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Add docs directory for github.
2018-12-29 03:53:09 +02:00
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<H1>BMZ Algorithm</H1>
</CENTER>
<HR NOSHADE SIZE=1>
<H2>History</H2>
<P>
At the end of 2003, professor <A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A> was
finishing the second edition of his <A HREF="http://www.dcc.ufmg.br/algoritmos/">book</A>.
During the <A HREF="http://www.dcc.ufmg.br/algoritmos/">book</A> writing,
professor <A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A> studied the problem of generating
<A HREF="concepts.html">minimal perfect hash functions</A>
(if you are not familiarized with this problem, see <A HREF="#papers">[1</A>]<A HREF="#papers">[2</A>]).
Professor <A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A> coded a modified version of
the <A HREF="chm.html">CHM algorithm</A>, which was proposed by
Czech, Havas and Majewski, and put it in his <A HREF="http://www.dcc.ufmg.br/algoritmos/">book</A>.
The <A HREF="chm.html">CHM algorithm</A> is based on acyclic random graphs to generate
<A HREF="concepts.html">order preserving minimal perfect hash functions</A> in linear time.
Professor <A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A>
argued himself, why must the random graph
be acyclic? In the modified version availalbe in his <A HREF="http://www.dcc.ufmg.br/algoritmos/">book</A> he got rid of this restriction.
</P>
<P>
The modification presented a problem, it was impossible to generate minimal perfect hash functions
for sets with more than 1000 keys.
At the same time, <A HREF="http://www.dcc.ufmg.br/~fbotelho">Fabiano C. Botelho</A>,
a master degree student at <A HREF="http://www.dcc.ufmg.br">Departament of Computer Science</A> in
<A HREF="http://www.ufmg.br">Federal University of Minas Gerais</A>,
started to be advised by <A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A> who presented the problem
to <A HREF="http://www.dcc.ufmg.br/~fbotelho">Fabiano</A>.
</P>
<P>
During the master, <A HREF="http://www.dcc.ufmg.br/~fbotelho">Fabiano</A> and
<A HREF="http://www.dcc.ufmg.br/~nivio">Nivio Ziviani</A> faced lots of problems.
In april of 2004, <A HREF="http://www.dcc.ufmg.br/~fbotelho">Fabiano</A> was talking with a
friend of him (David Menoti) about the problems
and many ideas appeared.
The ideas were implemented and a very fast algorithm to generate
minimal perfect hash functions had been designed.
We refer the algorithm to as <B>BMZ</B>, because it was conceived by Fabiano C. <B>B</B>otelho,
David <B>M</B>enoti and Nivio <B>Z</B>iviani. The algorithm is described in <A HREF="#papers">[1</A>].
To analyse BMZ algorithm we needed some results from the random graph theory, so
we invited professor <A HREF="http://www.ime.usp.br/~yoshi">Yoshiharu Kohayakawa</A> to help us.
The final description and analysis of BMZ algorithm is presented in <A HREF="#papers">[2</A>].
</P>
<HR NOSHADE SIZE=1>
<H2>The Algorithm</H2>
<P>
The BMZ algorithm shares several features with the <A HREF="chm.html">CHM algorithm</A>.
In particular, BMZ algorithm is also
based on the generation of random graphs <IMG ALIGN="middle" SRC="figs/img27.png" BORDER="0" ALT="">, where <IMG ALIGN="bottom" SRC="figs/img28.png" BORDER="0" ALT=""> is in
one-to-one correspondence with the key set <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> for which we wish to
generate a <A HREF="concepts.html">minimal perfect hash function</A>.
The two main differences between BMZ algorithm and CHM algorithm
are as follows: (<I>i</I>) BMZ algorithm generates random
graphs <IMG ALIGN="middle" SRC="figs/img27.png" BORDER="0" ALT=""> with <IMG ALIGN="middle" SRC="figs/img29.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img30.png" BORDER="0" ALT="">, where <IMG ALIGN="middle" SRC="figs/img31.png" BORDER="0" ALT="">,
and hence <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> necessarily contains cycles,
while CHM algorithm generates <I>acyclic</I> random
graphs <IMG ALIGN="middle" SRC="figs/img27.png" BORDER="0" ALT=""> with <IMG ALIGN="middle" SRC="figs/img29.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img30.png" BORDER="0" ALT="">,
with a greater number of vertices: <IMG ALIGN="middle" SRC="figs/img33.png" BORDER="0" ALT="">;
(<I>ii</I>) CHM algorithm generates <A HREF="concepts.html">order preserving minimal perfect hash functions</A>
while BMZ algorithm does not preserve order. Thus, BMZ algorithm improves
the space requirement at the expense of generating functions that are not
order preserving.
</P>
<P>
Suppose <IMG ALIGN="bottom" SRC="figs/img14.png" BORDER="0" ALT=""> is a universe of <I>keys</I>.
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="">.
Let us show how the BMZ algorithm constructs a minimal perfect hash function <IMG ALIGN="bottom" SRC="figs/img7.png" BORDER="0" ALT="">.
We make use of two auxiliary random functions <IMG ALIGN="middle" SRC="figs/img41.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img55.png" BORDER="0" ALT="">,
where <IMG ALIGN="middle" SRC="figs/img56.png" BORDER="0" ALT=""> for some suitably chosen integer <IMG ALIGN="bottom" SRC="figs/img57.png" BORDER="0" ALT="">,
where <IMG ALIGN="middle" SRC="figs/img58.png" BORDER="0" ALT="">.We build a random graph <IMG ALIGN="middle" SRC="figs/img59.png" BORDER="0" ALT=""> on <IMG ALIGN="bottom" SRC="figs/img60.png" BORDER="0" ALT="">,
whose edge set is <IMG ALIGN="middle" SRC="figs/img61.png" BORDER="0" ALT="">. There is an edge in <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> for each
key in the set of keys <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
</P>
<P>
In what follows, we shall be interested in the <I>2-core</I> of
the random graph <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, that is, the maximal subgraph
of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> with minimal degree at
least 2 (see <A HREF="#papers">[2</A>] for details).
Because of its importance in our context, we call the 2-core the
<I>critical</I> subgraph of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> and denote it by <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">.
The vertices and edges in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> are said to be <I>critical</I>.
We let <IMG ALIGN="middle" SRC="figs/img64.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img65.png" BORDER="0" ALT="">.
Moreover, we let <IMG ALIGN="middle" SRC="figs/img66.png" BORDER="0" ALT=""> be the set of <I>non-critical</I>
vertices in <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">.
We also let <IMG ALIGN="middle" SRC="figs/img67.png" BORDER="0" ALT=""> be the set of all critical
vertices that have at least one non-critical vertex as a neighbour.
Let <IMG ALIGN="middle" SRC="figs/img68.png" BORDER="0" ALT=""> be the set of <I>non-critical</I> edges in <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">.
Finally, we let <IMG ALIGN="middle" SRC="figs/img69.png" BORDER="0" ALT=""> be the <I>non-critical</I> subgraph
of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">.
The non-critical subgraph <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> corresponds to the <I>acyclic part</I>
of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">.
We have <IMG ALIGN="middle" SRC="figs/img71.png" BORDER="0" ALT="">.
</P>
<P>
We then construct a suitable labelling <IMG ALIGN="middle" SRC="figs/img72.png" BORDER="0" ALT=""> of the vertices
of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">: we choose <IMG ALIGN="middle" SRC="figs/img73.png" BORDER="0" ALT=""> for each <IMG ALIGN="middle" SRC="figs/img74.png" BORDER="0" ALT=""> in such
a way that <IMG ALIGN="middle" SRC="figs/img75.png" BORDER="0" ALT=""> (<IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">) is a
minimal perfect hash function for <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
This labelling <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT=""> can be found in linear time
if the number of edges in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> is at most <IMG ALIGN="middle" SRC="figs/img76.png" BORDER="0" ALT=""> (see <A HREF="#papers">[2</A>]
for details).
</P>
<P>
Figure 1 presents a pseudo code for the BMZ algorithm.
The procedure BMZ (<IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">) receives as input the set of
keys <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and produces the labelling <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">.
The method uses a mapping, ordering and searching approach.
We now describe each step.
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD>procedure BMZ (<IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">)</TD>
</TR>
<TR>
<TD>&nbsp;&nbsp;&nbsp;&nbsp;Mapping (<IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">);</TD>
</TR>
<TR>
<TD>&nbsp;&nbsp;&nbsp;&nbsp;Ordering (<IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">);</TD>
</TR>
<TR>
<TD>&nbsp;&nbsp;&nbsp;&nbsp;Searching (<IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">);</TD>
</TR>
<TR>
<TD><B>Figure 1</B>: Main steps of BMZ algorithm for constructing a minimal perfect hash function</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<H3>Mapping Step</H3>
<P>
The procedure Mapping (<IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">) receives as input the set
of keys <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and generates the random graph <IMG ALIGN="middle" SRC="figs/img59.png" BORDER="0" ALT="">, by generating
two auxiliary functions <IMG ALIGN="middle" SRC="figs/img41.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img78.png" BORDER="0" ALT="">.
</P>
<P>
The functions <IMG ALIGN="middle" SRC="figs/img41.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img42.png" BORDER="0" ALT=""> are constructed as follows.
We impose some upper bound <IMG ALIGN="bottom" SRC="figs/img79.png" BORDER="0" ALT=""> on the lengths of the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
To define <IMG ALIGN="middle" SRC="figs/img80.png" BORDER="0" ALT=""> (<IMG ALIGN="middle" SRC="figs/img81.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img62.png" BORDER="0" ALT="">), we generate
an <IMG ALIGN="middle" SRC="figs/img82.png" BORDER="0" ALT=""> table of random integers <IMG ALIGN="middle" SRC="figs/img83.png" BORDER="0" ALT="">.
For a key <IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT=""> of length <IMG ALIGN="middle" SRC="figs/img84.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img85.png" BORDER="0" ALT="">, we let
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><IMG ALIGN="middle" SRC="figs/img86.png" BORDER="0" ALT=""></TD>
</TR>
</TABLE>
<P>
The random graph <IMG ALIGN="middle" SRC="figs/img59.png" BORDER="0" ALT=""> has vertex set <IMG ALIGN="middle" SRC="figs/img56.png" BORDER="0" ALT=""> and
edge set <IMG ALIGN="middle" SRC="figs/img61.png" BORDER="0" ALT="">. We need <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> to be
simple, i.e., <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> should have neither loops nor multiple edges.
A loop occurs when <IMG ALIGN="middle" SRC="figs/img87.png" BORDER="0" ALT=""> for some <IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">.
We solve this in an ad hoc manner: we simply let <IMG ALIGN="middle" SRC="figs/img88.png" BORDER="0" ALT=""> in this case.
If we still find a loop after this, we generate another pair <IMG ALIGN="middle" SRC="figs/img89.png" BORDER="0" ALT="">.
When a multiple edge occurs we abort and generate a new pair <IMG ALIGN="middle" SRC="figs/img89.png" BORDER="0" ALT="">.
Although the function above causes <A HREF="concepts.html">collisions</A> with probability <I>1/t</I>,
in <A HREF="index.html">cmph library</A> we use faster hash
functions (<A HREF="http://www.cs.yorku.ca/~oz/hash.html">DJB2 hash</A>, <A HREF="http://www.isthe.com/chongo/tech/comp/fnv/">FNV hash</A>,
<A HREF="http://burtleburtle.net/bob/hash/doobs.html">Jenkins hash</A> and <A HREF="http://www.cs.yorku.ca/~oz/hash.html">SDBM hash</A>)
in which we do not need to impose any upper bound <IMG ALIGN="bottom" SRC="figs/img79.png" BORDER="0" ALT=""> on the lengths of the keys in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">.
</P>
<P>
As mentioned before, for us to find the labelling <IMG ALIGN="middle" SRC="figs/img72.png" BORDER="0" ALT=""> of the
vertices of <IMG ALIGN="middle" SRC="figs/img59.png" BORDER="0" ALT=""> in linear time,
we require that <IMG ALIGN="middle" SRC="figs/img108.png" BORDER="0" ALT="">.
The crucial step now is to determine the value
of <IMG ALIGN="bottom" SRC="figs/img1.png" BORDER="0" ALT=""> (in <IMG ALIGN="bottom" SRC="figs/img57.png" BORDER="0" ALT="">) to obtain a random
graph <IMG ALIGN="middle" SRC="figs/img71.png" BORDER="0" ALT=""> with <IMG ALIGN="middle" SRC="figs/img109.png" BORDER="0" ALT="">.
Botelho, Menoti an Ziviani determinded emprically in <A HREF="#papers">[1</A>] that
the value of <IMG ALIGN="bottom" SRC="figs/img1.png" BORDER="0" ALT=""> is <I>1.15</I>. This value is remarkably
close to the theoretical value determined in <A HREF="#papers">[2</A>],
which is around <IMG ALIGN="bottom" SRC="figs/img112.png" BORDER="0" ALT="">.
</P>
<HR NOSHADE SIZE=1>
<H3>Ordering Step</H3>
<P>
The procedure Ordering (<IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">) receives
as input the graph <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> and partitions <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> into the two
subgraphs <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">, so that <IMG ALIGN="middle" SRC="figs/img71.png" BORDER="0" ALT="">.
</P>
<P>
Figure 2 presents a sample graph with 9 vertices
and 8 edges, where the degree of a vertex is shown besides each vertex.
Initially, all vertices with degree 1 are added to a queue <IMG ALIGN="middle" SRC="figs/img136.png" BORDER="0" ALT="">.
For the example shown in Figure 2(a), <IMG ALIGN="middle" SRC="figs/img137.png" BORDER="0" ALT=""> after the initialization step.
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><IMG ALIGN="middle" SRC="figs/img138.png" BORDER="0" ALT=""></TD>
</TR>
<TR>
<TD><B>Figure 2:</B> Ordering step for a graph with 9 vertices and 8 edges.</TD>
</TR>
</TABLE>
<P>
Next, we remove one vertex <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> from the queue, decrement its degree and
the degree of the vertices with degree greater than 0 in the adjacent
list of <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">, as depicted in Figure 2(b) for <IMG ALIGN="bottom" SRC="figs/img140.png" BORDER="0" ALT="">.
At this point, the adjacencies of <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> with degree 1 are
inserted into the queue, such as vertex 1.
This process is repeated until the queue becomes empty.
All vertices with degree 0 are non-critical vertices and the others are
critical vertices, as depicted in Figure 2(c).
Finally, to determine the vertices in <IMG ALIGN="middle" SRC="figs/img141.png" BORDER="0" ALT=""> we collect all
vertices <IMG ALIGN="middle" SRC="figs/img142.png" BORDER="0" ALT=""> with at least one vertex <IMG ALIGN="bottom" SRC="figs/img143.png" BORDER="0" ALT=""> that
is in Adj<IMG ALIGN="middle" SRC="figs/img144.png" BORDER="0" ALT=""> and in <IMG ALIGN="middle" SRC="figs/img145.png" BORDER="0" ALT="">, as the vertex 8 in Figure 2(c).
</P>
<HR NOSHADE SIZE=1>
<H3>Searching Step</H3>
<P>
In the searching step, the key part is
the <I>perfect assignment problem</I>: find <IMG ALIGN="middle" SRC="figs/img153.png" BORDER="0" ALT=""> such that
the function <IMG ALIGN="middle" SRC="figs/img154.png" BORDER="0" ALT=""> defined by
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><IMG ALIGN="middle" SRC="figs/img155.png" BORDER="0" ALT=""></TD>
</TR>
</TABLE>
<P>
is a bijection from <IMG ALIGN="middle" SRC="figs/img156.png" BORDER="0" ALT=""> to <IMG ALIGN="middle" SRC="figs/img157.png" BORDER="0" ALT=""> (recall <IMG ALIGN="middle" SRC="figs/img158.png" BORDER="0" ALT="">).
We are interested in a labelling <IMG ALIGN="middle" SRC="figs/img72.png" BORDER="0" ALT=""> of
the vertices of the graph <IMG ALIGN="middle" SRC="figs/img59.png" BORDER="0" ALT=""> with
the property that if <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img22.png" BORDER="0" ALT=""> are keys
in <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT="">, then <IMG ALIGN="middle" SRC="figs/img159.png" BORDER="0" ALT="">; that is, if we associate
to each edge the sum of the labels on its endpoints, then these values
should be all distinct.
Moreover, we require that all the sums <IMG ALIGN="middle" SRC="figs/img160.png" BORDER="0" ALT=""> (<IMG ALIGN="middle" SRC="figs/img18.png" BORDER="0" ALT="">)
fall between <IMG ALIGN="bottom" SRC="figs/img115.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img161.png" BORDER="0" ALT="">, and thus we have a bijection
between <IMG ALIGN="bottom" SRC="figs/img20.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img157.png" BORDER="0" ALT="">.
</P>
<P>
The procedure Searching (<IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">)
receives as input <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> and finds a
suitable <IMG ALIGN="middle" SRC="figs/img162.png" BORDER="0" ALT=""> bit value for each vertex <IMG ALIGN="middle" SRC="figs/img74.png" BORDER="0" ALT="">, stored in the
array <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT="">.
This step is first performed for the vertices in the
critical subgraph <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> (the 2-core of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">)
and then it is performed for the vertices in <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> (the non-critical subgraph
of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT=""> that contains the "acyclic part" of <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">).
The reason the assignment of the <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT=""> values is first
performed on the vertices in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> is to resolve reassignments
as early as possible (such reassignments are consequences of the cycles
in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> and are depicted hereinafter).
</P>
<HR NOSHADE SIZE=1>
<H4>Assignment of Values to Critical Vertices</H4>
<P>
The labels <IMG ALIGN="middle" SRC="figs/img73.png" BORDER="0" ALT=""> (<IMG ALIGN="middle" SRC="figs/img142.png" BORDER="0" ALT="">)
are assigned in increasing order following a greedy
strategy where the critical vertices <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> are considered one at a time,
according to a breadth-first search on <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">.
If a candidate value <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> for <IMG ALIGN="middle" SRC="figs/img73.png" BORDER="0" ALT=""> is forbidden
because setting <IMG ALIGN="middle" SRC="figs/img163.png" BORDER="0" ALT=""> would create two edges with the same sum,
we try <IMG ALIGN="middle" SRC="figs/img164.png" BORDER="0" ALT=""> for <IMG ALIGN="middle" SRC="figs/img73.png" BORDER="0" ALT="">. This fact is referred to
as a <I>reassignment</I>.
</P>
<P>
Let <IMG ALIGN="middle" SRC="figs/img165.png" BORDER="0" ALT=""> be the set of addresses assigned to edges in <IMG ALIGN="middle" SRC="figs/img166.png" BORDER="0" ALT="">.
Initially <IMG ALIGN="middle" SRC="figs/img167.png" BORDER="0" ALT="">.
Let <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> be a candidate value for <IMG ALIGN="middle" SRC="figs/img73.png" BORDER="0" ALT="">.
Initially <IMG ALIGN="bottom" SRC="figs/img168.png" BORDER="0" ALT="">.
Considering the subgraph <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> in Figure 2(c),
a step by step example of the assignment of values to vertices in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT=""> is
presented in Figure 3.
Initially, a vertex <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> is chosen, the assignment <IMG ALIGN="middle" SRC="figs/img163.png" BORDER="0" ALT=""> is made
and <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is set to <IMG ALIGN="middle" SRC="figs/img164.png" BORDER="0" ALT="">.
For example, suppose that vertex <IMG ALIGN="bottom" SRC="figs/img169.png" BORDER="0" ALT=""> in Figure 3(a) is
chosen, the assignment <IMG ALIGN="middle" SRC="figs/img170.png" BORDER="0" ALT=""> is made and <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is set to <IMG ALIGN="bottom" SRC="figs/img96.png" BORDER="0" ALT="">.
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><IMG ALIGN="middle" SRC="figs/img171.png" BORDER="0" ALT=""></TD>
</TR>
<TR>
<TD><B>Figure 3:</B> Example of the assignment of values to critical vertices.</TD>
</TR>
</TABLE>
<P>
In Figure 3(b), following the adjacent list of vertex <IMG ALIGN="bottom" SRC="figs/img169.png" BORDER="0" ALT="">,
the unassigned vertex <IMG ALIGN="bottom" SRC="figs/img115.png" BORDER="0" ALT=""> is reached.
At this point, we collect in the temporary variable <IMG ALIGN="bottom" SRC="figs/img172.png" BORDER="0" ALT=""> all adjacencies
of vertex <IMG ALIGN="bottom" SRC="figs/img115.png" BORDER="0" ALT=""> that have been assigned an <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> value,
and <IMG ALIGN="middle" SRC="figs/img173.png" BORDER="0" ALT="">.
Next, for all <IMG ALIGN="middle" SRC="figs/img174.png" BORDER="0" ALT="">, we check if <IMG ALIGN="middle" SRC="figs/img175.png" BORDER="0" ALT="">.
Since <IMG ALIGN="middle" SRC="figs/img176.png" BORDER="0" ALT="">, then <IMG ALIGN="middle" SRC="figs/img177.png" BORDER="0" ALT=""> is set
to <IMG ALIGN="bottom" SRC="figs/img96.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is incremented
by 1 (now <IMG ALIGN="bottom" SRC="figs/img178.png" BORDER="0" ALT="">) and <IMG ALIGN="middle" SRC="figs/img179.png" BORDER="0" ALT="">.
Next, vertex <IMG ALIGN="bottom" SRC="figs/img180.png" BORDER="0" ALT=""> is reached, <IMG ALIGN="middle" SRC="figs/img181.png" BORDER="0" ALT=""> is set
to <IMG ALIGN="bottom" SRC="figs/img62.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is set to <IMG ALIGN="bottom" SRC="figs/img180.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img182.png" BORDER="0" ALT="">.
Next, vertex <IMG ALIGN="bottom" SRC="figs/img183.png" BORDER="0" ALT=""> is reached and <IMG ALIGN="middle" SRC="figs/img184.png" BORDER="0" ALT="">.
Since <IMG ALIGN="middle" SRC="figs/img185.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img186.png" BORDER="0" ALT="">, then <IMG ALIGN="middle" SRC="figs/img187.png" BORDER="0" ALT=""> is
set to <IMG ALIGN="bottom" SRC="figs/img180.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is set to <IMG ALIGN="bottom" SRC="figs/img183.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img188.png" BORDER="0" ALT="">.
Finally, vertex <IMG ALIGN="bottom" SRC="figs/img189.png" BORDER="0" ALT=""> is reached and <IMG ALIGN="middle" SRC="figs/img190.png" BORDER="0" ALT="">.
Since <IMG ALIGN="middle" SRC="figs/img191.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is incremented by 1 and set to 5, as depicted in
Figure 3(c).
Since <IMG ALIGN="middle" SRC="figs/img192.png" BORDER="0" ALT="">, <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> is again incremented by 1 and set to 6,
as depicted in Figure 3(d).
These two reassignments are indicated by the arrows in Figure 3.
Since <IMG ALIGN="middle" SRC="figs/img193.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img194.png" BORDER="0" ALT="">, then <IMG ALIGN="middle" SRC="figs/img195.png" BORDER="0" ALT=""> is set
to <IMG ALIGN="bottom" SRC="figs/img196.png" BORDER="0" ALT=""> and <IMG ALIGN="middle" SRC="figs/img197.png" BORDER="0" ALT="">. This finishes the algorithm.
</P>
<HR NOSHADE SIZE=1>
<H4>Assignment of Values to Non-Critical Vertices</H4>
<P>
As <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> is acyclic, we can impose the order in which addresses are
associated with edges in <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT="">, making this step simple to solve
by a standard depth first search algorithm.
Therefore, in the assignment of values to vertices in <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> we
benefit from the unused addresses in the gaps left by the assignment of values
to vertices in <IMG ALIGN="middle" SRC="figs/img63.png" BORDER="0" ALT="">.
For that, we start the depth-first search from the vertices in <IMG ALIGN="middle" SRC="figs/img141.png" BORDER="0" ALT=""> because
the <IMG ALIGN="middle" SRC="figs/img37.png" BORDER="0" ALT=""> values for these critical vertices were already assigned
and cannot be changed.
</P>
<P>
Considering the subgraph <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> in Figure 2(c),
a step by step example of the assignment of values to vertices in <IMG ALIGN="middle" SRC="figs/img70.png" BORDER="0" ALT=""> is
presented in Figure 4.
Figure 4(a) presents the initial state of the algorithm.
The critical vertex 8 is the only one that has non-critical vertices as
adjacent.
In the example presented in Figure 3, the addresses <IMG ALIGN="middle" SRC="figs/img198.png" BORDER="0" ALT=""> were not used.
So, taking the first unused address <IMG ALIGN="bottom" SRC="figs/img115.png" BORDER="0" ALT=""> and the vertex <IMG ALIGN="bottom" SRC="figs/img96.png" BORDER="0" ALT="">,
which is reached from the vertex <IMG ALIGN="bottom" SRC="figs/img169.png" BORDER="0" ALT="">, <IMG ALIGN="middle" SRC="figs/img199.png" BORDER="0" ALT=""> is set
to <IMG ALIGN="middle" SRC="figs/img200.png" BORDER="0" ALT="">, as shown in Figure 4(b).
The only vertex that is reached from vertex <IMG ALIGN="bottom" SRC="figs/img96.png" BORDER="0" ALT=""> is vertex <IMG ALIGN="bottom" SRC="figs/img62.png" BORDER="0" ALT="">, so
taking the unused address <IMG ALIGN="bottom" SRC="figs/img183.png" BORDER="0" ALT=""> we set <IMG ALIGN="middle" SRC="figs/img201.png" BORDER="0" ALT=""> to <IMG ALIGN="middle" SRC="figs/img202.png" BORDER="0" ALT="">,
as shown in Figure 4(c).
This process is repeated until the UnAssignedAddresses list becomes empty.
</P>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><IMG ALIGN="middle" SRC="figs/img203.png" BORDER="0" ALT=""></TD>
</TR>
<TR>
<TD><B>Figure 4:</B> Example of the assignment of values to non-critical vertices.</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<A NAME="heuristic"></A>
<H2>The Heuristic</H2>
<P>
We now present an heuristic for BMZ algorithm that
reduces the value of <IMG ALIGN="bottom" SRC="figs/img1.png" BORDER="0" ALT=""> to any given value between <I>1.15</I> and <I>0.93</I>.
This reduces the space requirement to store the resulting function
to any given value between <IMG ALIGN="bottom" SRC="figs/img12.png" BORDER="0" ALT=""> words and <IMG ALIGN="bottom" SRC="figs/img13.png" BORDER="0" ALT=""> words.
The heuristic reuses, when possible, the set
of <IMG ALIGN="bottom" SRC="figs/img11.png" BORDER="0" ALT=""> values that caused reassignments, just before
trying <IMG ALIGN="middle" SRC="figs/img164.png" BORDER="0" ALT="">.
Decreasing the value of <IMG ALIGN="bottom" SRC="figs/img1.png" BORDER="0" ALT=""> leads to an increase in the number of
iterations to generate <IMG ALIGN="bottom" SRC="figs/img32.png" BORDER="0" ALT="">.
For example, for <IMG ALIGN="bottom" SRC="figs/img244.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img6.png" BORDER="0" ALT="">, the analytical expected number
of iterations are <IMG ALIGN="bottom" SRC="figs/img245.png" BORDER="0" ALT=""> and <IMG ALIGN="bottom" SRC="figs/img246.png" BORDER="0" ALT="">, respectively (see <A HREF="#papers">[2</A>]
for details),
while for <IMG ALIGN="bottom" SRC="figs/img128.png" BORDER="0" ALT=""> the same value is around <I>2.13</I>.
</P>
<HR NOSHADE SIZE=1>
<H2>Memory Consumption</H2>
<P>
Now we detail the memory consumption to generate and to store minimal perfect hash functions
using the BMZ algorithm. The structures responsible for memory consumption are in the
following:
</P>
<UL>
<LI>Graph:
<OL>
<LI><B>first</B>: is a vector that stores <I>cn</I> integer numbers, each one representing
the first edge (index in the vector edges) in the list of
edges of each vertex.
The integer numbers are 4 bytes long. Therefore,
the vector first is stored in <I>4cn</I> bytes.
<P></P>
<LI><B>edges</B>: is a vector to represent the edges of the graph. As each edge
is compounded by a pair of vertices, each entry stores two integer numbers
of 4 bytes that represent the vertices. As there are <I>n</I> edges, the
vector edges is stored in <I>8n</I> bytes.
<P></P>
<LI><B>next</B>: given a vertex <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">, we can discover the edges that
contain <IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT=""> following its list of edges,
which starts on first[<IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">] and the next
edges are given by next[...first[<IMG ALIGN="bottom" SRC="figs/img139.png" BORDER="0" ALT="">]...]. Therefore, the vectors first and next represent
the linked lists of edges of each vertex. As there are two vertices for each edge,
when an edge is iserted in the graph, it must be inserted in the two linked lists
of the vertices in its composition. Therefore, there are <I>2n</I> entries of integer
numbers in the vector next, so it is stored in <I>4*2n = 8n</I> bytes.
<P></P>
<LI><B>critical vertices(critical_nodes vector)</B>: is a vector of <I>cn</I> bits,
where each bit indicates if a vertex is critical (1) or non-critical (0).
Therefore, the critical and non-critical vertices are represented in <I>cn/8</I> bytes.
<P></P>
<LI><B>critical edges (used_edges vector)</B>: is a vector of <I>n</I> bits, where each
bit indicates if an edge is critical (1) or non-critical (0). Therefore, the
critical and non-critical edges are represented in <I>n/8</I> bytes.
<P></P>
</OL>
<LI>Other auxiliary structures
<OL>
<LI><B>queue</B>: is a queue of integer numbers used in the breadth-first search of the
assignment of values to critical vertices. There is an entry in the queue for
each two critical vertices. Let <IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT=""> be the expected number of critical
vertices. Therefore, the queue is stored in <I>4*0.5*<IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT="">=2<IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT=""></I>.
<P></P>
<LI><B>visited</B>: is a vector of <I>cn</I> bits, where each bit indicates if the g value of
a given vertex was already defined. Therefore, the vector visited is stored
in <I>cn/8</I> bytes.
<P></P>
<LI><B>function <I>g</I></B>: is represented by a vector of <I>cn</I> integer numbers.
As each integer number is 4 bytes long, the function <I>g</I> is stored in
<I>4cn</I> bytes.
</OL>
</UL>
<P>
Thus, the total memory consumption of BMZ algorithm for generating a minimal
perfect hash function (MPHF) is: <I>(8.25c + 16.125)n +2<IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT=""> + O(1)</I> bytes.
As the value of constant <I>c</I> may be 1.15 and 0.93 we have:
</P>
<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
<TR>
<TH><I>c</I></TH>
<TH><IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT=""></TH>
<TH>Memory consumption to generate a MPHF</TH>
</TR>
<TR>
<TD>0.93</TD>
<TD ALIGN="center"><I>0.497n</I></TD>
<TD ALIGN="center"><I>24.80n + O(1)</I></TD>
</TR>
<TR>
<TD>1.15</TD>
<TD ALIGN="center"><I>0.401n</I></TD>
<TD ALIGN="center"><I>26.42n + O(1)</I></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 1:</B> Memory consumption to generate a MPHF using the BMZ algorithm.</TD>
</TR>
</TABLE>
<P>
The values of <IMG ALIGN="middle" SRC="figs/img110.png" BORDER="0" ALT=""> were calculated using Eq.(1) presented in <A HREF="#papers">[2</A>].
</P>
<P>
Now we present the memory consumption to store the resulting function.
We only need to store the <I>g</I> function. Thus, we need <I>4cn</I> bytes.
Again we have:
</P>
<TABLE ALIGN="center" BORDER="1" CELLPADDING="4">
<TR>
<TH><I>c</I></TH>
<TH>Memory consumption to store a MPHF</TH>
</TR>
<TR>
<TD>0.93</TD>
<TD ALIGN="center"><I>3.72n</I></TD>
</TR>
<TR>
<TD>1.15</TD>
<TD ALIGN="center"><I>4.60n</I></TD>
</TR>
</TABLE>
<TABLE ALIGN="center" CELLPADDING="4">
<TR>
<TD><B>Table 2:</B> Memory consumption to store a MPHF generated by the BMZ algorithm.</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<H2>Experimental Results</H2>
<P>
<A HREF="comparison.html">CHM x BMZ</A>
</P>
<HR NOSHADE SIZE=1>
<A NAME="papers"></A>
<H2>Papers</H2>
<OL>
<LI><A HREF="http://www.dcc.ufmg.br/~fbotelho">F. C. Botelho</A>, D. Menoti, <A HREF="http://www.dcc.ufmg.br/~nivio">N. Ziviani</A>. <A HREF="papers/bmz_tr004_04.ps">A New algorithm for constructing minimal perfect hash functions</A>, Technical Report TR004/04, Department of Computer Science, Federal University of Minas Gerais, 2004.
<P></P>
<LI><A HREF="http://www.dcc.ufmg.br/~fbotelho">F. C. Botelho</A>, Y. Kohayakawa, and <A HREF="http://www.dcc.ufmg.br/~nivio">N. Ziviani</A>. <A HREF="papers/wea05.pdf">A Practical Minimal Perfect Hashing Method</A>. <I>4th International Workshop on efficient and Experimental Algorithms (WEA05),</I> Springer-Verlag Lecture Notes in Computer Science, vol. 3505, Santorini Island, Greece, May 2005, 488-500.
</OL>
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<P>
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>
</P>
<P>
<A HREF="mailto:fc_botelho@users.sourceforge.net">Fabiano Cupertino Botelho</A>
</P>
<P>
<A HREF="mailto:nivio@dcc.ufmg.br">Nivio Ziviani</A>
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