be used as disk I/O buffer.
In this section we evaluate how much the parameter [figs/brz/img8.png] affects the runtime of our algorithm.
For that we fixed //n// in 1 billion of URLs,
set the main memory of the machine used for the experiments
to 1 gigabyte and used [figs/brz/img8.png] equal to 100, 200, 300, 400, 500 and 600
megabytes.
Table 3 presents the number of files //N//,
the buffer size used for all files, the number of seeks in the worst case considering
the pessimistic assumption mentioned in [[2, Section 5.1 #papers]], and
the time to generate a MPHF for 1 billion of keys as a function of the amount of internal
memory available. Observing Table 3 we noticed that the time spent in the construction
decreases as the value of [figs/brz/img8.png] increases. However, for [figs/brz/img213.png], the variation
on the time is not as significant as for [figs/brz/img214.png].
This can be explained by the fact that the kernel 2.6 I/O scheduler of Linux
has smart policies for avoiding seeks and diminishing the average seek time
(see [http://www.linuxjournal.com/article/6931 http://www.linuxjournal.com/article/6931]).
%!include(html): ''TABLEBRZ3.t2t''
| **Table 3:**Influence of the internal memory area size ([figs/brz/img8.png]) in the external memory based algorithm runtime.
----------------------------------------
==Papers==[papers]

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<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img5.png"
ALT="$n$"></SPAN> (millions) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 1 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 2 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 4 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 8 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 16 </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
Average time (s) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img187.png"
ALT="$6.9 \pm 0.3$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img188.png"
ALT="$13.8 \pm 0.2$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img189.png"
ALT="$31.9 \pm 0.7$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img190.png"
ALT="$69.9 \pm 1.1$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img191.png"
ALT="$140.6 \pm 2.5$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
SD </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img192.png"
ALT="$0.4$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img193.png"
ALT="$0.2$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img194.png"
ALT="$0.9$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img195.png"
ALT="$1.5$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img196.png"
ALT="$3.5$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img5.png"
ALT="$n$"></SPAN> (millions) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 32 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 64 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 128 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 512 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 1000 </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
Average time (s) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img197.png"
ALT="$284.3 \pm 1.1$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img198.png"
ALT="$587.9 \pm 3.9$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
$1223.6 \pm 4.9$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="88" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img199.png"
ALT="$1223.6 \pm 4.9$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
$5966.4 \pm 9.5$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="88" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img200.png"
ALT="$5966.4 \pm 9.5$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
$13229.5 \pm 12.7$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="104" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img201.png"
ALT="$13229.5 \pm 12.7$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
SD </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img202.png"
ALT="$1.6$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img203.png"
ALT="$5.5$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img204.png"
ALT="$6.8$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img205.png"
ALT="$13.2$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img206.png"
ALT="$18.6$"></SPAN> </SMALL></TD>
</TR>
<TR><TD></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD></TD>
</TR>
</TABLE>

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<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img8.png"
ALT="$\mu $"></SPAN> (MB) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img215.png"
ALT="$100$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img216.png"
ALT="$200$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img217.png"
ALT="$300$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img218.png"
ALT="$400$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img219.png"
ALT="$500$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img212.png"
ALT="$600$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img58.png"
ALT="$N$"></SPAN> (files) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img220.png"
ALT="$619$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img221.png"
ALT="$310$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img222.png"
ALT="$207$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img223.png"
ALT="$155$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img224.png"
ALT="$124$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img225.png"
ALT="$104$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
&nbsp;(buffer size in KB) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img226.png"
ALT="$165$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img227.png"
ALT="$661$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img228.png"
ALT="$1,484$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img229.png"
ALT="$2,643$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img230.png"
ALT="$4,129$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img231.png"
ALT="$5,908$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img135.png"
ALT="$\beta$"></SPAN>/&nbsp;(# of seeks in the worst case) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="59" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img232.png"
ALT="$384,478$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img233.png"
ALT="$95,974$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img234.png"
ALT="$42,749$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img235.png"
ALT="$24,003$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img236.png"
ALT="$15,365$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="figs/brz/img237.png"
ALT="$10,738$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
Time (hours) </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img238.png"
ALT="$4.04$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img239.png"
ALT="$3.64$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img240.png"
ALT="$3.34$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img241.png"
ALT="$3.20$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/brz/img242.png"
ALT="$3.13$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img243.png"
ALT="$3.09$"></SPAN> </SMALL></TD>
</TR>
</TABLE>