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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="SECTION0013211000000000000000">Basic Operations</A></H3>
<P>
Program <A HREF="page395.html#progbitset4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page395.html#progbitset4c"><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 constructor for the <tt>SetAsBitVector</tt>
class as well as the three basic operations--<tt>Insert</tt>, <tt>IsMember</tt>, and <tt>Withdraw</tt>.
The constructor takes a single argument
<IMG WIDTH=144 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline67308" SRC="img1608.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1608.gif" >,
which specifies the universe and, consequently,
the number of bits needed in the bit array.
The constructor creates a vector of <tt>Word</tt>s of length <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" >,
where <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" > is the number of bits in a word,
and sets the elements of the vector to zero.
The running time of the constructor 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>
<P><A NAME="28751"> </A><A NAME="progbitset4c"> </A> <IMG WIDTH=575 HEIGHT=506 ALIGN=BOTTOM ALT="program28400" SRC="img1627.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1627.gif" ><BR>
<STRONG>Program:</STRONG> <tt>SetAsBitVector</tt> Class Constructor, <tt>Insert</tt>, <tt>Withdraw</tt> and <tt>IsMember</tt> Member Function Definitions<BR>
<P>
<P>
To insert an item into the set,
we need to change the appropriate bit in the array of bits to one.
The <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > bit of the bit array
is bit <IMG WIDTH=56 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline67416" SRC="img1628.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1628.gif" > of word <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline67418" SRC="img1629.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1629.gif" >.
Thus, the <tt>Insert</tt> function is implemented using
a <em>bitwise or</em> operation to change the <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > bit to one
as shown in Program <A HREF="page395.html#progbitset4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page395.html#progbitset4c"><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>.
Even though it is slightly more complicated than the corresponding operation
for the <tt>SetAsArray</tt> class,
the running time for this operation is still <I>O</I>(1).
Since <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" > is a power of two,
it is possible to replace the division and modulo operations,
<tt>/</tt> and <tt>%</tt>,
with shifts and masks like this:
<PRE>vector [item >> shift] |= 1 << (item & mask);</PRE>
for a suitable definition of the constants <tt>shift</tt> and <tt>mask</tt>.
Depending on the compiler and machine architecture,
doing so may improve the performance of the <tt>Insert</tt> operation by
a constant factor.
Of course, its asymptotic performance is still <I>O</I>(1).
<P>
To withdraw an item from the set,
we need to clear the appropriate bit in the array of bits
and to test if an item is a member of the set,
we test the corresponding bit.
The <tt>IsMember</tt> and <tt>Withdraw</tt> routines in Program <A HREF="page395.html#progbitset4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page395.html#progbitset4c"><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>
show how this can be done.
Like <tt>Insert</tt>,
both these routines have constant worst-case running times.
<P>
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