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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="SECTION0012440000000000000000">Merging Binomial Queues</A></H2>
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Merging two binomial queues is like doing binary addition.
For example, consider the addition of <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif" > and <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66914" SRC="img1550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1550.gif" >:
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+</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66738" SRC="img1520.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1520.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66914" SRC="img1550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1550.gif" > </TD></TR>
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</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66916" SRC="img1551.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1551.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66922" SRC="img1552.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1552.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif" ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66594" SRC="img1501.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1501.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66928" SRC="img1553.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1553.gif" > </TD></TR>
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The usual algorithm for addition begins with the least-significant ``bit.''
Since <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif" > contains a <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66594" SRC="img1501.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1501.gif" > tree and <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66914" SRC="img1550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1550.gif" > does not,
the result is simply the <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66594" SRC="img1501.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1501.gif" > tree from <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif" >.
<P>
In the next step,
we add the <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif" > from <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif" > and the <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif" > from <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66914" SRC="img1550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1550.gif" >.
Combining the two <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif" >s we get a <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66922" SRC="img1552.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1552.gif" > which we <em>carry</em><A NAME=27579> </A>
to the next column.
Since there are no <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif" >s left,
the result does not contain any.
The addition continues in a similar manner
until all the columns have been added up.
<P>
Program <A HREF="page379.html#progbinom4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page379.html#progbinom4c"><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> gives an implementation of this addition algorithm.
The <tt>Merge</tt> member function of the <tt>BinomialQueue</tt> class
takes a reference to a <tt>BinomialQueue</tt> and adds its subtrees
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