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<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION0015110000000000000000">Example-Counting Change</A></H2>
<A NAME="secalgschange">&#160;</A>
<P>
Consider the problem a cashier solves
every time she counts out some amount of currency.
The cashier has at her disposal a collection of
notes and coins of various denominations
and is required to count out a specified sum
using the smallest possible number of pieces.
<P>
The problem can be expressed mathematically as follows:
Let there be <I>n</I> pieces of money (notes or coins),  <IMG WIDTH=139 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68403" SRC="img1804.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1804.gif"  >,
and let  <IMG WIDTH=11 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64564" SRC="img1246.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1246.gif"  > be the denomination of  <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline58415" SRC="img85.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img85.gif"  >.
E.g., if  <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline58415" SRC="img85.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img85.gif"  > is a dime, then  <IMG WIDTH=50 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68411" SRC="img1805.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1805.gif"  >.
To count out a given sum of money <I>A</I>
we find the smallest subset of <I>P</I>, say  <IMG WIDTH=43 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline68417" SRC="img1806.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1806.gif"  >,
such that  <IMG WIDTH=94 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline68419" SRC="img1807.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1807.gif"  >.
<P>
One way to represent the subset <I>S</I> is to use <I>n</I> variables
 <IMG WIDTH=145 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68425" SRC="img1808.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1808.gif"  >, such that
<P> <IMG WIDTH=308 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath68395" SRC="img1809.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1809.gif"  ><P>
Given  <IMG WIDTH=105 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68427" SRC="img1810.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1810.gif"  > our <em>objective</em>
is to minimize
<P> <IMG WIDTH=268 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath68396" SRC="img1811.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1811.gif"  ><P>
subject to the constraint
<P> <IMG WIDTH=294 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath68397" SRC="img1812.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1812.gif"  ><P><BR> <HR>
<UL> 
<LI> <A NAME="tex2html7387" HREF="page443.html#SECTION0015111000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page443.html#SECTION0015111000000000000000">Brute-Force Algorithm</A>
<LI> <A NAME="tex2html7388" HREF="page444.html#SECTION0015112000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page444.html#SECTION0015112000000000000000">Greedy Algorithm</A>
</UL>
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