*** empty log message ***
This commit is contained in:
parent
a1296d2fdd
commit
94957a85ba
387
BDZ.t2t
387
BDZ.t2t
|
@ -1,401 +1,112 @@
|
|||
BMZ Algorithm
|
||||
BDZ Algorithm
|
||||
|
||||
|
||||
%!includeconf: CONFIG.t2t
|
||||
|
||||
----------------------------------------
|
||||
==History==
|
||||
==Introduction==
|
||||
|
||||
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/].
|
||||
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 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
|
||||
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/].
|
||||
The [CHM algorithm chm.html] is based on acyclic random graphs to generate
|
||||
[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
|
||||
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
|
||||
for sets with more than 1000 keys.
|
||||
At the same time, [Fabiano C. Botelho http://www.dcc.ufmg.br/~fbotelho],
|
||||
a master degree student at [Departament of Computer Science http://www.dcc.ufmg.br] in
|
||||
[Federal University of Minas Gerais http://www.ufmg.br],
|
||||
started to be advised by [Nivio Ziviani http://www.dcc.ufmg.br/~nivio] who presented the problem
|
||||
to [Fabiano http://www.dcc.ufmg.br/~fbotelho].
|
||||
|
||||
During the master, [Fabiano http://www.dcc.ufmg.br/~fbotelho] and
|
||||
[Nivio Ziviani http://www.dcc.ufmg.br/~nivio] faced lots of problems.
|
||||
In april of 2004, [Fabiano http://www.dcc.ufmg.br/~fbotelho] 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 **BMZ**, because it was conceived by Fabiano C. **B**otelho,
|
||||
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
|
||||
we invited professor [Yoshiharu Kohayakawa http://www.ime.usp.br/~yoshi] to help us.
|
||||
The final description and analysis of BMZ algorithm is presented in [[2 #papers]].
|
||||
|
||||
----------------------------------------
|
||||
|
||||
==The Algorithm==
|
||||
|
||||
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],
|
||||
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],
|
||||
whose edge set is [figs/img61.png]. There is an edge in [figs/img32.png] for each
|
||||
key in the set of keys [figs/img20.png].
|
||||
|
||||
In what follows, we shall be interested in the //2-core// of
|
||||
the random graph [figs/img32.png], that is, the maximal subgraph
|
||||
of [figs/img32.png] with minimal degree at
|
||||
least 2 (see [[2 #papers]] for details).
|
||||
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].
|
||||
The vertices and edges in [figs/img63.png] are said to be //critical//.
|
||||
We let [figs/img64.png] and [figs/img65.png].
|
||||
Moreover, we let [figs/img66.png] be the set of //non-critical//
|
||||
vertices in [figs/img32.png].
|
||||
We also let [figs/img67.png] be the set of all critical
|
||||
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].
|
||||
Finally, we let [figs/img69.png] be the //non-critical// subgraph
|
||||
of [figs/img32.png].
|
||||
The non-critical subgraph [figs/img70.png] corresponds to the //acyclic part//
|
||||
of [figs/img32.png].
|
||||
We have [figs/img71.png].
|
||||
|
||||
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
|
||||
a way that [figs/img75.png] ([figs/img18.png]) is a
|
||||
minimal perfect hash function for [figs/img20.png].
|
||||
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] (see [[2 #papers]]
|
||||
for details).
|
||||
|
||||
Figure 1 presents a pseudo code for the BMZ algorithm.
|
||||
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].
|
||||
The method uses a mapping, ordering and searching approach.
|
||||
We now describe each step.
|
||||
| procedure BMZ ([figs/img20.png], [figs/img37.png])
|
||||
| Mapping ([figs/img20.png], [figs/img32.png]);
|
||||
| Ordering ([figs/img32.png], [figs/img63.png], [figs/img70.png]);
|
||||
| Searching ([figs/img32.png], [figs/img63.png], [figs/img70.png], [figs/img37.png]);
|
||||
| **Figure 1**: Main steps of BMZ algorithm for constructing a minimal perfect hash function
|
||||
|
||||
----------------------------------------
|
||||
|
||||
===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://www.cs.yorku.ca/~oz/hash.html], [FNV hash http://www.isthe.com/chongo/tech/comp/fnv/],
|
||||
[Jenkins hash http://burtleburtle.net/bob/hash/doobs.html] and [SDBM hash http://www.cs.yorku.ca/~oz/hash.html])
|
||||
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===
|
||||
===Assigning 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===
|
||||
===Ranking 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//.
|
||||
|
||||
----------------------------------------
|
||||
|
||||
==Memory Consumption==
|
||||
|
||||
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
|
||||
using the BDZ algorithm. The structures responsible for memory consumption are in the
|
||||
following:
|
||||
- Graph:
|
||||
- 3-graph:
|
||||
+ **first**: is a vector that stores //cn// 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,
|
||||
incident edges of each vertex. The integer numbers are 4 bytes long. Therefore,
|
||||
the vector first is stored in //4cn// bytes.
|
||||
|
||||
+ **edges**: 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
|
||||
is compounded by three vertices, each entry stores three integer numbers
|
||||
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 //12n// bytes.
|
||||
|
||||
+ **next**: given a vertex [figs/img139.png], we can discover the edges that
|
||||
contain [figs/img139.png] following its list of edges,
|
||||
contain [figs/img139.png] following its list of incident edges,
|
||||
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,
|
||||
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
|
||||
numbers in the vector next, so it is stored in //4*2n = 8n// bytes.
|
||||
the linked lists of edges of each vertex. As there are three vertices for each edge,
|
||||
when an edge is iserted in the 3-graph, it must be inserted in the three linked lists
|
||||
of the vertices in its composition. Therefore, there are //3n// entries of integer
|
||||
numbers in the vector next, so it is stored in //4*3n = 12n// bytes.
|
||||
|
||||
+ **critical vertices(critical_nodes vector)**: is a vector of //cn// 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 //cn/8// bytes.
|
||||
+ **Vertices degree (vert_degree vector)**: is a vector of //cn// bytes
|
||||
that represents the degree of each vertex. We can use just one byte for each
|
||||
vertex because the 3-graph is sparse, once it has more vertices than edges.
|
||||
Therefore, the vertices degree is represented in //cn// bytes.
|
||||
|
||||
+ **critical edges (used_edges vector)**: is a vector of //n// 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 //n/8// bytes.
|
||||
- Acyclicity test:
|
||||
+ **List of deleted edges obtained when we test whether the 3-graph is a forest (queue vector)**:
|
||||
is a vector of //n// integer numbers containing indexes of vector edges. Therefore, it
|
||||
requires //4n// bytes in internal memory.
|
||||
|
||||
- Other auxiliary structures
|
||||
+ **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
|
||||
each two critical vertices. Let [figs/img110.png] be the expected number of critical
|
||||
vertices. Therefore, the queue is stored in //4*0.5*[figs/img110.png]=2[figs/img110.png]//.
|
||||
+ **Marked edges in the acyclicity test (marked_edges vector)**:
|
||||
is a bit vector of //n// bits to indicate the edges that have already been deleted during
|
||||
the acyclicity test. Therefore, it requires //n/8// bytes in internal memory.
|
||||
|
||||
+ **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
|
||||
in //cn/8// bytes.
|
||||
|
||||
+ **function //g//**: is represented by a vector of //cn// integer numbers.
|
||||
As each integer number is 4 bytes long, the function //g// is stored in
|
||||
//4cn// bytes.
|
||||
- MPHF description
|
||||
+ **function //g//**: is represented by a vector of //2cn// bits. Therefore, it is
|
||||
stored in //0.25cn// bytes
|
||||
+ **ranktable**: is a lookup table used to store some precomputed ranking information.
|
||||
It has //(cn)/(2^b)// entries of 4-byte integer numbers. Therefore it is stored in
|
||||
//(4cn)/(2^b)// bytes. The larger is b, the more compact is the resulting MPHFs and
|
||||
the slower are the functions. So b imposes a trade-of between space and time.
|
||||
+ **Total**: 0.25cn + (4cn)/(2^b) bytes
|
||||
|
||||
|
||||
Thus, the total memory consumption of BMZ algorithm for generating a minimal
|
||||
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:
|
||||
|| //c// | [figs/img110.png] | Memory consumption to generate a MPHF |
|
||||
| 0.93 | //0.497n// | //24.80n + O(1)// |
|
||||
| 1.15 | //0.401n// | //26.42n + O(1)// |
|
||||
Thus, the total memory consumption of BDZ algorithm for generating a minimal
|
||||
perfect hash function (MPHF) is: //(28.125 + 5c)n + 0.25cn + (4cn)/(2^b) + O(1)// bytes.
|
||||
As the value of constant //c// may be larger than or equal to 1.23 we have:
|
||||
|| //c// | //b// | Memory consumption to generate a MPHF (in bytes) |
|
||||
| 1.23 | //7// | //34.62n + O(1)// |
|
||||
| 1.23 | //8// | //34.60n + O(1)// |
|
||||
|
||||
| **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]].
|
||||
| **Table 1:** Memory consumption to generate a MPHF using the BDZ algorithm.
|
||||
|
||||
Now we present the memory consumption to store the resulting function.
|
||||
We only need to store the //g// function. Thus, we need //4cn// bytes.
|
||||
Again we have:
|
||||
|| //c// | Memory consumption to store a MPHF |
|
||||
| 0.93 | //3.72n// |
|
||||
| 1.15 | //4.60n// |
|
||||
So we have:
|
||||
|| //c// | //b// | Memory consumption to store a MPHF (in bits) |
|
||||
| 1.23 | //7// | //2.77n + O(1)// |
|
||||
| 1.23 | //8// | //2.61n + O(1)// |
|
||||
|
||||
| **Table 2:** Memory consumption to store a MPHF generated by the BMZ algorithm.
|
||||
| **Table 2:** Memory consumption to store a MPHF generated by the BDZ algorithm.
|
||||
----------------------------------------
|
||||
|
||||
==Experimental Results==
|
||||
|
||||
[CHM x BMZ comparison.html]
|
||||
|
||||
Experimental results to compare the BDZ algorithm with the other ones in the CMPH
|
||||
library are presented in Botelho, Pagh and Ziviani [[1 #papers],[2 #papers]].
|
||||
----------------------------------------
|
||||
|
||||
==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], R. Pagh, [N. Ziviani http://www.dcc.ufmg.br/~nivio]. [Simple and space-efficient minimal perfect hash functions papers/wads07.pdf]. //10th International Workshop on Algorithms and Data Structures (WADs'07),// Springer-Verlag Lecture Notes in Computer Science, vol. 4619, Halifax, Canada, August 2007, 139-150.
|
||||
|
||||
+ [F. C. Botelho http://www.dcc.ufmg.br/~fbotelho], Y. Kohayakawa, and [N. Ziviani http://www.dcc.ufmg.br/~nivio]. [A Practical Minimal Perfect Hashing Method papers/wea05.pdf]. //4th International Workshop on efficient and Experimental Algorithms (WEA05),// Springer-Verlag Lecture Notes in Computer Science, vol. 3505, Santorini Island, Greece, May 2005, 488-500.
|
||||
+ [F. C. Botelho http://www.dcc.ufmg.br/~fbotelho]. [Near Space-Optimal Perfect Hashing Algorithms papers/thesis.pdf]. //Thesis Proposal//, //Department of Computer Science//, //Federal University of Minas Gerais//, July 2007.
|
||||
|
||||
|
||||
%!include: ALGORITHMS.t2t
|
||||
|
|
Before Width: | Height: | Size: 21 KiB After Width: | Height: | Size: 5.0 KiB |
|
@ -48,7 +48,8 @@ The CMPH Library encapsulates the newest and more efficient algorithms in an eas
|
|||
generalization of a graph where each edge connects 3 vertices instead of only 2. The
|
||||
resulting functions are not order preserving and can be stored in only //(2 + x)cn//
|
||||
bits, where //c// should be larger than or equal to //1.23// and //x// is a constant
|
||||
larger than //0// (actually, x = 1/b and b is a parameter that should be larger 2).
|
||||
larger than //0// (actually, x = 1/b and b is a parameter that should be larger than 2).
|
||||
For //c = 1.23// and //b = 8//, the resulting functions are stored in approximately 2.6 bits per key.
|
||||
%html% - [BMZ Algorithm bmz.html].
|
||||
%txt% - BMZ Algorithm.
|
||||
A very fast algorithm based on cyclic random graphs to construct minimal
|
||||
|
@ -81,15 +82,17 @@ The CMPH Library encapsulates the newest and more efficient algorithms in an eas
|
|||
|
||||
==News for version 0.8==
|
||||
|
||||
- [An algorithm to generate MPHFs that require around 2.62 bits per key to be stored bdz.html], which is referred to as BDZ algorithm. The algorithm is the fastest one available in the literature for sets that can be treated in internal memory.
|
||||
- [An algorithm to generate MPHFs that require around 2.6 bits per key to be stored bdz.html], which is referred to as BDZ algorithm. The algorithm is the fastest one available in the literature for sets that can be treated in internal memory.
|
||||
- The hash functions djb2, fnv and sdbm were removed because they do not use random seeds and therefore are not useful for MPHFs algorithms.
|
||||
- All reported bugs and suggestions have been corrected and included as well.
|
||||
|
||||
|
||||
==News for version 0.7==
|
||||
|
||||
- Added man pages and a pkgconfig file.
|
||||
|
||||
|
||||
[News log newslog.html]
|
||||
----------------------------------------
|
||||
|
||||
==Examples==
|
||||
|
@ -234,7 +237,7 @@ Use the project page at sourceforge: http://sf.net/projects/cmph
|
|||
|
||||
==License Stuff==
|
||||
|
||||
Code is under the LGPL.
|
||||
Code is under the LGPL and the MPL 1.1.
|
||||
----------------------------------------
|
||||
|
||||
%!include: FOOTER.t2t
|
||||
|
|
2
gendocs
2
gendocs
|
@ -1,6 +1,7 @@
|
|||
#!/bin/sh
|
||||
|
||||
txt2tags -t html --mask-email -i README.t2t -o index.html
|
||||
txt2tags -t html -i BDZ.t2t -o bdz.html
|
||||
txt2tags -t html -i BMZ.t2t -o bmz.html
|
||||
txt2tags -t html -i BRZ.t2t -o brz.html
|
||||
txt2tags -t html -i CHM.t2t -o chm.html
|
||||
|
@ -12,6 +13,7 @@ txt2tags -t html -i CONCEPTS.t2t -o concepts.html
|
|||
txt2tags -t html -i NEWSLOG.t2t -o newslog.html
|
||||
|
||||
txt2tags -t txt --mask-email -i README.t2t -o README
|
||||
txt2tags -t txt -i BDZ.t2t -o BDZ
|
||||
txt2tags -t txt -i BMZ.t2t -o BMZ
|
||||
txt2tags -t txt -i BRZ.t2t -o BRZ
|
||||
txt2tags -t txt -i CHM.t2t -o CHM
|
||||
|
|
Binary file not shown.
Binary file not shown.
Binary file not shown.
Loading…
Reference in New Issue