ch17.6.htm

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R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
) (<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
3</SUB>
 + <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
n</SUB>
) .</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85463">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85465">
 </A>
<A NAME="36161">
 </A>
(17.21)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85467">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85469">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85471">
 </A>
node <SPAN CLASS="Symbol">
&#8712;</SPAN>
 <SPAN CLASS="EquationVariables">
k</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85473">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85475">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85477">
 </A>
&nbsp;</P>
</TD>
</TR>
</TABLE>
<P CLASS="Exercise">
<A NAME="pgfId=52707">
 </A>
In this equation there are two types of capacitors: those due to the interconnect, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
0</SUB>
, and those due to the gate loads at each sink, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
n</SUB>
. <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
 is the driving resistance of the driving gate (the pull-up or pull-down resistance); <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
0 </SUB>
is the resistance of a one-grid-long piece of interconnect; and <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
0</SUB>
 is the capacitance of a one-grid-long piece of interconnect. Thus,  </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85734">
 </A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
k</SUB>
<SUB CLASS="Subscript">
</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85736">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85738">
 </A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
0</SUB>
 + <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
n</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85740">
 </A>
and</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85742">
 </A>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85744">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85746">
 </A>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
L</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
 + <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
 ,</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85748">
 </A>
(17.22)</P>
</TD>
</TR>
</TABLE>
<P CLASS="ExerciseNoIndent">
<A NAME="pgfId=32905">
 </A>
since every path to ground must pass through <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
. <SPAN CLASS="EquationVariables">
L</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
 is the path length (in routing-grid units) between a node <SPAN CLASS="EquationVariables">
k</SPAN>
 and one of the <SPAN CLASS="EquationVariables">
n</SPAN>
 sink nodes. </P>
<P CLASS="Exercise">
<A NAME="pgfId=64300">
 </A>
With these definitions we can expand Eq.&nbsp;<A HREF="CH17.6.htm#36161" CLASS="XRef">
17.21</A>
 to the following:  </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85895">
 </A>
<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
p</SUB>
<SUB CLASS="Subscript">
</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85897">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86004">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85899">
 </A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85901">
 </A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
0</SUB>
 <SPAN CLASS="EquationVariables">
L</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85903">
 </A>
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85905">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85907">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86006">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85909">
 </A>
node <SPAN CLASS="Symbol">
&#8712;</SPAN>
 <SPAN CLASS="EquationVariables">
k</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85911">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85913">
 </A>
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85947">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85949">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86008">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85951">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85953">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85955">
 </A>
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85965">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85967">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86010">
 </A>
+</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86024">
 </A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=86026">
 </A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
n</SUB>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
L</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
 + <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
0</SUB>
 + <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
d</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
n</SUB>
 .</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85973">
 </A>
<A NAME="42475">
 </A>
(17.23)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85983">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85985">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86012">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=86028">
 </A>
node <SPAN CLASS="Symbol">
&#8712;</SPAN>
 <SPAN CLASS="EquationVariables">
k</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=86030">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85991">
 </A>
&nbsp;</P>
</TD>
</TR>
</TABLE>
<P CLASS="Exercise">
<A NAME="pgfId=52955">
 </A>
<A HREF="CH17.6.htm#11836" CLASS="XRef">
Figure&nbsp;17.24</A>
 shows examples of three different types of trees. The MRST minimizes the rectilinear path length. The <SPAN CLASS="Definition">
shortest-path tree</SPAN>
<A NAME="marker=52962">
 </A>
 (<SPAN CLASS="Definition">
SPT</SPAN>
<A NAME="marker=52963">
 </A>
<A NAME="marker=52964">
 </A>
) minimizes the sum of path lengths to all sinks. The <SPAN CLASS="Definition">
quadratic minimum Steiner tree</SPAN>
<A NAME="marker=52965">
 </A>
 (<SPAN CLASS="Definition">
QMST</SPAN>
<A NAME="marker=52966">
 </A>
<A NAME="marker=52967">
 </A>
) minimizes the sum of path lengths to all nodes (every grid-point on the tree). </P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=71152">
 </A>
a.&nbsp;Find the measures for the MRST, SPT, and QMST for each of the three different tree types shown in <A HREF="CH17.6.htm#11836" CLASS="XRef">
Figure&nbsp;17.24</A>
. </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=71158">
 </A>
b.&nbsp;Explain how to apply these trees to Eq.&nbsp;<A HREF="CH17.6.htm#42475" CLASS="XRef">
17.23</A>
. </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=71159">
 </A>
c.&nbsp;Compare Eqs.<A HREF="CH17.6.htm#14070" CLASS="XRef">
17.19</A>
 and <A HREF="CH17.6.htm#21031" CLASS="XRef">
17.20</A>
 for the purposes of timing-driven routing.</LI>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFigure">
<A NAME="pgfId=52638">
 </A>
<IMG SRC="CH17-24.gif" ALIGN="BASELINE">
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFigureTitle">
<A NAME="pgfId=52640">
 </A>
FIGURE&nbsp;17.24&nbsp;<A NAME="11836">
 </A>
Examples of trees for timing-driven layout. (a)&nbsp;The MRST. (b)&nbsp;The shortest-path tree (SPT). (c)&nbsp;The quadratic minimum Steiner tree (QMST). (Problem <A HREF="CH17.6.htm#21193" CLASS="XRef">
17.21</A>
)</P>
</TD>
</TR>
</TABLE>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=37812">
 </A>
17.22&nbsp;<SPAN CLASS="Emphasis">
</SPAN>
<A NAME="24110">
 </A>
(**Elmore delay, 120 min.)&nbsp;<A HREF="CH17.6.htm#33728" CLASS="XRef">
Figure&nbsp;17.25</A>
 shows an RC tree. The <SPAN CLASS="EquationVariables">
m</SPAN>
th moment of the impulse response for node <SPAN CLASS="EquationVariables">
i</SPAN>
 in an RC t