181 lines
5.5 KiB
HTML
181 lines
5.5 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>CHM Algorithm</TITLE>
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</HEAD><BODY BGCOLOR="white" TEXT="black">
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<CENTER>
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<H1>CHM Algorithm</H1>
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</CENTER>
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<HR NOSHADE SIZE=1>
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<H2>The Algorithm</H2>
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<P>
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The algorithm is presented in <A HREF="#papers">[1,2,3</A>].
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</P>
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<HR NOSHADE SIZE=1>
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<H2>Memory Consumption</H2>
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<P>
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Now we detail the memory consumption to generate and to store minimal perfect hash functions
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using the CHM algorithm. The structures responsible for memory consumption are in the
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following:
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</P>
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<UL>
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<LI>Graph:
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<OL>
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<LI><B>first</B>: is a vector that stores <I>cn</I> integer numbers, each one representing
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the first edge (index in the vector edges) in the list of
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edges of each vertex.
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The integer numbers are 4 bytes long. Therefore,
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the vector first is stored in <I>4cn</I> bytes.
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<P></P>
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<LI><B>edges</B>: is a vector to represent the edges of the graph. As each edge
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is compounded by a pair of vertices, each entry stores two integer numbers
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of 4 bytes that represent the vertices. As there are <I>n</I> edges, the
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vector edges is stored in <I>8n</I> bytes.
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<P></P>
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<LI><B>next</B>: given a vertex <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">, we can discover the edges that
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contain <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> following its list of edges, which starts on
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first[<IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">] and the next
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edges are given by next[...first[<IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">]...]. Therefore,
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the vectors first and next represent
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the linked lists of edges of each vertex. As there are two vertices for each edge,
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when an edge is iserted in the graph, it must be inserted in the two linked lists
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of the vertices in its composition. Therefore, there are <I>2n</I> entries of integer
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numbers in the vector next, so it is stored in <I>4*2n = 8n</I> bytes.
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<P></P>
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</OL>
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<LI>Other auxiliary structures
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<OL>
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<LI><B>visited</B>: is a vector of <I>cn</I> bits, where each bit indicates if the g value of
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a given vertex was already defined. Therefore, the vector visited is stored
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in <I>cn/8</I> bytes.
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<P></P>
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<LI><B>function <I>g</I></B>: is represented by a vector of <I>cn</I> integer numbers.
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As each integer number is 4 bytes long, the function <I>g</I> is stored in
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<I>4cn</I> bytes.
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</OL>
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</UL>
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<P>
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Thus, the total memory consumption of CHM algorithm for generating a minimal
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perfect hash function (MPHF) is: <I>(8.125c + 16)n + O(1)</I> bytes.
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As the value of constant <I>c</I> must be at least 2.09 we have:
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</P>
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<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
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<TR>
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<TH><I>c</I></TH>
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<TH>Memory consumption to generate a MPHF</TH>
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</TR>
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<TR>
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<TD>2.09</TD>
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<TD ALIGN="center"><I>33.00n + O(1)</I></TD>
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</TR>
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</TABLE>
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<TABLE ALIGN="center" CELLPADDING="4">
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<TR>
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<TD><B>Table 1:</B> Memory consumption to generate a MPHF using the CHM algorithm.</TD>
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</TR>
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</TABLE>
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<P>
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Now we present the memory consumption to store the resulting function.
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We only need to store the <I>g</I> function. Thus, we need <I>4cn</I> bytes.
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Again we have:
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</P>
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<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
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<TR>
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<TH><I>c</I></TH>
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<TH>Memory consumption to store a MPHF</TH>
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</TR>
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<TR>
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<TD>2.09</TD>
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<TD ALIGN="center"><I>8.36n</I></TD>
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</TR>
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</TABLE>
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<TABLE ALIGN="center" CELLPADDING="4">
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<TR>
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<TD><B>Table 2:</B> Memory consumption to store a MPHF generated by the CHM algorithm.</TD>
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</TR>
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</TABLE>
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<HR NOSHADE SIZE=1>
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<H2>Experimental Results</H2>
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<P>
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<A HREF="comparison.html">CHM x BMZ</A>
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</P>
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<HR NOSHADE SIZE=1>
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<A NAME="papers"></A>
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<H2>Papers</H2>
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<OL>
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<LI>Z.J. Czech, G. Havas, and B.S. Majewski. <A HREF="papers/chm92.pdf">An optimal algorithm for generating minimal perfect hash functions.</A>, Information Processing Letters, 43(5):257-264, 1992.
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<P></P>
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<LI>Z.J. Czech, G. Havas, and B.S. Majewski. Fundamental study perfect hashing.
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Theoretical Computer Science, 182:1-143, 1997.
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<P></P>
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<LI>B.S. Majewski, N.C. Wormald, G. Havas, and Z.J. Czech. A family of perfect hashing methods.
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The Computer Journal, 39(6):547--554, 1996.
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</OL>
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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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<TD><A HREF="chd.html">CHD</A></TD>
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<TD><A HREF="bdz.html">BDZ</A></TD>
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<TD><A HREF="bmz.html">BMZ</A></TD>
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<TD><A HREF="chm.html">CHM</A></TD>
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<TD><A HREF="brz.html">BRZ</A></TD>
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<TD><A HREF="fch.html">FCH</A></TD>
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</TR>
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</TABLE>
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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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<!-- html code generated by txt2tags 2.6 (http://txt2tags.org) -->
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<!-- cmdline: txt2tags -t html -i CHM.t2t -o docs/chm.html -->
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