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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="SECTION0016100000000000000000">Basics</A></H1>
<P>
Consider an arbitrary sequence <IMG WIDTH=157 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69689" SRC="img2088.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2088.gif" >
comprised of of <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" > elements drawn from a some universal set <I>U</I>.
The goal of <em>sorting</em> is to rearrange the elements of <I>S</I>
to produce a new sequence, say <I>S</I>',
in which the elements of <I>S</I> appear <em>in order</em>.
<P>
But what does it mean for the elements of <I>S</I>' to be <em>in order</em>?
We shall assume that there is a relation, <I><</I>,
defined over the universe <I>U</I>.
The relation <I><</I> must be a <em>total order</em>,
which is defined as follows:
<P>
<BLOCKQUOTE> <b>Definition</b><A NAME="defntotalorder"> </A>
A <em>total order</em><A NAME=34520> </A> is a relation, say <I><</I>,
defined on the elements of some universal set <I>U</I>
with the following properties:
<OL><LI>
For all pairs of elements <IMG WIDTH=95 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69713" SRC="img2089.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2089.gif" >,
<em>exactly one</em> of the following is true: <I>i</I><I><</I><I>j</I>, <I>i</I>=<I>j</I>, or <I>j</I><I><</I><I>i</I>.
<P>
(All elements are commensurate<A NAME=34523> </A>).<LI>
For all triples <IMG WIDTH=142 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69721" SRC="img2090.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2090.gif" >,
<IMG WIDTH=162 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline61946" SRC="img846.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img846.gif" >.
<P>
(The relation <I><</I> is transitive<A NAME=34524> </A>).
</OL></BLOCKQUOTE>
<P>
In order to <em>sort</em> the elements of the sequence <I>S</I>,
we determine the <em>permutation</em> <IMG WIDTH=162 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69729" SRC="img2091.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2091.gif" >
of the elements of <I>S</I> such that
<P> <IMG WIDTH=345 HEIGHT=15 ALIGN=BOTTOM ALT="displaymath69687" SRC="img2092.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2092.gif" ><P>
In practice, we are not interested in the permutation <I>P</I>, <em>per se</em>.
Instead, our objective is to compute the sorted sequence
<IMG WIDTH=162 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline69735" SRC="img2093.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2093.gif" >
in which <IMG WIDTH=51 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline69737" SRC="img2094.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2094.gif" > for <IMG WIDTH=64 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline69739" SRC="img2095.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2095.gif" >.
<P>
Sometimes the sequence to be sorted, <I>S</I>, contains duplicates.
I.e., <IMG WIDTH=134 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69743" SRC="img2096.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2096.gif" > such that <IMG WIDTH=46 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline69745" SRC="img2097.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2097.gif" >.
In general when a sequence that contains duplicates is sorted,
there is no guarantee that the duplicated elements
retain their relative positions.
I.e., <IMG WIDTH=9 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline69747" SRC="img2098.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2098.gif" > could appear either before or after <IMG WIDTH=12 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline69749" SRC="img2099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2099.gif" >
in the sorted sequence <I>S</I>'.
If duplicates retain their relative positions in the sorted sequence
the sort is said to be <em>stable</em><A NAME=34536> </A>.
In order for <IMG WIDTH=9 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline69747" SRC="img2098.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2098.gif" > and <IMG WIDTH=12 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline69749" SRC="img2099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2099.gif" > to retain their relative order in the
sorted sequence,
we require that <IMG WIDTH=17 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline69757" SRC="img2100.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2100.gif" > precedes <IMG WIDTH=18 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline69759" SRC="img2101.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2101.gif" > in <I>S</I>'.
Therefore, the sort is stable if <IMG WIDTH=49 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline69763" SRC="img2102.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2102.gif" >.
<P>
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