external memory based algorithm documentation updated
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@ -16,7 +16,7 @@ a daily task.
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For instance, the simple assignment of number identifiers to web pages of a collection
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can be a challenging task.
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While traditional databases simply cannot handle more traffic once the working
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set of URLs does not fit in main memory anymorei[[4 #papers]], the algorithm we propose here to
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set of URLs does not fit in main memory anymore[[4 #papers]], the algorithm we propose here to
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construct MPHFs can easily scale to billions of entries.
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As there are many applications for MPHFs, it is
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@ -301,6 +301,121 @@ as shown in [[3 #papers]].
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%!include(html): ''TABLEBRZ1.t2t''
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| **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 %.
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Figure 6 presents the runtime for each trial. In addition,
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the solid line corresponds to a linear regression model
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obtained from the experimental measurements.
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As we can see, the runtime for a given //n// has a considerable
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fluctuation. However, the fluctuation also grows linearly with //n//.
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| [figs/brz/bmz_temporegressao.png]
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| **Figure 6:** Time versus number of keys in //S// for the internal memory based algorithm. The solid line corresponds to a linear regression model.
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The observed fluctuation in the runtimes is as expected; recall that this
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runtime has the form [figs/brz/img159.png] with //Z// a geometric random variable with
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mean //1/p=e//. Thus, the runtime has mean [figs/brz/img181.png] and standard
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deviation [figs/brz/img182.png].
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Therefore, the standard deviation also grows
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linearly with //n//, as experimentally verified
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in Table 1 and in Figure 6.
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----------------------------------------
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===Performance of the External Memory Based Algorithm===
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The runtime of the external memory based algorithm is also a random variable,
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but now it follows a (highly concentrated) normal distribution, as we discuss at the end of this
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section. Again, we are interested in verifying the linearity claim made in
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[[2, Section 5.1 #papers]]. Therefore, we ran the algorithm for
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several numbers //n// of keys in //S//.
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The values chosen for //n// were 1, 2, 4, 8, 16, 32, 64, 128, 512 and 1000
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million.
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We limited the main memory in 500 megabytes for the experiments.
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The size [figs/brz/img8.png] of the a priori reserved internal memory area
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was set to 250 megabytes, the parameter //b// was set to //175// and
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the building block algorithm parameter //c// was again set to //1//.
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We show later on how [figs/brz/img8.png] affects the runtime of the algorithm. The other two parameters
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have insignificant influence on the runtime.
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We again use a statistical method for determining a suitable sample size
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to estimate the number of trials to be run for each value of //n//. We got that
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just one trial for each //n// would be enough with a confidence level of 95 %.
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However, we made 10 trials. This number of trials seems rather small, but, as
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shown below, the behavior of our algorithm is very stable and its runtime is
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almost deterministic (i.e., the standard deviation is very small).
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Table 2 presents the runtime average for each //n//,
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the respective standard deviations, and
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the respective confidence intervals given by
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the average time [figs/brz/img167.png] the distance from average time
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considering a confidence level of 95 %.
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Observing the runtime averages we noticed that
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the algorithm runs in expected linear time,
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as shown in [[2, Section 5.1 #papers]]. Better still,
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it is only approximately 60 % slower than the BMZ algorithm.
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To get that value we used the linear regression model obtained for the runtime of
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the internal memory based algorithm to estimate how much time it would require
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for constructing a MPHF for a set of 1 billion keys.
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We got 2.3 hours for the internal memory based algorithm and we measured
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3.67 hours on average for the external memory based algorithm.
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Increasing the size of the internal memory area
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from 250 to 600 megabytes,
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we have brought the time to 3.09 hours. In this case, the external memory based algorithm is
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just 34 % slower in this setup.
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%!include(html): ''TABLEBRZ2.t2t''
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| **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 %.
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Figure 7 presents the runtime for each trial. In addition,
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the solid line corresponds to a linear regression model
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obtained from the experimental measurements.
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As we were expecting the runtime for a given //n// has almost no
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variation.
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| [figs/brz/brz_temporegressao.png]
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| **Figure 7:** Time versus number of keys in //S// for our algorithm. The solid line corresponds to a linear regression model.
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An intriguing observation is that the runtime of the algorithm is almost
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deterministic, in spite of the fact that it uses as building block an
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algorithm with a considerable fluctuation in its runtime. A given bucket
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//i//, [figs/brz/img47.png], is a small set of keys (at most 256 keys) and,
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as argued in last Section, the runtime of the
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building block algorithm is a random variable [figs/brz/img207.png] with high fluctuation.
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However, the runtime //Y// of the searching step of the external memory based algorithm is given
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by [figs/brz/img209.png]. Under the hypothesis that
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the [figs/brz/img207.png] are independent and bounded, the {\it law of large numbers} (see,
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e.g., [[6 #papers]]) implies that the random variable [figs/brz/img210.png] converges
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to a constant as [figs/brz/img83.png]. This explains why the runtime of our
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algorithm is almost deterministic.
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----------------------------------------
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=== Controlling disk accesses ===
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In order to bring down the number of seek operations on disk
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we benefit from the fact that our algorithm leaves almost all main
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memory available to be used as disk I/O buffer.
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In this section we evaluate how much the parameter [figs/brz/img8.png] affects the runtime of our algorithm.
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For that we fixed //n// in 1 billion of URLs,
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set the main memory of the machine used for the experiments
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to 1 gigabyte and used [figs/brz/img8.png] equal to 100, 200, 300, 400, 500 and 600
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megabytes.
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Table 3 presents the number of files //N//,
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the buffer size used for all files, the number of seeks in the worst case considering
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the pessimistic assumption mentioned in [[2, Section 5.1 #papers]], and
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the time to generate a MPHF for 1 billion of keys as a function of the amount of internal
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memory available. Observing Table 3 we noticed that the time spent in the construction
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decreases as the value of [figs/brz/img8.png] increases. However, for [figs/brz/img213.png], the variation
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on the time is not as significant as for [figs/brz/img214.png].
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This can be explained by the fact that the kernel 2.6 I/O scheduler of Linux
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has smart policies for avoiding seeks and diminishing the average seek time
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(see [http://www.linuxjournal.com/article/6931 http://www.linuxjournal.com/article/6931]).
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%!include(html): ''TABLEBRZ3.t2t''
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| **Table 3:**Influence of the internal memory area size ([figs/brz/img8.png]) in the external memory based algorithm runtime.
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----------------------------------------
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==Papers==[papers]
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@ -0,0 +1,133 @@
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<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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<SPAN CLASS="MATH"><IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img5.png"
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ALT="$n$"></SPAN> (millions) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 1 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 2 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 4 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 8 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 16 </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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Average time (s) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img187.png"
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ALT="$6.9 \pm 0.3$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img188.png"
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ALT="$13.8 \pm 0.2$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img189.png"
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ALT="$31.9 \pm 0.7$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img190.png"
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ALT="$69.9 \pm 1.1$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img191.png"
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ALT="$140.6 \pm 2.5$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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SD </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img192.png"
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ALT="$0.4$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img193.png"
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ALT="$0.2$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img194.png"
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ALT="$0.9$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img195.png"
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ALT="$1.5$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img196.png"
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ALT="$3.5$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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<SPAN CLASS="MATH"><IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img5.png"
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ALT="$n$"></SPAN> (millions) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 32 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 64 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 128 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 512 </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> 1000 </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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Average time (s) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img197.png"
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ALT="$284.3 \pm 1.1$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="80" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img198.png"
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ALT="$587.9 \pm 3.9$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
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$1223.6 \pm 4.9$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="88" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img199.png"
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ALT="$1223.6 \pm 4.9$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
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$5966.4 \pm 9.5$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="88" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img200.png"
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ALT="$5966.4 \pm 9.5$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <!-- MATH
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$13229.5 \pm 12.7$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="104" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img201.png"
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ALT="$13229.5 \pm 12.7$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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SD </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img202.png"
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ALT="$1.6$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img203.png"
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ALT="$5.5$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img204.png"
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ALT="$6.8$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img205.png"
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ALT="$13.2$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img206.png"
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ALT="$18.6$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD></TD>
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<TD></TD>
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<TD></TD>
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<TD></TD>
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<TD></TD>
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<TD></TD>
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</TR>
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</TABLE>
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@ -0,0 +1,147 @@
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<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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<SPAN CLASS="MATH"><IMG
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WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="figs/brz/img8.png"
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ALT="$\mu $"></SPAN> (MB) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img215.png"
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ALT="$100$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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ALT="$200$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img217.png"
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ALT="$300$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img218.png"
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ALT="$400$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img219.png"
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ALT="$500$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img212.png"
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ALT="$600$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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<SPAN CLASS="MATH"><IMG
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WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img58.png"
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ALT="$N$"></SPAN> (files) </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img220.png"
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ALT="$619$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img221.png"
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ALT="$310$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="figs/brz/img222.png"
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ALT="$207$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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ALT="$155$"></SPAN> </SMALL></TD>
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<TD ALIGN="CENTER"><SMALL CLASS="SCRIPTSIZE"> <SPAN CLASS="MATH"><IMG
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SRC="figs/brz/img224.png"
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ALT="$124$"></SPAN> </SMALL></TD>
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SRC="figs/brz/img225.png"
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ALT="$104$"></SPAN> </SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT"><SMALL CLASS="SCRIPTSIZE">
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(buffer size in KB) </SMALL></TD>
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ALT="$165$"></SPAN> </SMALL></TD>
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<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>/ (# 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
|
||||
WIDTH="32" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="figs/brz/img243.png"
|
||||
ALT="$3.09$"></SPAN> </SMALL></TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
Loading…
Reference in New Issue