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<TITLE>Removing Items from a Leftist Heap</TITLE>
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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="SECTION0012350000000000000000">Removing Items from a Leftist Heap</A></H2>
<P>
The <tt>FindMin</tt> member function locates the item with the smallest key
in a given priority queue
and the <tt>DequeueMin</tt> member function removes it from the queue.
Since the smallest item in a heap is found at the root,
the <tt>FindMin</tt> operation is easy to implement.
Program <A HREF="page369.html#progleftist3c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page369.html#progleftist3c"><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> shows how it can be done.
Clearly, the running time of the <tt>FindMin</tt> operation is <I>O</I>(1).
<P>
<P><A NAME="26665"> </A><A NAME="progleftist3c"> </A> <IMG WIDTH=575 HEIGHT=124 ALIGN=BOTTOM ALT="program26570" SRC="img1493.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1493.gif" ><BR>
<STRONG>Program:</STRONG> <tt>LeftistHeap</tt> Class <tt>FindMin</tt> Member Function Definitions<BR>
<P>
<P>
Since the smallest item in a heap is at the root,
the <tt>DequeueMin</tt> operation must delete the root node.
Since a leftist heap is a binary heap,
the root has at most two children.
In general when the root is deleted,
we are left with two non-empty leftist heaps.
Since we already have an efficient way to merge leftist heaps,
the solution is to simply merge the two children of the root
to obtain a single heap again!
Program <A HREF="page369.html#progleftist4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page369.html#progleftist4c"><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> shows how the <tt>DequeueMin</tt> operation
of the <tt>LeftistHeap</tt> class can be implemented.
<P>
<P><A NAME="26667"> </A><A NAME="progleftist4c"> </A> <IMG WIDTH=575 HEIGHT=315 ALIGN=BOTTOM ALT="program26583" SRC="img1494.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1494.gif" ><BR>
<STRONG>Program:</STRONG> <tt>LeftistHeap</tt> Class <tt>DequeueMin</tt> Member Function Definition<BR>
<P>
<P>
The running time of Program <A HREF="page369.html#progleftist4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page369.html#progleftist4c"><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> is determined
by the time required to merge the two children of the root (line 13)
since the rest of the work in <tt>DequeueMin</tt> can be done in constant time.
Consider the running time to delete the root of a leftist heap <I>T</I>
with <I>n</I> internal nodes.
The running time to merge the left and right subtrees of <I>T</I>
<P> <IMG WIDTH=425 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath66540" SRC="img1495.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1495.gif" ><P>
where <IMG WIDTH=16 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66448" SRC="img1476.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1476.gif" > and <IMG WIDTH=18 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66450" SRC="img1477.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1477.gif" > are the null path lengths of the left and right
subtrees <I>T</I>, respectively.
In the worst case, <IMG WIDTH=47 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66556" SRC="img1496.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1496.gif" > and <IMG WIDTH=91 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline66558" SRC="img1497.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1497.gif" >.
If we assume that <IMG WIDTH=179 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline64824" SRC="img1296.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1296.gif" >,
the running time for <tt>DequeueMin</tt> is <IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59891" SRC="img403.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img403.gif" >.
<P>
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