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<CAPTION ALIGN=BOTTOM><STRONG>Table:</STRONG> Pascal's Triangle</CAPTION></TABLE>
</P></DIV><P>
<P>
Program&nbsp;<A HREF="page467.html#progalgs5c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page467.html#progalgs5c"><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> defines the function <tt>Binom</tt>
which takes two integer arguments <I>n</I> and <I>m</I>
and computes the binomial coefficient  <IMG WIDTH=21 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline69021" SRC="img1950.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1950.gif"  >
by computing Pascal's triangle.
According to Equation&nbsp;<A HREF="page467.html#eqnalgsbinomii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page467.html#eqnalgsbinomii"><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>,
each subsequent row depends only on the preceding row--it is only necessary to keep track of one row of data.
The implementation shown uses an array of length <I>n</I>
to represent a row of Pascal's triangle.
Consequently, instead of a table of size  <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif"  >,
the algorithm gets by with <I>O</I>(<I>n</I>) space.
The implementation has been coded carefully
so that the computation can be done in place.
I.e., the elements of  <IMG WIDTH=29 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68987" SRC="img1943.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1943.gif"  > are computed
in reverse so that they can be written over the elements of  <IMG WIDTH=12 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline67646" SRC="img1662.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1662.gif"  >
that are no longer needed.
<P>
<P><A NAME="33431">&#160;</A><A NAME="progalgs5c">&#160;</A> <IMG WIDTH=575 HEIGHT=238 ALIGN=BOTTOM ALT="program33428" SRC="img1964.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1964.gif"  ><BR>
<STRONG>Program:</STRONG> Dynamic Programming Example--Computing Binomial Coefficients<BR>
<P>
<P>
The worst-case running time of the <tt>Binom</tt>
routine given in Program&nbsp;<A HREF="page467.html#progalgs5c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page467.html#progalgs5c"><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 clearly  <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif"  >.
<P>
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