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<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION0015111000000000000000">Brute-Force Algorithm</A></H3>
<P>
Since each of the elements of  <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"  >
is either a zero or a one,
there are  <IMG WIDTH=14 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline60823" SRC="img563.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img563.gif"  > possible values for <I>X</I>.
A brute-force algorithm to solve this problem
finds the best solution by enumerating all the possible values of <I>X</I>.
<P>
For each possible value of <I>X</I> we check first
if the constraint  <IMG WIDTH=101 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline68439" SRC="img1813.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1813.gif"  > is satisfied.
A value which satisfies the constraint is called
a <em>feasible solution</em><A NAME=32311>&#160;</A>.
The solution to the problem is the feasible solution which
minimizes  <IMG WIDTH=52 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline68441" SRC="img1814.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1814.gif"  > which is called the
<em>objective function</em><A NAME=32315>&#160;</A>.
<P>
Since there are  <IMG WIDTH=14 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline60823" SRC="img563.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img563.gif"  > possible values of <I>X</I>
the running time of a brute-force solution is  <IMG WIDTH=38 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60725" SRC="img544.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img544.gif"  >.
The running time needed to determine whether a possible value is
a feasible solution is <I>O</I>(<I>n</I>)
and the time required to evaluate the objective function is also <I>O</I>(<I>n</I>).
Therefore, the running time of the brute-force algorithm is  <IMG WIDTH=49 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68453" SRC="img1815.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1815.gif"  >.
<P>
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