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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="SECTION009610000000000000000">Linear Probing</A></H2>
<P>
The simplest collision resolution strategy in open addressing
is called <em>linear probing</em><A NAME=13215> </A>.
In linear probing, the function <I>c</I>(<I>i</I>) is a linear function in <I>i</I>.
I.e., it is of the form
<P> <IMG WIDTH=296 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath63020" SRC="img1030.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1030.gif" ><P>
Property 1 requires that <I>c</I>(0)=0.
Therefore, <IMG WIDTH=10 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline63030" SRC="img1031.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1031.gif" > must be zero.
<P>
In order for <IMG WIDTH=61 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline63032" SRC="img1032.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1032.gif" > to satisfy Property 2,
<IMG WIDTH=9 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline63034" SRC="img1033.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1033.gif" > and <I>M</I> must be relatively prime.
If we know the <I>M</I> will always be a prime number,
then any <IMG WIDTH=9 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline63034" SRC="img1033.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1033.gif" > will do.
On the other hand,
if we cannot be certain that <I>M</I> is prime,
then <IMG WIDTH=9 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline63034" SRC="img1033.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1033.gif" > must be one.
Therefore, linear probing sequence that is usually used is
<P> <IMG WIDTH=330 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath63021" SRC="img1034.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1034.gif" ><P>
for <IMG WIDTH=147 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline63046" SRC="img1035.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1035.gif" >.
<P>
Figure <A HREF="page238.html#fighash3" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page238.html#fighash3"><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 an example of
a scatter table using open addressing together with linear probing.
For example, consider the string <tt>"åtta"</tt>.
This string hashes to array position <IMG WIDTH=12 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline62904" SRC="img996.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img996.gif" >.
The corresponding linear probing sequence begins at position
<IMG WIDTH=12 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline62904" SRC="img996.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img996.gif" > and goes on to positions <IMG WIDTH=14 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline62908" SRC="img997.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img997.gif" >, <IMG WIDTH=13 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline62910" SRC="img998.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img998.gif" >,....
In this case,
the search for the string <tt>"åtta"</tt> succeeds after three probes.
<P>
<P><A NAME="13865"> </A><A NAME="fighash3"> </A> <IMG WIDTH=575 HEIGHT=380 ALIGN=BOTTOM ALT="figure13223" SRC="img1036.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1036.gif" ><BR>
<STRONG>Figure:</STRONG> Scatter Table using Open Addressing and Linear Probing<BR>
<P>
<P>
To insert an item <I>x</I> into the scatter table,
an empty cell is found by following the same probe sequence that
would be used in a search for item <I>x</I>.
Thus, linear probing finds an empty cell by doing a linear search
beginning from array position <I>h</I>(<I>x</I>).
<P>
An unfortunate characteristic of linear probing arises from the fact
that as the table fills,
clusters of consecutive cells form
and the time required for a search increases with the size of the cluster.
Furthermore, when we attempt to insert an item in the table
at a position which is already occupied,
that item is ultimately inserted at the end of the cluster--thereby increasing its length.
This by itself is not inherently a bad thing.
After all, when using the chained approach,
every insertion increase the length of some chain by one.
However, whenever an insertion is made between two clusters
that are separated by one unoccupied position,
the two clusters become one,
thereby potentially increasing the cluster length by an amount
much greater than one--a bad thing!
This phenomenon is called <em>primary clustering</em><A NAME=13869> </A>.
<P>
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