115 lines
5.6 KiB
HTML
115 lines
5.6 KiB
HTML
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<HTML>
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<HEAD>
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<META NAME="generator" CONTENT="http://txt2tags.org">
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<LINK REL="stylesheet" TYPE="text/css" HREF="DOC.css">
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<TITLE>Minimal Perfect Hash Functions - Introduction</TITLE>
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</HEAD><BODY BGCOLOR="white" TEXT="black">
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<CENTER>
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<H1>Minimal Perfect Hash Functions - Introduction</H1>
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</CENTER>
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<HR NOSHADE SIZE=1>
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<H2>Basic Concepts</H2>
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<P>
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Suppose <IMG ALIGN="bottom" SRC="figs/img14.png" BORDER="0" ALT=""> is a universe of <I>keys</I>.
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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="">.
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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="">.
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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
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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
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a <I>hash table</I>.
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Hashing methods for <I>non-static sets</I> of keys can be used to construct
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data structures storing <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and supporting membership queries
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"<IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">?" in expected time <IMG ALIGN="middle" SRC="figs/img21.png" BORDER="0" ALT="">.
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However, they involve a certain amount of wasted space owing to unused
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locations in the table and waisted time to resolve collisions when
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two keys are hashed to the same table location.
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</P>
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<P>
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For <I>static sets</I> of keys it is possible to compute a function
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to find any key in a table in one probe; such hash functions are called
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<I>perfect</I>.
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More precisely, given a set of keys <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, we shall say that a
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hash function <IMG ALIGN="bottom" SRC="figs/img15.png" BORDER="0" ALT=""> is a <I>perfect hash function</I>
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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="">,
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that is, there are no <I>collisions</I> among the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">:
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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="">,
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then <IMG ALIGN="middle" SRC="figs/img24.png" BORDER="0" ALT="">.
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Figure 1(a) illustrates a perfect hash function.
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Since no collisions occur, each key can be retrieved from the table
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with a single probe.
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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="">,
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then we say that <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> is a <I>minimal perfect hash function</I>
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for <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
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Figure 1(b) illustrates a minimal perfect hash function.
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Minimal perfect hash functions totally avoid the problem of wasted
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space and time. A perfect hash function <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> is <I>order preserving</I>
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if the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> are arranged in some given order
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and <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT=""> preserves this order in the hash table.
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</P>
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<TABLE ALIGN="center" CELLPADDING="4">
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<TR>
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<TD ALIGN="right"><center><IMG ALIGN="middle" SRC="figs/img26.png" BORDER="0" ALT=""></center></TD>
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</TR>
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<TR>
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<TD><B>Figure 1:</B> (a) Perfect hash function. (b) Minimal perfect hash function.</TD>
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</TR>
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</TABLE>
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<P>
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Minimal perfect hash functions are widely used for memory efficient
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storage and fast retrieval of items from static sets, such as words in natural
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languages, reserved words in programming languages or interactive systems,
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universal resource locations (URLs) in Web search engines, or item sets in
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data mining techniques.
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</P>
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<HR NOSHADE SIZE=1>
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<TABLE ALIGN="center" CELLPADDING="4">
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<TR>
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<TD><A HREF="index.html">Home</A></TD>
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<HR NOSHADE SIZE=1>
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<P>
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Enjoy!
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</P>
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<P>
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<A HREF="mailto:davi@users.sourceforge.net">Davi de Castro Reis</A>
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</P>
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<P>
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<A HREF="mailto:db8192@users.sourceforge.net">Djamel Belazzougui</A>
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</P>
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<P>
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<A HREF="mailto:fc_botelho@users.sourceforge.net">Fabiano Cupertino Botelho</A>
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</P>
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<P>
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<A HREF="mailto:nivio@dcc.ufmg.br">Nivio Ziviani</A>
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</P>
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