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@ -9,15 +9,17 @@ BMZ Algorithm
At the end of 2003, professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] was At the end of 2003, professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] was
finishing the second edition of his [book http://www.dcc.ufmg.br/algoritmos/]. finishing the second edition of his [book http://www.dcc.ufmg.br/algoritmos/].
During the [book http://www.dcc.ufmg.br/algoritmos/] writing, During the [book http://www.dcc.ufmg.br/algoritmos/] writing,
professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] studied the problem of generating minimal perfect hash professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] studied the problem of generating
functions (if you are not familiarized with this problem, see [1][2]). [minimal perfect hash functions concepts.html]
(if you are not familiarized with this problem, see [[1 #papers]][[2 #papers]]).
Professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] coded a modified version of Professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] coded a modified version of
the [CHM algorithm chm.html], which was proposed by the [CHM algorithm chm.html], which was proposed by
Czech, Havas and Majewski and put it in his [book http://www.dcc.ufmg.br/algoritmos/]. Czech, Havas and Majewski, and put it in his [book http://www.dcc.ufmg.br/algoritmos/].
The [CHM algorithm chm.html] is based on acyclic random graphs to generate order preserving The [CHM algorithm chm.html] is based on acyclic random graphs to generate
minimal perfect hash functions in linear time. Professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] [order preserving minimal perfect hash functions concepts.html] in linear time.
Professor [Nivio Ziviani http://www.dcc.ufmg.br/~nivio]
argued himself, why must the random graph argued himself, why must the random graph
be acyclic? In the modified version availalbe in his [book http://www.dcc.ufmg.br/algoritmos/] he got rid of such restriction. be acyclic? In the modified version availalbe in his [book http://www.dcc.ufmg.br/algoritmos/] he got rid of this restriction.
The modification presented a problem, it was impossible to generate minimal perfect hash functions The modification presented a problem, it was impossible to generate minimal perfect hash functions
for sets with more than 1000 keys. for sets with more than 1000 keys.
@ -32,19 +34,38 @@ During the master, [Fabiano http://www.dcc.ufmg.br/~fbotelho] and
In april of 2004, [Fabiano http://www.dcc.ufmg.br/~fbotelho] was talking with a In april of 2004, [Fabiano http://www.dcc.ufmg.br/~fbotelho] was talking with a
friend of him (David Menoti) about the problems friend of him (David Menoti) about the problems
and many ideas appeared. and many ideas appeared.
The ideas were implemented and we noticed that a very fast algorithm to generate The ideas were implemented and a very fast algorithm to generate
minimal perfect hash functions had been designed. minimal perfect hash functions had been designed.
We refer the algorithm to as **BMZ**, because it was conceived by Fabiano C. **B**otelho We refer the algorithm to as **BMZ**, because it was conceived by Fabiano C. **B**otelho,
David **M**enoti and Nivio **Z**iviani. The algorithm is described in [1]. David **M**enoti and Nivio **Z**iviani. The algorithm is described in [[1 #papers]].
To analyse BMZ algorithm we needed some results from the random graph theory, so To analyse BMZ algorithm we needed some results from the random graph theory, so
we invite professor [Yoshiharu Kohayakawa http://www.ime.usp.br/~yoshi] to help us. we invite professor [Yoshiharu Kohayakawa http://www.ime.usp.br/~yoshi] to help us.
The final description and analysis of BMZ algorithm is presented in [2]. The final description and analysis of BMZ algorithm is presented in [[2 #papers]].
---------------------------------------- ----------------------------------------
==The Algorithm== ==The Algorithm==
Let us show how the minimal perfect hash function [figs/img7.png] will be constructed. The BMZ algorithm shares several features with the [CHM algorithm chm.html].
In particular, BMZ algorithm is also
based on the generation of random graphs [figs/img27.png], where [figs/img28.png] is in
one-to-one correspondence with the key set [figs/img20.png] for which we wish to
generate a [minimal perfect hash function concepts.html].
The two main differences between BMZ algorithm and CHM algorithm
are as follows: (//i//) BMZ algorithm generates random
graphs [figs/img27.png] with [figs/img29.png] and [figs/img30.png], where [figs/img31.png],
and hence [figs/img32.png] necessarily contains cycles,
while CHM algorithm generates //acyclic// random
graphs [figs/img27.png] with [figs/img29.png] and [figs/img30.png],
with a greater number of vertices: [figs/img33.png];
(//ii//) CHM algorithm generates [order preserving minimal perfect hash functions concepts.html]
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.
Suppose [figs/img14.png] is a universe of //keys//.
Let [figs/img17.png] be a set of [figs/img8.png] keys from [figs/img14.png].
Let us show how the BMZ algorithm constructs a minimal perfect hash function [figs/img7.png].
We make use of two auxiliary random functions [figs/img41.png] and [figs/img55.png], We make use of two auxiliary random functions [figs/img41.png] and [figs/img55.png],
where [figs/img56.png] for some suitably chosen integer [figs/img57.png], where [figs/img56.png] for some suitably chosen integer [figs/img57.png],
where [figs/img58.png].We build a random graph [figs/img59.png] on [figs/img60.png], where [figs/img58.png].We build a random graph [figs/img59.png] on [figs/img60.png],
@ -54,7 +75,7 @@ key in the set of keys [figs/img20.png].
In what follows, we shall be interested in the //2-core// of In what follows, we shall be interested in the //2-core// of
the random graph [figs/img32.png], that is, the maximal subgraph the random graph [figs/img32.png], that is, the maximal subgraph
of [figs/img32.png] with minimal degree at of [figs/img32.png] with minimal degree at
least 2 (see, e.g., [2] for details). least 2 (see [[2 #papers]] for details).
Because of its importance in our context, we call the 2-core the Because of its importance in our context, we call the 2-core the
//critical// subgraph of [figs/img32.png] and denote it by [figs/img63.png]. //critical// subgraph of [figs/img32.png] and denote it by [figs/img63.png].
The vertices and edges in [figs/img63.png] are said to be //critical//. The vertices and edges in [figs/img63.png] are said to be //critical//.
@ -65,7 +86,7 @@ We also let [figs/img67.png] be the set of all critical
vertices that have at least one non-critical vertex as a neighbour. vertices that have at least one non-critical vertex as a neighbour.
Let [figs/img68.png] be the set of //non-critical// edges in [figs/img32.png]. Let [figs/img68.png] be the set of //non-critical// edges in [figs/img32.png].
Finally, we let [figs/img69.png] be the //non-critical// subgraph Finally, we let [figs/img69.png] be the //non-critical// subgraph
of [figs/img32.png. of [figs/img32.png].
The non-critical subgraph [figs/img70.png] corresponds to the //acyclic part// The non-critical subgraph [figs/img70.png] corresponds to the //acyclic part//
of [figs/img32.png]. of [figs/img32.png].
We have [figs/img71.png]. We have [figs/img71.png].
@ -74,33 +95,222 @@ We then construct a suitable labelling [figs/img72.png] of the vertices
of [figs/img32.png]: we choose [figs/img73.png] for each [figs/img74.png] in such of [figs/img32.png]: we choose [figs/img73.png] for each [figs/img74.png] in such
a way that [figs/img75.png] ([figs/img18.png]) is a a way that [figs/img75.png] ([figs/img18.png]) is a
minimal perfect hash function for [figs/img20.png]. minimal perfect hash function for [figs/img20.png].
We will see later on that this labelling [figs/img37.png] can be found in linear time This labelling [figs/img37.png] can be found in linear time
if the number of edges in [figs/img63.png] is at most [figs/img76.png]. if the number of edges in [figs/img63.png] is at most [figs/img76.png] (see [[2 #papers]]
for details).
Figure 2 presents a pseudo code for the algorithm. Figure 1 presents a pseudo code for the BMZ algorithm.
The procedure GenerateMPHF ([figs/img20.png], [figs/img37.png]) receives as input the set of The procedure BMZ ([figs/img20.png], [figs/img37.png]) receives as input the set of
keys [figs/img20.png] and produces the labelling [figs/img37.png]. keys [figs/img20.png] and produces the labelling [figs/img37.png].
The method uses a mapping, ordering and searching approach. The method uses a mapping, ordering and searching approach.
We now describe each step. We now describe each step.
| procedure GenerateMPHF ([figs/img20.png], [figs/img37.png]) | procedure BMZ ([figs/img20.png], [figs/img37.png])
|     Mapping ([figs/img20.png], [figs/img32.png]); |     Mapping ([figs/img20.png], [figs/img32.png]);
|     Ordering ([figs/img32.png], [figs/img63.png], [figs/img70.png]); |     Ordering ([figs/img32.png], [figs/img63.png], [figs/img70.png]);
|     Searching ([figs/img32.png], [figs/img63.png], [figs/img70.png], [figs/img37.png]); |     Searching ([figs/img32.png], [figs/img63.png], [figs/img70.png], [figs/img37.png]);
**Figure 2**: Main steps of the algorithm for constructing a minimal perfect hash function | **Figure 1**: Main steps of BMZ algorithm for constructing a minimal perfect hash function
===Mapping Step===
===Ordering Step===
===Searching Step===
====Assignment of Values to Critical Vertices====
====Assignment of Values to Non-Critical Vertices====
---------------------------------------- ----------------------------------------
==The Heuristic== ===Mapping Step===
The procedure Mapping ([figs/img20.png], [figs/img32.png]) receives as input the set
of keys [figs/img20.png] and generates the random graph [figs/img59.png], by generating
two auxiliary functions [figs/img41.png], [figs/img78.png].
The functions [figs/img41.png] and [figs/img42.png] are constructed as follows.
We impose some upper bound [figs/img79.png] on the lengths of the keys in [figs/img20.png].
To define [figs/img80.png] ([figs/img81.png], [figs/img62.png]), we generate
an [figs/img82.png] table of random integers [figs/img83.png].
For a key [figs/img18.png] of length [figs/img84.png] and [figs/img85.png], we let
| [figs/img86.png]
The random graph [figs/img59.png] has vertex set [figs/img56.png] and
edge set [figs/img61.png]. We need [figs/img32.png] to be
simple, i.e., [figs/img32.png] should have neither loops nor multiple edges.
A loop occurs when [figs/img87.png] for some [figs/img18.png].
We solve this in an ad hoc manner: we simply let [figs/img88.png] in this case.
If we still find a loop after this, we generate another pair [figs/img89.png].
When a multiple edge occurs we abort and generate a new pair [figs/img89.png].
Although the function above causes [collisions concepts.html] with probability //1/t//,
in [cmph library index.html] we use faster hash
functions ([DJB2 hash http://], [FNV hash http://], [Jenkins hash http://]
and [SDBM hash http://]) in which we do not need to impose any upper bound [figs/img79.png] on the lengths of the keys in [figs/img20.png].
As mentioned before, for us to find the labelling [figs/img72.png] of the
vertices of [figs/img59.png] in linear time,
we require that [figs/img108.png].
The crucial step now is to determine the value
of [figs/img1.png] (in [figs/img57.png]) to obtain a random
graph [figs/img71.png] with [figs/img109.png].
Botelho, Menoti an Ziviani determinded emprically in [[1 #papers]] that
the value of [figs/img1.png] is //1.15//. This value is remarkably
close to the theoretical value determined in [[2 #papers]],
which is around [figs/img112.png].
----------------------------------------
===Ordering Step===
The procedure Ordering ([figs/img32.png], [figs/img63.png], [figs/img70.png]) receives
as input the graph [figs/img32.png] and partitions [figs/img32.png] into the two
subgraphs [figs/img63.png] and [figs/img70.png], so that [figs/img71.png].
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 [figs/img136.png].
For the example shown in Figure 2(a), [figs/img137.png] after the initialization step.
| [figs/img138.png]
| **Figure 2:** Ordering step for a graph with 9 vertices and 8 edges.
Next, we remove one vertex [figs/img139.png] from the queue, decrement its degree and
the degree of the vertices with degree greater than 0 in the adjacent
list of [figs/img139.png], as depicted in Figure 2(b) for [figs/img140.png].
At this point, the adjacencies of [figs/img139.png] 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 [figs/img141.png] we collect all
vertices [figs/img142.png] with at least one vertex [figs/img143.png] that
is in Adj[figs/img144.png] and in [figs/img145.png], as the vertex 8 in Figure 2(c).
----------------------------------------
===Searching Step===
In the searching step, the key part is
the //perfect assignment problem//: find [figs/img153.png] such that
the function [figs/img154.png] defined by
| [figs/img155.png]
is a bijection from [figs/img156.png] to [figs/img157.png] (recall [figs/img158.png]).
We are interested in a labelling [figs/img72.png] of
the vertices of the graph [figs/img59.png] with
the property that if [figs/img11.png] and [figs/img22.png] are keys
in [figs/img20.png], then [figs/img159.png]; 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 [figs/img160.png] ([figs/img18.png])
fall between [figs/img115.png] and [figs/img161.png], and thus we have a bijection
between [figs/img20.png] and [figs/img157.png].
The procedure Searching ([figs/img32.png], [figs/img63.png], [figs/img70.png], [figs/img37.png])
receives as input [figs/img32.png], [figs/img63.png], [figs/img70.png] and finds a
suitable [figs/img162.png] bit value for each vertex [figs/img74.png], stored in the
array [figs/img37.png].
This step is first performed for the vertices in the
critical subgraph [figs/img63.png] of [figs/img32.png] (the 2-core of [figs/img32.png])
and then it is performed for the vertices in [figs/img70.png] (the non-critical subgraph
of [figs/img32.png] that contains the "acyclic part" of [figs/img32.png]).
The reason the assignment of the [figs/img37.png] values is first
performed on the vertices in [figs/img63.png] is to resolve reassignments
as early as possible (such reassignments are consequences of the cycles
in [figs/img63.png] and are depicted hereinafter).
----------------------------------------
====Assignment of Values to Critical Vertices====
The labels [figs/img73.png] ([figs/img142.png])
are assigned in increasing order following a greedy
strategy where the critical vertices [figs/img139.png] are considered one at a time,
according to a breadth-first search on [figs/img63.png].
If a candidate value [figs/img11.png] for [figs/img73.png] is forbidden
because setting [figs/img163.png] would create two edges with the same sum,
we try [figs/img164.png] for [figs/img73.png]. This fact is referred to
as a //reassignment//.
Let [figs/img165.png] be the set of addresses assigned to edges in [figs/img166.png].
Initially [figs/img167.png].
Let [figs/img11.png] be a candidate value for [figs/img73.png].
Initially [figs/img168.png].
Considering the subgraph [figs/img63.png] in Figure 2(c),
a step by step example of the assignment of values to vertices in [figs/img63.png] is
presented in Figure 3.
Initially, a vertex [figs/img139.png] is chosen, the assignment [figs/img163.png] is made
and [figs/img11.png] is set to [figs/img164.png].
For example, suppose that vertex [figs/img169.png] in Figure 3(a) is
chosen, the assignment [figs/img170.png] is made and [figs/img11.png] is set to [figs/img96.png].
| [figs/img171.png]
| **Figure 3:** Example of the assignment of values to critical vertices.
In Figure 3(b), following the adjacent list of vertex [figs/img169.png],
the unassigned vertex [figs/img115.png] is reached.
At this point, we collect in the temporary variable [figs/img172.png] all adjacencies
of vertex [figs/img115.png] that have been assigned an [figs/img11.png] value,
and [figs/img173.png].
Next, for all [figs/img174.png], we check if [figs/img175.png].
Since [figs/img176.png], then [figs/img177.png] is set
to [figs/img96.png], [figs/img11.png] is incremented
by 1 (now [figs/img178.png]) and [figs/img179.png].
Next, vertex [figs/img180.png] is reached, [figs/img181.png] is set
to [figs/img62.png], [figs/img11.png] is set to [figs/img180.png] and [figs/img182.png].
Next, vertex [figs/img183.png] is reached and [figs/img184.png].
Since [figs/img185.png] and [figs/img186.png], then [figs/img187.png] is
set to [figs/img180.png], [figs/img11.png] is set to [figs/img183.png] and [figs/img188.png].
Finally, vertex [figs/img189.png] is reached and [figs/img190.png].
Since [figs/img191.png], [figs/img11.png] is incremented by 1 and set to 5, as depicted in
Figure 3(c).
Since [figs/img192.png], [figs/img11.png] 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 [figs/img193.png] and [figs/img194.png], then [figs/img195.png] is set
to [figs/img196.png] and [figs/img197.png]. This finishes the algorithm.
----------------------------------------
====Assignment of Values to Non-Critical Vertices====
As [figs/img70.png] is acyclic, we can impose the order in which addresses are
associated with edges in [figs/img70.png], making this step simple to solve
by a standard depth first search algorithm.
Therefore, in the assignment of values to vertices in [figs/img70.png] we
benefit from the unused addresses in the gaps left by the assignment of values
to vertices in [figs/img63.png].
For that, we start the depth-first search from the vertices in [figs/img141.png] because
the [figs/img37.png] values for these critical vertices were already assigned
and cannot be changed.
Considering the subgraph [figs/img70.png] in Figure 2(c),
a step by step example of the assignment of values to vertices in [figs/img70.png] 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 [figs/img198.png] were not used.
So, taking the first unused address [figs/img115.png] and the vertex [figs/img96.png],
which is reached from the vertex [figs/img169.png], [figs/img199.png] is set
to [figs/img200.png], as shown in Figure 4(b).
The only vertex that is reached from vertex [figs/img96.png] is vertex [figs/img62.png], so
taking the unused address [figs/img183.png] we set [figs/img201.png] to [figs/img202.png],
as shown in Figure 4(c).
This process is repeated until the UnAssignedAddresses list becomes empty.
| [figs/img203.png]
| **Figure 4:** Example of the assignment of values to non-critical vertices.
----------------------------------------
==The Heuristic==[heuristic]
We now present an heuristic for BMZ algorithm that
reduces the value of [figs/img1.png] to any given value between //1.15// and //0.93//.
This reduces the space requirement to store the resulting function
to any given value between [figs/img12.png] words and [figs/img13.png] words.
The heuristic reuses, when possible, the set
of [figs/img11.png] values that caused reassignments, just before
trying [figs/img164.png].
Decreasing the value of [figs/img1.png] leads to an increase in the number of
iterations to generate [figs/img32.png].
For example, for [figs/img244.png] and [figs/img6.png], the analytical expected number
of iterations are [figs/img245.png] and [figs/img246.png], respectively (see [[2 #papers]]
for details),
while for [figs/img128.png] the same value is around //2.13//.
---------------------------------------- ----------------------------------------
@ -121,9 +331,10 @@ following:
of 4 bytes that represent the vertices. As there are //n// edges, the of 4 bytes that represent the vertices. As there are //n// edges, the
vector edges is stored in //8n// bytes. vector edges is stored in //8n// bytes.
+ **next**: given a vertex //v//, we can discover the edges that contain //v// + **next**: given a vertex [figs/img139.png], we can discover the edges that
following its list of edges, which starts on first[//v//] and the next contain [figs/img139.png] following its list of edges,
edges are given by next[...first[//v//]...]. Therefore, the vectors first and next represent which starts on first[[figs/img139.png]] and the next
edges are given by next[...first[[figs/img139.png]]...]. Therefore, the vectors first and next represent
the linked lists of edges of each vertex. As there are two vertices for each edge, 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 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 //2n// entries of integer of the vertices in its composition. Therefore, there are //2n// entries of integer
@ -140,8 +351,8 @@ following:
- Other auxiliary structures - Other auxiliary structures
+ **queue**: is a queue of integer numbers used in the breadth-first search of the + **queue**: 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 assignment of values to critical vertices. There is an entry in the queue for
each two critical vertices. Let //|Vcrit|// be the expected number of critical each two critical vertices. Let [figs/img110.png] be the expected number of critical
vertices. Therefore, the queue is stored in //4*0.5*|Vcrit|=2|Vcrit|//. vertices. Therefore, the queue is stored in //4*0.5*[figs/img110.png]=2[figs/img110.png]//.
+ **visited**: is a vector of //cn// bits, where each bit indicates if the g value of + **visited**: is a vector of //cn// bits, where each bit indicates if the g value of
a given vertex was already defined. Therefore, the vector visited is stored a given vertex was already defined. Therefore, the vector visited is stored
@ -153,12 +364,15 @@ following:
Thus, the total memory consumption of BMZ algorithm for generating a minimal Thus, the total memory consumption of BMZ algorithm for generating a minimal
perfect hash function (MPHF) is: //(8.25c + 16.125)n +2|Vcrit| + O(1)// bytes. perfect hash function (MPHF) is: //(8.25c + 16.125)n +2[figs/img110.png] + O(1)// bytes.
As the value of constant //c// may be 1.15 and 0.93 we have: As the value of constant //c// may be 1.15 and 0.93 we have:
|| //c// | //|Vcrit|// | Memory consumption to generate a MPHF | || //c// | [figs/img110.png] | Memory consumption to generate a MPHF |
| 0.93 | //0.497n// | //24.80n + O(1)// | | 0.93 | //0.497n// | //24.80n + O(1)// |
| 1.15 | //0.401n// | //26.42n + O(1)// | | 1.15 | //0.401n// | //26.42n + O(1)// |
The values of |Vcrit| were calculated using Eq.(1) presented in [2].
| **Table 1:** Memory consumption to generate a MPHF using the BMZ algorithm.
The values of [figs/img110.png] were calculated using Eq.(1) presented in [[2 #papers]].
Now we present the memory consumption to store the resulting function. Now we present the memory consumption to store the resulting function.
We only need to store the //g// function. Thus, we need //4cn// bytes. We only need to store the //g// function. Thus, we need //4cn// bytes.
@ -167,9 +381,16 @@ Again we have:
| 0.93 | //3.72n// | | 0.93 | //3.72n// |
| 1.15 | //4.60n// | | 1.15 | //4.60n// |
| **Table 2:** Memory consumption to store a MPHF generated by the BMZ algorithm.
---------------------------------------- ----------------------------------------
==Papers== ==Experimental Results==
[CHM x BMZ comparison.html]
----------------------------------------
==Papers==[papers]
+ [F. C. Botelho http://www.dcc.ufmg.br/~fbotelho], D. Menoti, [N. Ziviani http://www.dcc.ufmg.br/~nivio]. [A New algorithm for constructing minimal perfect hash functions papers/bmz_tr004_04.ps], Technical Report TR004/04, Department of Computer Science, Federal University of Minas Gerais, 2004. + [F. C. Botelho http://www.dcc.ufmg.br/~fbotelho], D. Menoti, [N. Ziviani http://www.dcc.ufmg.br/~nivio]. [A New algorithm for constructing minimal perfect hash functions papers/bmz_tr004_04.ps], Technical Report TR004/04, Department of Computer Science, Federal University of Minas Gerais, 2004.
@ -177,7 +398,7 @@ Again we have:
---------------------------------------- ----------------------------------------
[Home index.html] | [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
---------------------------------------- ----------------------------------------
%!include: FOOTER.t2t %!include: FOOTER.t2t

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24
CHM.t2t
View File

@ -4,8 +4,11 @@ CHM Algorithm
%!includeconf: CONFIG.t2t %!includeconf: CONFIG.t2t
---------------------------------------- ----------------------------------------
==The Algorithm== ==The Algorithm==
----------------------------------------
==Memory Consumption== ==Memory Consumption==
Now we detail the memory consumption to generate and to store minimal perfect hash functions Now we detail the memory consumption to generate and to store minimal perfect hash functions
@ -23,9 +26,11 @@ following:
of 4 bytes that represent the vertices. As there are //n// edges, the of 4 bytes that represent the vertices. As there are //n// edges, the
vector edges is stored in //8n// bytes. vector edges is stored in //8n// bytes.
+ **next**: given a vertex //v//, we can discover the edges that contain //v// + **next**: given a vertex [figs/img139.png], we can discover the edges that
following its list of edges, which starts on first[//v//] and the next contain [figs/img139.png] following its list of edges, which starts on
edges are given by next[...first[//v//]...]. Therefore, the vectors first and next represent first[[figs/img139.png]] and the next
edges are given by next[...first[[figs/img139.png]]...]. Therefore,
the vectors first and next represent
the linked lists of edges of each vertex. As there are two vertices for each edge, 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 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 //2n// entries of integer of the vertices in its composition. Therefore, there are //2n// entries of integer
@ -47,12 +52,23 @@ As the value of constant //c// must be at least 2.09 we have:
|| //c// | Memory consumption to generate a MPHF | || //c// | Memory consumption to generate a MPHF |
| 2.09 | //33.00n + O(1)// | | 2.09 | //33.00n + O(1)// |
| **Table 1:** Memory consumption to generate a MPHF using the CHM algorithm.
Now we present the memory consumption to store the resulting function. Now we present the memory consumption to store the resulting function.
We only need to store the //g// function. Thus, we need //4cn// bytes. We only need to store the //g// function. Thus, we need //4cn// bytes.
Again we have: Again we have:
|| //c// | Memory consumption to store a MPHF | || //c// | Memory consumption to store a MPHF |
| 2.09 | //8.36n// | | 2.09 | //8.36n// |
| **Table 2:** Memory consumption to store a MPHF generated by the CHM algorithm.
----------------------------------------
==Experimental Results==
[CHM x BMZ comparison.html]
----------------------------------------
==Papers== ==Papers==
@ -66,7 +82,7 @@ Again we have:
---------------------------------------- ----------------------------------------
[Home index.html] | [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
---------------------------------------- ----------------------------------------
%!include: FOOTER.t2t %!include: FOOTER.t2t

97
COMPARISON.t2t
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@ -5,17 +5,106 @@ Comparison Between BMZ And CHM Algorithms
---------------------------------------- ----------------------------------------
==Features== ==Characteristics==
Table 1 presents the main characteristics of the two algorithms.
The number of edges in the graph [figs/img27.png] is [figs/img236.png],
the number of keys in the input set [figs/img20.png].
The number of vertices of [figs/img32.png] is equal
to [figs/img12.png] and [figs/img237.png] for BMZ algorithm and the CHM algorithm, respectively.
This measure is related to the amount of space to store the array [figs/img37.png].
This improves the space required to store a function in BMZ algorithm to [figs/img238.png] of the space required by the CHM algorithm.
The number of critical edges is [figs/img76.png] and 0, for BMZ algorithm and the CHM algorithm,
respectively.
BMZ algorithm generates random graphs that necessarily contains cycles and the
CHM algorithm
generates
acyclic random graphs.
Finally, the CHM algorithm generates [order preserving functions concepts.html]
while BMZ algorithm does not preserve order.
==Constructing Minimal Perfect Hash Functions== %!include(html): ''TABLE1.t2t''
| **Table 1:** Main characteristics of the algorithms.
----------------------------------------
==Memory Consumption== ==Memory Consumption==
- Memory consumption to generate the minimal perfect hash function (MPHF):
|| Algorithm | //c// | Memory consumption to generate a MPHF |
| BMZ | 0.93 | //24.80n + O(1)// |
| BMZ | 1.15 | //26.42n + O(1)// |
| CHM | 2.09 | //33.00n + O(1)// |
==Run times== | **Table 2:** Memory consumption to generate a MPHF using the algorithms BMZ and CHM.
- Memory consumption to store the resulting minimal perfect hash function (MPHF):
|| Algorithm | //c// | Memory consumption to store a MPHF |
| BMZ | 0.93 | //3.72n// |
| BMZ | 1.15 | //4.60n// |
| CHM | 2.09 | //8.36n// |
| **Table 3:** Memory consumption to store a MPHF generated by the algorithms BMZ and CHM.
---------------------------------------- ----------------------------------------
[Home index.html]
==Run times==
We now present some experimental results to compare the BMZ and CHM algorithms.
The data consists of a collection of 100 million universe resource locations
(URLs) collected from the Web.
The average length of a URL in the collection is 63 bytes.
All experiments were carried on
a computer running the Linux operating system, version 2.6.7,
with a 2.4 gigahertz processor and
4 gigabytes of main memory.
Table 4 presents time measurements.
All times are in seconds.
The table entries represent averages over 50 trials.
The column labelled as [figs/img243.png] represents
the number of iterations to generate the random graph [figs/img32.png] in the
mapping step of the algorithms.
The next columns represent the run times
for the mapping plus ordering steps together and the searching
step for each algorithm.
The last column represents the percent gain of our algorithm
over the CHM algorithm.
%!include(html): ''TABLE4.t2t''
| **Table 4:** Time measurements for BMZ and the CHM algorithm.
The mapping step of the BMZ algorithm is faster because
the expected number of iterations in the mapping step to generate [figs/img32.png] are
2.13 and 2.92 for BMZ algorithm and the CHM algorithm, respectively
(see [[2 bmz.html#papers]] for details).
The graph [figs/img32.png] generated by BMZ algorithm
has [figs/img12.png] vertices, against [figs/img237.png] for the CHM algorithm.
These two facts make BMZ algorithm faster in the mapping step.
The ordering step of BMZ algorithm is approximately equal to
the time to check if [figs/img32.png] is acyclic for the CHM algorithm.
The searching step of the CHM algorithm is faster, but the total
time of BMZ algorithm is, on average, approximately 59 % faster
than the CHM algorithm.
It is important to notice the times for the searching step:
for both algorithms they are not the dominant times,
and the experimental results clearly show
a linear behavior for the searching step.
We now present run times for BMZ algorithm using a [heuristic bmz.html#heuristic] that
reduces the space requirement
to any given value between [figs/img12.png] words and [figs/img13.png] words.
For example, for [figs/img244.png] and [figs/img6.png], the analytical expected number
of iterations are [figs/img245.png] and [figs/img246.png], respectively
(for [figs/img247.png], the number of iterations are 2.78 for [figs/img244.png] and 3.04
for [figs/img6.png]).
Table 5 presents the total times to construct a
function for [figs/img247.png], with an increase from [figs/img248.png] seconds
for [figs/img128.png] (see Table 4) to [figs/img249.png] seconds for [figs/img244.png] and
to [figs/img250.png] seconds for [figs/img6.png].
%!include(html): ''TABLE5.t2t''
| **Table 5:** Time measurements for BMZ tuned algorithm with [figs/img5.png] and [figs/img6.png].
----------------------------------------
| [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
---------------------------------------- ----------------------------------------
%!include: FOOTER.t2t %!include: FOOTER.t2t

56
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@ -0,0 +1,56 @@
Minimal Perfect Hash Functions - Introduction
%!includeconf: CONFIG.t2t
----------------------------------------
==Basic Concepts==
Suppose [figs/img14.png] is a universe of //keys//.
Let [figs/img15.png] be a //hash function// that maps the keys from [figs/img14.png] to a given interval of integers [figs/img16.png].
Let [figs/img17.png] be a set of [figs/img8.png] keys from [figs/img14.png].
Given a key [figs/img18.png], the hash function [figs/img7.png] computes an
integer in [figs/img19.png] for the storage or retrieval of [figs/img11.png] in
a //hash table//.
Hashing methods for //non-static sets// of keys can be used to construct
data structures storing [figs/img20.png] and supporting membership queries
"[figs/img18.png]?" in expected time [figs/img21.png].
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.
For //static sets// of keys it is possible to compute a function
to find any key in a table in one probe; such hash functions are called
//perfect//.
More precisely, given a set of keys [figs/img20.png], we shall say that a
hash function [figs/img15.png] is a //perfect hash function//
for [figs/img20.png] if [figs/img7.png] is an injection on [figs/img20.png],
that is, there are no //collisions// among the keys in [figs/img20.png]:
if [figs/img11.png] and [figs/img22.png] are in [figs/img20.png] and [figs/img23.png],
then [figs/img24.png].
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 [figs/img25.png], that is, the table has the same size as [figs/img20.png],
then we say that [figs/img7.png] is a //minimal perfect hash function//
for [figs/img20.png].
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 [figs/img7.png] is //order preserving//
if the keys in [figs/img20.png] are arranged in some given order
and [figs/img7.png] preserves this order in the hash table.
| [figs/img26.png]
| **Figure 1:** (a) Perfect hash function. (b) Minimal perfect hash function.
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.
----------------------------------------
| [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
----------------------------------------
%!include: FOOTER.t2t

42
CONFIG.t2t
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@ -1,4 +1,46 @@
%! style(html): DOC.css
%! PreProc(html): '^%html% ' '' %! PreProc(html): '^%html% ' ''
%! PreProc(txt): '^%txt% ' '' %! PreProc(txt): '^%txt% ' ''
%! PostProc(html): "&" "&" %! PostProc(html): "&" "&"
%! PostProc(txt): " " " " %! PostProc(txt): " " " "
%! PostProc(html): 'ALIGN="middle" SRC="figs/img7.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img7.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img57.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img57.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img32.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img32.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img20.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img20.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img60.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img60.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img62.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img62.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img79.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img79.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img139.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img139.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img140.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img140.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img143.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img143.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img115.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img115.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img11.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img11.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img169.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img169.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img96.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img96.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img178.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img178.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img180.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img180.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img183.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img183.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img189.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img189.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img196.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img196.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img172.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img172.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img8.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img8.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img1.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img1.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img14.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img14.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img128.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img128.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img112.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img112.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img12.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img12.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img13.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img13.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img244.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img244.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img245.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img245.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img246.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img246.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img15.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img15.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img25.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img25.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img168.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img168.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img6.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img6.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img5.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img5.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img28.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img28.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img237.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img237.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img248.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img237.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img248.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img237.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img249.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img249.png"\1>'
%! PostProc(html): 'ALIGN="middle" SRC="figs/img250.png"(.*?)>' 'ALIGN="bottom" SRC="figs/img250.png"\1>'

33
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@ -0,0 +1,33 @@
/* implement both fixed-size and relative sizes */
SMALL.XTINY { }
SMALL.TINY { }
SMALL.SCRIPTSIZE { }
BODY { font-size: 13 }
TD { font-size: 13 }
SMALL.FOOTNOTESIZE { font-size: 13 }
SMALL.SMALL { }
BIG.LARGE { }
BIG.XLARGE { }
BIG.XXLARGE { }
BIG.HUGE { }
BIG.XHUGE { }
/* heading styles */
H1 { }
H2 { }
H3 { }
H4 { }
H5 { }
/* mathematics styles */
DIV.displaymath { } /* math displays */
TD.eqno { } /* equation-number cells */
/* document-specific styles come next */
DIV.navigation { }
DIV.center { }
SPAN.textit { font-style: italic }
SPAN.arabic { }
SPAN.eqn-number { }

3
FAQ.t2t
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@ -1,6 +1,7 @@
CMPH FAQ CMPH FAQ
%!includeconf: CONFIG.t2t
- How do I define the ids of the keys? - How do I define the ids of the keys?
- You don't. The ids will be assigned by the algorithm creating the minimal - You don't. The ids will be assigned by the algorithm creating the minimal
@ -26,7 +27,7 @@ one is executed?
---------------------------------------- ----------------------------------------
[Home index.html] | [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
---------------------------------------- ----------------------------------------
%!include: FOOTER.t2t %!include: FOOTER.t2t

3
GPERF.t2t
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@ -1,6 +1,7 @@
GPERF versus CMPH GPERF versus CMPH
%!includeconf: CONFIG.t2t
You might ask why cmph if [gperf http://www.gnu.org/software/gperf/gperf.html] You might ask why cmph if [gperf http://www.gnu.org/software/gperf/gperf.html]
already works perfectly. Actually, gperf and cmph have different goals. already works perfectly. Actually, gperf and cmph have different goals.
@ -32,7 +33,7 @@ assigning ids to millions of documents), while the former is usually found in
the compiler programming area (detect reserved keywords). the compiler programming area (detect reserved keywords).
---------------------------------------- ----------------------------------------
[Home index.html] | [Home index.html] | [CHM chm.html] | [BMZ bmz.html]
---------------------------------------- ----------------------------------------
%!include: FOOTER.t2t %!include: FOOTER.t2t

1
LOGO.t2t Normal file
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@ -0,0 +1 @@
<a href="http://sourceforge.net"><img src="http://sourceforge.net/sflogo.php?group_id=96251&amp;type=1" width="88" height="31" border="0" alt="SourceForge.net Logo" /> </a>

7
README.t2t
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@ -8,7 +8,8 @@ CMPH - C Minimal Perfect Hashing Library
==Description== ==Description==
C Minimal Perfect Hashing Library is a portable LGPLed library to create and C Minimal Perfect Hashing Library is a portable LGPLed library to create and
to work with minimal perfect hash functions. The cmph library encapsulates the newest to work with [minimal perfect hash functions concepts.html].
The cmph library encapsulates the newest
and more efficient algorithms (available in the literature) in an easy-to-use, and more efficient algorithms (available in the literature) in an easy-to-use,
production-quality and fast API. The library is designed to work with big entries that production-quality and fast API. The library is designed to work with big entries that
can not fit in the main memory. It has been used successfully for constructing minimal perfect can not fit in the main memory. It has been used successfully for constructing minimal perfect
@ -54,7 +55,7 @@ of the distinguishable features of cmph:
- New heuristic added to the bmz algorithm permits to generate a mphf with only - New heuristic added to the bmz algorithm permits to generate a mphf with only
//24.6n + O(1)// bytes. The resulting function can be stored in //3.72n// bytes. //24.6n + O(1)// bytes. The resulting function can be stored in //3.72n// bytes.
%html% [click here bmz.html] for details. %html% [click here bmz.html#heuristic] for details.
---------------------------------------- ----------------------------------------
@ -173,5 +174,5 @@ Code is under the LGPL.
%!include: FOOTER.t2t %!include: FOOTER.t2t
%!include(html): ''LOGO.html'' %!include(html): ''LOGO.t2t''
Last Updated: %%date(%c) Last Updated: %%date(%c)

76
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@ -0,0 +1,76 @@
<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Characteristics </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=2><SMALL CLASS="FOOTNOTESIZE"> <SPAN>Algorithms</SPAN></SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> BMZ </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> CHM </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img1.png"
ALT="$c$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1.15 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.09 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img239.png"
ALT="$\vert E(G)\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="89" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img240.png"
ALT="$\vert V(G)\vert=\vert g\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="20" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img241.png"
ALT="$cn$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="20" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img241.png"
ALT="$cn$"></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<!-- MATH
$|E(G_{\rm crit})|$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img111.png"
ALT="$\vert E(G_{\rm crit})\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img242.png"
ALT="$0.5\vert E(G)\vert$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 0</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img32.png"
ALT="$G$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> cyclic </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> acyclic </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Order preserving </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> no </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> yes </SMALL></TD>
</TR>
</TABLE>

109
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<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE"> <SPAN> BMZ </SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE">
<SPAN>CHM algorithm</SPAN></SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> Gain</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> (%)</SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,562,500 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.28 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 8.54 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.37 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 10.91 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.70 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 14.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1.57 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 16.13 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 48 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3,125,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.16 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 15.92 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 4.88 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 20.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.85 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 30.36 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3.20 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 61 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 6,250,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.20 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.09 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 10.48 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 43.57 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.90 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 62.26 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 6.76 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 69.02 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 58 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 12,500,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.00 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 63.26 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 23.04 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 86.30 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.60 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 117.99 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 14.94 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 132.92 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 54 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.00 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 130.79 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 51.55 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 182.34 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 262.05 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 33.68 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 295.73 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 62 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 50,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.07 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 273.75 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 114.12 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 387.87 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.90 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 577.59 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 73.97 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 651.56 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 68 </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 100,000,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.07 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 567.47 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 243.13 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 810.60 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,131.06 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 157.23 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 1,288.29 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 59 </SMALL></TD>
</TR>
</TABLE>

46
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@ -0,0 +1,46 @@
<TABLE CELLPADDING=3 BORDER="1" ALIGN="center">
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img8.png"
ALT="$n$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE"> <SPAN> BMZ <SPAN CLASS="MATH"><IMG
WIDTH="60" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img5.png"
ALT="$c=1.00$"></SPAN></SPAN> </SMALL></TD>
<TD ALIGN="CENTER" COLSPAN=4><SMALL CLASS="FOOTNOTESIZE">
<SPAN> BMZ <SPAN CLASS="MATH"><IMG
WIDTH="60" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="figs/img6.png"
ALT="$c=0.93$"></SPAN></SPAN> </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
</SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Total </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
<SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="figs/img243.png"
ALT="$N_i$"></SPAN> </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Map+Ord </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">Search </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE">
Total </SMALL></TD>
</TR>
<TR><TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 12,500,000 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 2.78 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 76.68 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25.06 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 101.74 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 3.04 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 76.39 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 25.80 </SMALL></TD>
<TD ALIGN="CENTER"><SMALL CLASS="FOOTNOTESIZE"> 102.19 </SMALL></TD>
</TR>
</TABLE>