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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="SECTION0013212000000000000000">Union, Intersection and Difference</A></H3>
<P>
The implementations of the union, intersection, and difference operators
(<tt>+</tt>, <tt>*</tt>, and <tt>-</tt>, respectively)
for operands of type <tt>SetAsBitVector</tt> are shown in Program&nbsp;<A HREF="page396.html#progbitset5c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page396.html#progbitset5c"><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>.
The code is quite similar to that for the <tt>SetAsArray</tt> class
given in Program&nbsp;<A HREF="page392.html#progbitset2c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page392.html#progbitset2c"><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="28758">&#160;</A><A NAME="progbitset5c">&#160;</A> <IMG WIDTH=575 HEIGHT=620 ALIGN=BOTTOM ALT="program28441" SRC="img1630.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1630.gif"  ><BR>
<STRONG>Program:</STRONG> <tt>SetAsBitVector</tt> Class Union, Intersection and Difference 	Operator Definitions<BR>
<P>
<P>
Instead of using the Boolean operators <tt>&amp;&amp;</tt>, <tt>||</tt> and <tt>!</tt>,
we have used the bitwise operators <tt>&amp;</tt>, <tt>|</tt> and <tt></tt>.
By using the bitwise operators,
 <IMG WIDTH=99 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline67410" SRC="img1625.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1625.gif"  > bits of the result are computed
in each iteration of the loop.
Therefore, the number of iterations
required is  <IMG WIDTH=43 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline67408" SRC="img1624.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1624.gif"  > instead of <I>N</I>.
The worst-case running time of each of these operations
is  <IMG WIDTH=130 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline67412" SRC="img1626.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1626.gif"  >.
<P>
Notice that the asymptotic performance of these <tt>SetAsBitVector</tt>
class operations is the same as the asymptotic performance of
the <tt>SetAsArray</tt> class operations.
I.e., both of them are <I>O</I>(<I>N</I>).
Nevertheless, the <tt>SetAsBitVector</tt> class operations are faster.
In fact, the bit-vector approach is asymptotically faster than the 
the array approach by the factor <I>w</I>.
<P>
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