BDZ Algorithm


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==Introduction==


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==The Algorithm==


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===Mapping Step===


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===Assigning Step===

 
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===Ranking Step===


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==Memory Consumption==

Now we detail the memory consumption to generate and to store minimal perfect hash functions
using the BDZ algorithm. The structures responsible for memory consumption are in the 
following:
- 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 
    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 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 //12n// bytes. 
    
  + **next**: given a vertex [figs/img139.png], we can discover the edges that 
    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 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.
    
  + **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.

- 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. 
    
  + **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. 

- 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 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 BDZ algorithm.
     
Now we present the memory consumption to store the resulting function.
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 BDZ algorithm.  
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==Experimental Results==

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]].
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==Papers==[papers]

+ [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]. [Near Space-Optimal Perfect Hashing Algorithms papers/thesis.pdf]. //Thesis Proposal//, //Department of Computer Science//, //Federal University of Minas Gerais//, July 2007.


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