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<TITLE>Data Structures and Algorithms: Sorting</TITLE>

<META name="description" content="Data Structures and Algorithms Course Notes,
PLDS210 University of Western Australia">
<META name="keywords" content="data structures,algorithms,abstract data types,
sorting, bubble sort, selection sort, insertion sort">
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<FONT FACE=helvetica SIZE=+1><I>Data Structures and Algorithms</I></FONT>
</TD></TR>

<TR><TD><FONT FACE=helvetica SIZE=+2><B>7 Sorting</B></FONT>
</TD></TR>
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<P>

Sorting is one of the most important operations performed
by computers.
In the days of magnetic tape storage before modern data-bases,
it was almost certainly the <I>most</I> common operation
performed by computers as most "database" updating was done
by sorting transactions and merging them with a master file.
It's still important for presentation of data extracted from
databases: most people prefer to get reports sorted into some
relevant order before wading through pages of data!

<A NAME=bubble></A>
<H3>7.1 Bubble, Selection, Insertion Sorts</H3>
There are a large number of variations of one basic
strategy for sorting.
It's the same strategy that you use for sorting your
bridge hand. You pick up a card, 
start at the beginning of your hand and find the place
to insert the new card, insert it and move all the others
up one place.
<FONT COLOR=green>
<PRE>/* Insertion sort for integers */

void insertion( int a[], int n ) {
/* Pre-condition: a contains n items to be sorted */
    int i, j, v;
    /* Initially, the first item is considered 'sorted' */
    /* i divides a into a sorted region, x&lt;i, and an
       unsorted one, x &gt;= i */
    for(i=1;i&lt;n;i++) {
        /* Select the item at the beginning of the
           as yet unsorted section */
        v = a[i];
        /* Work backwards through the array, finding where v 
           should go */
        j = i;
        /* If this element is greater than v,
              move it up one */
        while ( a[j-1] &gt; v ) {
          a[j] = a[j-1]; j = j-1;
          if ( j &lt;= 0 ) break;
          }
        /* Stopped when a[j-1] &lt;= v, so put v at position j */
        a[j] = v;
        }
    }    
</PRE></FONT>

<A NAME=insert_anim>
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<FONT COLOR=blue FACE=helvetica>
<B>Insertion Sort Animation</B><BR>
This animation was written by Woi Ang.</FONT></TD>
<TD ALIGN=center>
  <TABLE BORDER=0>
  <TR><TD>
    <applet CODEBASE="tppjava" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/Java/ins_sort/" code = "AlgAnimApp.class" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/Java/ins_sort/AlgAnimApp.class" width = 200 height = 35>
    <param name = "filename" value = "AlgThread.java">
    <param name = "buttonname" value = "Run the Animation">
    <param name = "algname" value = "Insertion Sort">
    </applet>
    </TD>
  </TR>
</TABLE>
</TD>
<TD><FONT FACE=helvetica>Please email comments to:<BR>
<A HREF=mailto:morris@ee.uwa.edu.au>morris@ee.uwa.edu.au</A>
</TR>
</TABLE>
<P>
<H3>Bubble Sort</H3>

Another variant of this procedure, called bubble sort, is commonly taught:
<FONT COLOR=green>
<PRE>/* Bubble sort for integers */
#define SWAP(a,b)   { int t; t=a; a=b; b=t; }

void bubble( int a[], int n )
/* Pre-condition: a contains n items to be sorted */
    {
    int i, j;
    /* Make n passes through the array */
    for(i=0;i&lt;n;i++)
        {
        /* From the first element to the end
           of the unsorted section */
        for(j=1;j&lt;(n-i);j++)
           {
           /* If adjacent items are out of order, swap them */
           if( a[j-1]&gt;a[j] ) SWAP(a[j-1],a[j]);
           }
        }
    }    
</PRE></FONT>

<H3>Analysis</H3>

Each of these algorithms requires <B><I>n</I></B>-1 passes:
each pass places one item in its correct place.
(The <B><I>n<SUP>th</SUP></I></B> is then in the correct place also.)
The <B><I>i<SUP>th</SUP></I></B> pass makes either <B><I>i</I></B>or 
<B><I>n - i</I></B> comparisons and moves.
So:
<BLOCKQUOTE>
<IMG SRC="sumi.gif" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/fig/sumi.gif"></IMG>
</BLOCKQUOTE>
or <B>O(<I>n</I><SUP>2</SUP>)</B> -
but we already know we can use heaps to get an
<B>O(<I>n</I> log<I>n</I>)</B>
algorithm.
Thus these algorithms are only suitable for small
problems where their simple code makes them faster than
the more complex code of the
<B>O(<I>n</I> log<I>n</I>)</B>
algorithm.
As a rule of thumb, expect to find an
<B>O(<I>n</I> log<I>n</I>)</B>
algorithm faster for <I>n</I>&gt;10 -
<I>but the exact value depends very much on individual machines!</I>.
<P>
They can be used to squeeze a little bit more performance out
of fast sort algorithms - <A HREF="qsort_perf.html" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/qsort_perf.html">see later</A>.

<P>
<TABLE WIDTH="100%" BGCOLOR="#00c0f0">
<TR><TD><H3>Key terms</H3></TD></TR></TABLE>
<DL>
<DT><FONT COLOR="#fa0000"><B>Bubble, Insertion, Selection Sorts</B></FONT>
   <DD>Simple sorting algorithms with <B><I>O(n<SUP>2</SUP>)</I></B>
      complexity - suitable for sorting small numbers of items only.
</DL>

<P>

<TABLE CELLPADDING=5 WIDTH="100%" BGCOLOR="#00f0f0" CELLSPACING=0>
<TR><TD>
Continue on to <A HREF="heapsort.html" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/heapsort.html">Heap Sort</A><BR>
Back to the <A HREF="ds_ToC.html" tppabs="http://www.ee.uwa.edu.au/~plsd210/ds/ds_ToC.html">Table of Contents</A>
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<SMALL>
&copy; <A HREF=mailto:morris@ee.uwa.edu.au>John Morris</A>, 1998
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