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<TITLE>Minimal Perfect Hash Functions - Introduction</TITLE>
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<H1>Minimal Perfect Hash Functions - Introduction</H1>
</CENTER>
<HR NOSHADE SIZE=1>
<H2>Basic Concepts</H2>
<P>
Suppose <IMG ALIGN="bottom" SRC="figs/img14.png" BORDER="0" ALT=""> is a universe of <I>keys</I>.
Let <IMG ALIGN="bottom" SRC="figs/img15.png" BORDER="0" ALT=""> be a <I>hash function</I> that maps the keys from <IMG ALIGN="bottom" SRC="figs/img14.png" BORDER="0" ALT=""> to a given interval of integers <IMG ALIGN="middle" SRC="figs/img16.png" BORDER="0" ALT="">.
Let <IMG ALIGN="middle" SRC="figs/img17.png" BORDER="0" ALT=""> be a set of <IMG ALIGN="bottom" SRC="figs/img8.png" BORDER="0" ALT=""> keys from <IMG ALIGN="bottom" SRC="figs/img14.png" BORDER="0" ALT="">.
Given a key <IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">, the hash function <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> computes an
integer in <IMG ALIGN="middle" SRC="figs/img19.png" BORDER="0" ALT=""> for the storage or retrieval of <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> in
a <I>hash table</I>.
Hashing methods for <I>non-static sets</I> of keys can be used to construct
data structures storing <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and supporting membership queries
"<IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">?" in expected time <IMG ALIGN="middle" SRC="figs/img21.png" BORDER="0" ALT="">.
However, they involve a certain amount of wasted space owing to unused
locations in the table and waisted time to resolve collisions when
two keys are hashed to the same table location.
</P>
<P>
For <I>static sets</I> of keys it is possible to compute a function
to find any key in a table in one probe; such hash functions are called
<I>perfect</I>.
More precisely, given a set of keys <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, we shall say that a
hash function <IMG ALIGN="bottom" SRC="figs/img15.png" BORDER="0" ALT=""> is a <I>perfect hash function</I>
for <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> if <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> is an injection on <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">,
that is, there are no <I>collisions</I> among the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">:
if <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img22.png" BORDER="0" ALT=""> are in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img23.png" BORDER="0" ALT="">,
then <IMG ALIGN="middle" SRC="figs/img24.png" BORDER="0" ALT="">.
Figure 1(a) illustrates a perfect hash function.
Since no collisions occur, each key can be retrieved from the table
with a single probe.
If <IMG ALIGN="bottom" SRC="figs/img25.png" BORDER="0" ALT="">, that is, the table has the same size as <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">,
then we say that <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> is a <I>minimal perfect hash function</I>
for <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
Figure 1(b) illustrates a minimal perfect hash function.
Minimal perfect hash functions totally avoid the problem of wasted
space and time. A perfect hash function <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> is <I>order preserving</I>
if the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> are arranged in some given order
and <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> preserves this order in the hash table.
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD ALIGN="right"><center><IMG ALIGN="middle" SRC="figs/img26.png" BORDER="0" ALT=""></center></TD>
</TR>
<TR>
<TD><B>Figure 1:</B> (a) Perfect hash function. (b) Minimal perfect hash function.</TD>
</TR>
</TABLE>
<P>
Minimal perfect hash functions are widely used for memory efficient
storage and fast retrieval of items from static sets, such as words in natural
languages, reserved words in programming languages or interactive systems,
universal resource locations (URLs) in Web search engines, or item sets in
data mining techniques.
</P>
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Enjoy!
</P>
<P>
<A HREF="mailto:davi@users.sourceforge.net">Davi de Castro Reis</A>
</P>
<P>
<A HREF="mailto:db8192@users.sourceforge.net">Djamel Belazzougui</A>
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<A HREF="mailto:fc_botelho@users.sourceforge.net">Fabiano Cupertino Botelho</A>
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<A HREF="mailto:nivio@dcc.ufmg.br">Nivio Ziviani</A>
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