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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="SECTION0016530000000000000000">Building the Heap</A></H2>
<P>
The <tt>BuildHeap</tt> routine shown in Program&nbsp;<A HREF="page506.html#progsorter9c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page506.html#progsorter9c"><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>
transforms an unsorted array into a max heap.
It does so by calling the <tt>PercolateDown</tt> routine
for  <IMG WIDTH=260 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline70355" SRC="img2189.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2189.gif"  >.
<P>
<P><A NAME="40216">&#160;</A><A NAME="progsorter9c">&#160;</A> <IMG WIDTH=575 HEIGHT=123 ALIGN=BOTTOM ALT="program40134" SRC="img2190.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2190.gif"  ><BR>
<STRONG>Program:</STRONG> <tt>HeapSorter&lt;T&gt;</tt> Class 	<tt>BuildHeap</tt> Member Function Definition<BR>
<P>
<P>
Why does <tt>BuildHeap</tt> start percolating at  <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58823" SRC="img180.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img180.gif"  >?
A complete binary tree with <I>n</I> nodes
has exactly  <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline68743" SRC="img1883.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1883.gif"  > leaves.
Therefore, the last node in the array which has a child
is in position  <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline58823" SRC="img180.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img180.gif"  >.
Consequently, the <tt>BuildHeap</tt> routine starts doing percolate down
operations from that point.
<P>
The <tt>BuildHeap</tt> visits the array elements in reverse order.
In effect the algorithm starts at the deepest node that has a child
and works toward the root of the tree.
Each array position visited is the root of a subtree.
As each such subtree is visited,
it is transformed into a max heap.
Figure&nbsp;<A HREF="page506.html#figsort5" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page506.html#figsort5"><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> illustrates how the <tt>BuildHeap</tt> routine
heapifies an array that is initially unsorted.
<P>
<P><A NAME="41041">&#160;</A><A NAME="figsort5">&#160;</A> <IMG WIDTH=575 HEIGHT=631 ALIGN=BOTTOM ALT="figure40148" SRC="img2191.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2191.gif"  ><BR>
<STRONG>Figure:</STRONG> Building A Heap<BR>
<P><BR> <HR>
<UL> 
<LI> <A NAME="tex2html8169" HREF="page507.html#SECTION0016531000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page507.html#SECTION0016531000000000000000">Running Time Analysis</A>
<LI> <A NAME="tex2html8170" HREF="page508.html#SECTION0016532000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page508.html#SECTION0016532000000000000000">The Sorting Phase</A>
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