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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H1><A NAME="SECTION0016600000000000000000">Merge Sorting</A></H1>
<P>
The fourth class of sorting algorithm we consider comprises
algorithms that sort <em>by merging</em><A NAME=42838> </A><A NAME=42839> </A>.
Merging is the combination of two or more sorted sequences
into a single sorted sequence.
<P>
Figure <A HREF="page509.html#figmerge" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page509.html#figmerge"><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 the basic, two-way merge operation.
In a two-way merge, two sorted sequences are merged into one.
Clearly, two sorted sequences each of length <I>n</I>
can be merged into a sorted sequence of length 2<I>n</I>
in <I>O</I>(2<I>n</I>)=<I>O</I>(<I>n</I>) steps.
However in order to do this,
we need space in which to store the result.
I.e., it is not possible to merge the two sequences <em>in place</em>
in <I>O</I>(<I>n</I>) steps.
<P>
<P><A NAME="44010"> </A><A NAME="figmerge"> </A> <IMG WIDTH=575 HEIGHT=192 ALIGN=BOTTOM ALT="figure42842" SRC="img2199.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2199.gif" ><BR>
<STRONG>Figure:</STRONG> Two-Way Merging<BR>
<P>
<P>
Sorting by merging is a recursive, divide-and-conquer strategy.
In the base case, we have a sequence with exactly one element in it.
Since such a sequence is already sorted, there is nothing to be done.
To sort a sequence of <I>n</I><I>></I>1 elements:
<OL><LI>
Divide the sequence into two sequences 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" > and <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" >;<LI>
recursively sort each of the two subsequences; and then,<LI>
merge the sorted subsequences to obtain the final result.
</OL>
Figure <A HREF="page509.html#figsort7" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page509.html#figsort7"><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 the operation
of the two-way merge sort algorithm.
<P>
<P><A NAME="44537"> </A><A NAME="figsort7"> </A> <IMG WIDTH=575 HEIGHT=479 ALIGN=BOTTOM ALT="figure44016" SRC="img2200.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2200.gif" ><BR>
<STRONG>Figure:</STRONG> Two-Way Merge Sorting<BR>
<P><BR> <HR>
<UL>
<LI> <A NAME="tex2html8205" HREF="page510.html#SECTION0016601000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page510.html#SECTION0016601000000000000000">Implementation</A>
<LI> <A NAME="tex2html8206" HREF="page511.html#SECTION0016602000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page511.html#SECTION0016602000000000000000">Merging</A>
<LI> <A NAME="tex2html8207" HREF="page512.html#SECTION0016603000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page512.html#SECTION0016603000000000000000">Two-Way Merge Sorting</A>
<LI> <A NAME="tex2html8208" HREF="page513.html#SECTION0016604000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page513.html#SECTION0016604000000000000000">Running Time Analysis</A>
</UL>
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