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<TITLE>Example-Computing Powers</TITLE>
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<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
<A NAME="tex2html2535" HREF="page53.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page53.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html2533" HREF="page47.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page47.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html2527" HREF="page51.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page51.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html2537" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html2538" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <BR><HR>
<H2><A NAME="SECTION003250000000000000000">Example-Computing Powers</A></H2>
<P>
In this section we consider the running time
to raise a number to a given integer power.
I.e., given a value <I>x</I> and non-negative integer <I>n</I>,
we wish to compute the  <IMG WIDTH=15 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline58739" SRC="img159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img159.gif"  >.
A na&#305;ve way to calculate  <IMG WIDTH=15 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline58739" SRC="img159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img159.gif"  > would be to use a loop such as
<PRE>int result = 1;
for (unsigned int i = 0; i &lt;= n; ++i)
    result *= x;</PRE>
While this may be fine for small values of <I>n</I>,
for large values of <I>n</I> the running time may become prohibitive.
As an alternative, consider the following recursive definition
<P><A NAME="eqnmodelpow">&#160;</A> <IMG WIDTH=500 HEIGHT=66 ALIGN=BOTTOM ALT="equation975" SRC="img160.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img160.gif"  ><P>
<P>
For example, using Equation&nbsp;<A HREF="page52.html#eqnmodelpow" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#eqnmodelpow"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>,
we would determine  <IMG WIDTH=21 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline58751" SRC="img161.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img161.gif"  > as follows
<P> <IMG WIDTH=337 HEIGHT=51 ALIGN=BOTTOM ALT="displaymath58727" SRC="img162.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img162.gif"  ><P>
which requires a total of five multiplication operations.
Similarly, we would compute  <IMG WIDTH=21 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline58753" SRC="img163.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img163.gif"  > as follows
<P> <IMG WIDTH=351 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath58728" SRC="img164.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img164.gif"  ><P>
which requires a total of eight multiplication operations.
<P>
A recursive algorithm to compute  <IMG WIDTH=15 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline58739" SRC="img159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img159.gif"  >
based on the direct implementation of Equation&nbsp;<A HREF="page52.html#eqnmodelpow" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#eqnmodelpow"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
is given in Program&nbsp;<A HREF="page52.html#progpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#progpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
Table&nbsp;<A HREF="page52.html#tblpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#tblpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> gives the running time,
as predicted by the simplified model,
for each of the executable statements in Program&nbsp;<A HREF="page52.html#progpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#progpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
<P>
<P><A NAME="996">&#160;</A><A NAME="progpowerc">&#160;</A> <IMG WIDTH=575 HEIGHT=181 ALIGN=BOTTOM ALT="program993" SRC="img165.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img165.gif"  ><BR>
<STRONG>Program:</STRONG> Program to compute  <IMG WIDTH=15 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline58739" SRC="img159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img159.gif"  ><BR>
<P>
<P>
<P><A NAME="1156">&#160;</A>
<P>
    <A NAME="tblpowerc">&#160;</A>
    <DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=4 BORDER FRAME=HSIDES RULES=GROUPS>
<COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
	    </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP COLSPAN=3> time</TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<P>
	    statement </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>n</I>=0 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>n</I><I>&gt;</I>0 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>n</I><I>&gt;</I>0 </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	    </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>n</I> is even </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>n</I> is odd </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>3 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 3 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 3 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 3 </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	    4 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 2 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	    5 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 5 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 5 </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	    6 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>  <IMG WIDTH=96 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58781" SRC="img166.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img166.gif"  > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	    8 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -- </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>  <IMG WIDTH=97 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58783" SRC="img167.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img167.gif"  > </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>TOTAL </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 5 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>  <IMG WIDTH=96 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58787" SRC="img168.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img168.gif"  > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
		 <IMG WIDTH=97 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58789" SRC="img169.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img169.gif"  > </TD></TR>
</TBODY>
<CAPTION ALIGN=BOTTOM><STRONG>Table:</STRONG> Computing the running time of Program&nbsp;<A HREF="page52.html#progpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#progpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A></CAPTION></TABLE>
</P></DIV><P>
<P>
By summing the columns in Table&nbsp;<A HREF="page52.html#tblpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#tblpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> we get
the following recurrence for the running time of Program&nbsp;<A HREF="page52.html#progpowerc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#progpowerc"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
<P><A NAME="eqnmodelpowtime">&#160;</A> <IMG WIDTH=500 HEIGHT=66 ALIGN=BOTTOM ALT="equation1015" SRC="img170.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img170.gif"  ><P>
As the first attempt at solving this recurrence,
let us suppose that  <IMG WIDTH=45 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58795" SRC="img171.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img171.gif"  > for some <I>k</I><I>&gt;</I>0.
Clearly, since <I>n</I> is a power of two, it is even.
Therefore,  <IMG WIDTH=137 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline58801" SRC="img172.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img172.gif"  >.
<P>
For  <IMG WIDTH=45 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58795" SRC="img171.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img171.gif"  >, Equation&nbsp;<A HREF="page52.html#eqnmodelpowtime" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html#eqnmodelpowtime"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> gives
<P> <IMG WIDTH=358 HEIGHT=19 ALIGN=BOTTOM ALT="displaymath58729" SRC="img173.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img173.gif"  ><P>
This can be solved by repeated substitution:
12

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