From 94957a85ba1f405f1867688e5c6cd85755d40995 Mon Sep 17 00:00:00 2001 From: fc_botelho Date: Sun, 23 Mar 2008 03:45:01 +0000 Subject: [PATCH] *** empty log message *** --- BDZ.t2t | 395 +++++++--------------------------------------- README.t2t | 9 +- gendocs | 2 + papers/fch92.pdf | Bin 0 -> 718507 bytes papers/thesis.pdf | Bin 0 -> 632075 bytes papers/wads07.pdf | Bin 0 -> 169593 bytes 6 files changed, 61 insertions(+), 345 deletions(-) create mode 100755 papers/fch92.pdf create mode 100755 papers/thesis.pdf create mode 100755 papers/wads07.pdf diff --git a/BDZ.t2t b/BDZ.t2t index 72921a6..68fc11b 100755 --- a/BDZ.t2t +++ b/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. - - + **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. - -- 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]//. - - + **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. + + **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. -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)// | - - | **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]]. + + **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. -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 diff --git a/README.t2t b/README.t2t index 4131fa8..f0d58d5 100644 --- a/README.t2t +++ b/README.t2t @@ -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 diff --git a/gendocs b/gendocs index d3c05a4..e8f34d0 100755 --- a/gendocs +++ b/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 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