The BDZ algorithm was designed by Fabiano C. Botelho, Djamal Belazzougui, Rasmus Pagh and Nivio Ziviani. It is a simple, efficient, near-optimal space and practical algorithm to generate a family [figs/bdz/img8.png] of PHFs and MPHFs. It is also referred to as BPZ algorithm because the work presented by Botelho, Pagh and Ziviani in [[2 #papers]]. In the Botelho's PhD. dissertation [[1 #papers]] it is also referred to as RAM algorithm because it is more suitable for key sets that can be handled in internal memory.
The BDZ algorithm uses //r//-uniform random hypergraphs given by function values of //r// uniform random hash functions on the input key set //S// for generating PHFs and MPHFs that require //O(n)// bits to be stored. A hypergraph is the generalization of a standard undirected graph where each edge connects [figs/bdz/img12.png] vertices. This idea is not new, see e.g. [[8 #papers]], but we have proceeded differently to achieve a space usage of //O(n)// bits rather than //O(n log n)// bits. Evaluation time for all schemes considered is constant. For //r=3// we obtain a space usage of approximately //2.6n// bits for an MPHF. More compact, and even simpler, representations can be achieved for larger //m//. For example, for //m=1.23n// we can get a space usage of //1.95n// bits.
Our best MPHF space upper bound is within a factor of //2// from the information theoretical lower bound of approximately //1.44// bits. We have shown that the BDZ algorithm is far more practical than previous methods with proven space complexity, both because of its simplicity, and because the constant factor of the space complexity is more than //6// times lower than its closest competitor, for plausible problem sizes. We verify the practicality experimentally, using slightly more space than in the mentioned theoretical bounds.
The BDZ algorithm is a three-step algorithm that generates PHFs and MPHFs based on random //r//-partite hypergraphs. This is an approach that provides a much tighter analysis and is much more simple than the one presented in [[3 #papers]], where it was implicit how to construct similar PHFs.The fastest and most compact functions are generated when //r=3//. In this case a PHF can be stored in approximately //1.95// bits per key and an MPHF in approximately //2.62// bits per key.
Figure 1 gives an overview of the algorithm for //r=3//, taking as input a key set [figs/bdz/img22.png] containing three English words, i.e., //S={who,band,the}//. The edge-oriented data structure proposed in [[4 #papers]] is used to represent hypergraphs, where each edge is explicitly represented as an array of //r// vertices and, for each vertex //v//, there is a list of edges that are incident on //v//.
+ It assumes that it is possible to find three uniform hash functions //h,,0,,//, //h,,1,,// and //h,,2,,//, with ranges //{0,1}//, //{2,3}// and //{4,5}//, respectively. These functions build an one-to-one mapping of the key set //S// to the edge set //E// of a random acyclic //3//-partite hypergraph //G=(V,E)//, where //|V|=m=6// and //|E|=n=3//. In [[1,2 #papers]] it is shown that it is possible to obtain such a hypergraph with probability tending to //1// as //n// tends to infinity whenever //m=cn// and //c > 1.22//. The value of that minimizes the hypergraph size (and thereby the amount of bits to represent the resulting functions) is in the range //(1.22,1.23)//. To illustrate the mapping, key "who" is mapped to edge //{h,,0,,("who"), h,,1,,("who"), h,,2,,("who")} = {1,3,5}//, key "band" is mapped to edge //{h,,0,,("band"), h,,1,,("band"), h,,2,,("band")} = {1,2,4}//, and key "the" is mapped to edge //{h,,0,,("the"), h,,1,,("the"), h,,2,,("the")} = {0,2,5}//.
+ It tests whether the resulting random //3//-partite hypergraph contains cycles by iteratively deleting edges connecting vertices of degree 1. The deleted edges are stored in the order of deletion in a list [figs/bdz/img4.png] to be used in the assigning step. The first deleted edge in Figure 1(a) was //{1,2,4}//, the second one was //{1,3,5}// and the third one was //{0,2,5}//. If it ends with an empty graph, then the test succeeds, otherwise it fails.
We now show how to use the Jenkins hash functions [[7 #papers]] to implement the three hash functions //h,,i,,//, which map values from //S// to //V,,i,,//, where [figs/bdz/img52.png]. These functions are used to build a random //3//-partite hypergraph, where [figs/bdz/img53.png] and [figs/bdz/img54.png]. Let [figs/bdz/img55.png] be a Jenkins hash function for [figs/bdz/img56.png], where
//w=32 or 64// for 32-bit and 64-bit architectures, respectively.
Let //H'// be an array of 3 //w//-bit values. The Jenkins hash function
allow us to compute in parallel the three entries in //H'//
and thereby the three hash functions //h,,i,,//, as follows:
The //Assigning Step// in Figure 1(b) outputs a PHF that maps the key set //S// into the range //[0,m-1]// and is represented by an array //g// storing values from the range //[0,3]//. The array //g// allows to select one out of the //3// vertices of a given edge, which is associated with a key //k//. A vertex for a key //k// is given by either //h,,0,,(k)//, //h,,1,,(k)// or //h,,2,,(k)//. The function //h,,i,,(k)// to be used for //k// is chosen by calculating //i = (g[h,,0,,(k)] + g[h,,1,,(k)] + g[h,,2,,(k)]) mod 3//. For instance, the values 1 and 4 represent the keys "who" and "band" because //i = (g[1] + g[3] + g[5]) mod 3 = 0// and //h,,0,,("who") = 1//, and //i = (g[1] + g[2] + g[4]) mod 3 = 2// and //h,,2,,("band") = 4//, respectively. The assigning step firstly initializes //g[i]=3// to mark every vertex as unassigned and //Visited[i]= false//, [figs/bdz/img88.png]. Let //Visited// be a boolean vector of size //m// to indicate whether a vertex has been visited. Then, for each edge [figs/bdz/img90.png] from tail to head, it looks for the first vertex //u// belonging //e// not yet visited. This is a sufficient condition for success [[1,2,8 #papers]]. Let //j// be the index of //u// in //e// for //j// in the range //[0,2]//. Then, it assigns [figs/bdz/img95.png]. Whenever it passes through a vertex //u// from //e//, if //u// has not yet been visited, it sets //Visited[u] = true//.
If we stop the BDZ algorithm in the assigning step we obtain a PHF with range //[0,m-1]//. The PHF has the following form: //phf(x) = h,,i(x),,(x)//, where key //x// is in //S// and //i(x) = (g[h,,0,,(x)] + g[h,,1,,(x)] + g[h,,2,,(x)]) mod 3//. In this case we do not need information for ranking and can set //g[i] = 0// whenever //g[i]// is equal to //3//, where //i// is in the range //[0,m-1]//. Therefore, the range of the values stored in //g// is narrowed from //[0,3]// to //[0,2]//. By using arithmetic coding as block of values (see [[1,2 #papers]] for details), or any compression technique that allows to perform random access in constant time to an array of compressed values [[5,6,12 #papers]], we can store the resulting PHFs in //mlog 3 = cnlog 3// bits, where //c > 1.22//. For //c = 1.23//, the space requirement is //1.95n// bits.
The //Ranking Step// in Figure 1 (c) outputs a data structure that permits to narrow the range of a PHF generated in the assigning step from //[0,m-1]// to //[0,n-1]// and thereby an MPHF is produced. The data structure allows to compute in constant time a function //rank// from //[0,m-1]// to //[0,n-1]// that counts the number of assigned positions before a given position //v// in //g//. For instance, //rank(4) = 2// because the positions //0// and //1// are assigned since //g[0]// and //g[1]// are not equal to //3//.
For the implementation of the ranking step we have borrowed a simple and efficient implementation from [[10 #papers]]. It requires [figs/bdz/img111.png] additional bits of space, where [figs/bdz/img112.png], and is obtained by storing explicitly the //rank// of every //k//th index in a rankTable, where [figs/bdz/img114.png]. The larger is //k// the more compact is the resulting MPHF. Therefore, the users can tradeoff space for evaluation time by setting //k// appropriately in the implementation. We only allow values for //k// that are power of two (i.e., //k=2^^b,,k,,^^// for some constant //b,,k,,// in order to replace the expensive division and modulo operations by bit-shift and bitwise "and" operations, respectively. We have used //k=256// in the experiments for generating more succinct MPHFs. We remark that it is still possible to obtain a more compact data structure by using the results presented in [[9,11 #papers]], but at the cost of a much more complex implementation.
We need to use an additional lookup table //T,,r,,// to guarantee the constant evaluation time of //rank(u)//. Let us illustrate how //rank(u)// is computed using both the rankTable and the lookup table //T,,r,,//. We first look up the rank of the largest precomputed index //v// lower than or equal to //u// in the rankTable, and use //T,,r,,// to count the number of assigned vertices from position //v// to //u-1//. The lookup table //T_r// allows us to count in constant time the number of assigned vertices in [figs/bdz/img122.png] bits, where [figs/bdz/img112.png]. Thus the actual evaluation time is [figs/bdz/img123.png]. For simplicity and without loss of generality we let [figs/bdz/img124.png] be a multiple of the number of bits [figs/bdz/img125.png] used to encode each entry of //g//. As the values in //g// come from the range //[0,3]//,
then [figs/bdz/img126.png] bits and we have tried [figs/bdz/img124.png] equal to //8// and //16//. We would expect that [figs/bdz/img124.png] equal to 16 should provide a faster evaluation time because we would need to carry out fewer lookups in //T,,r,,//. But, for both values the lookup table //T,,r,,// fits entirely in the CPU cache and we did not realize any significant difference in the evaluation times. Therefore we settle for the value //8//. We remark that each value of //r// requires a different lookup table //T,,r,, that can be generated a priori.
The resulting MPHFs have the following form: //h(x) = rank(phf(x))//. Then, we cannot get rid of the raking information by replacing the values 3 by 0 in the entries of //g//. In this case each entry in the array //g// is encoded with //2// bits and we need [figs/bdz/img133.png] additional bits to compute function //rank// in constant time. Then, the total space to store the resulting functions is [figs/bdz/img134.png] bits. By using //c = 1.23// and [figs/bdz/img135.png] we have obtained MPHFs that require approximately //2.62// bits per key to be stored.
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