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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="SECTION0015330000000000000000">Example-Merge Sorting</A></H2>
<A NAME="secalgsmergesort"> </A>
<P>
Sorting algorithms and sorters are covered in detail in Chapter <A HREF="page483.html#chapsorting" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page483.html#chapsorting"><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>.
In this section we consider a divide-and-conquer sorting algorithm--<em>merge sort</em><A NAME=32954> </A>.
Given an array of <I>n</I> items in arbitrary order,
the objective is to rearrange the elements of the array
so that they are ordered from the smallest element to the largest one.
<P>
The merge sort algorithm sorts a sequence of length <I>n</I><I>></I>1
by splitting it into to subsequences--one of length <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" >,
the other of length <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" >.
Each subsequence is sorted and then the two sorted sequences
are merged into one.
<P>
Program <A HREF="page458.html#progalgs3c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page458.html#progalgs3c"><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 function <tt>MergeSort</tt> which
takes three arguments,
<tt>array</tt>, <tt>i</tt> and <tt>n</tt>.
The routine sorts the following <tt>n</tt> elements:
<P> <IMG WIDTH=450 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath68669" SRC="img1866.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1866.gif" ><P>
The <tt>MergeSort</tt> routine calls itself
as well as the <tt>Merge</tt> routine.
The purpose of the <tt>Merge</tt> routine is to merge two sorted sequences,
one of length <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" >,
the other of length <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" >,
into a single sorted sequence of length <I>n</I>.
This can easily be done in <I>O</I>(<I>n</I>) time.
(See Program <A HREF="page511.html#progsorter11c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page511.html#progsorter11c"><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="32977"> </A><A NAME="progalgs3c"> </A> <IMG WIDTH=575 HEIGHT=199 ALIGN=BOTTOM ALT="program32974" SRC="img1884.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1884.gif" ><BR>
<STRONG>Program:</STRONG> Divide-and-Conquer Example--Merge Sorting<BR>
<P>
<P>
The running time of the <tt>MergeSort</tt> routine
depends on the number of items to be sorted, <I>n</I>.
Although Program <A HREF="page458.html#progalgs3c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page458.html#progalgs3c"><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> works correctly for arbitrary values of <I>n</I>,
it is much easier to determine the running time
if we assume that <I>n</I> is a power of two.
In this case,
the running time is given by the recurrence
<P><A NAME="eqnalgsms"> </A> <IMG WIDTH=500 HEIGHT=48 ALIGN=BOTTOM ALT="equation32983" SRC="img1885.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1885.gif" ><P>
<P>
Equation <A HREF="page458.html#eqnalgsms" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page458.html#eqnalgsms"><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 easily solved using repeated substitution:
<P> <IMG WIDTH=500 HEIGHT=121 ALIGN=BOTTOM ALT="eqnarray32989" SRC="img1886.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1886.gif" ><P>
Setting <IMG WIDTH=61 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline68699" SRC="img1870.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1870.gif" > gives <IMG WIDTH=219 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68761" SRC="img1887.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1887.gif" >.
<P>
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