Michel Habib and Lhouari Nourine. Tree structure for distributive lattices and its applications. Theoretical Computer Science,165:391--405, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....partition lattice it is O(n(log 2 n) There are much faster algorithms for for certain restricted classes of lattices. For distributive lattices, at least for the recognition problem. J. P. Bordat [1] has an O(e# log log n) algorithm. O. Karpushev improves this to a linear time algorithm. In [18] M. Habib and L. Nourine give fast algorithms for computing in distributive lattices. 2. Upper Covers Consider Algorithm 1. This is a straightforward algorithm which finds the covering relation from the relation. 1 P is topologically sorted. 2 Q P 3 for a P do 4 pop(Q) 5 S S ....

....the theorem follows. While the size N of Con L may be exponential in the size of L, Con L is distributive and so is isomorphic to the lattice of order ideals of J(Con L) It is straightforward to compute this in time O(mN) but there is a faster algorithm in R. Medina and L. Nourine [26] see also [18]. 4. Free and Finitely Presented Lattices Equational complexity. Consider the problem of deciding if two lattice terms s(x 1 , x n ) and t(x 1 , x n ) are always equal when evaluated in every lattice in some class of lattices K. For the three classical classes of lattices, all ....

Michel Habib and Lhouari Nourine, Tree structure for distributive lattices and its applications, Theoret. Comput. Sci. 165 (1996), no. 2, 391--405.

....method 3 of A t Kaci et al. 1] and the gene encoding technique of Caseau [2] which try to minimize the range of integers used to construct the sets, thus shortening the corresponding bit vectors. It is well known that nding an optimal bit vector encoding for partial ordered sets is NP hard [10] and that there exist classes of partial ordered sets (distributive and simplicial lattices) where an optimal encoding is as large as the number of types with only one supertype [10] Fortunately, type hierarchies can be encoded much more compactly than distributive lattices. In a previous paper, ....

....bit vectors. It is well known that nding an optimal bit vector encoding for partial ordered sets is NP hard [10] and that there exist classes of partial ordered sets (distributive and simplicial lattices) where an optimal encoding is as large as the number of types with only one supertype [10]. Fortunately, type hierarchies can be encoded much more compactly than distributive lattices. In a previous paper, we have developed a new and improved version of the Caseau approach [2] which we call Near Optimal Hierarchical Encoding (NHE) 12] This version generalizes Caseau s algorithm by ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Theoretical Computer Science, 165:391-405, 1996.

....fg f2g f2,3g f2,4g Gamma Gamma Gamma Gamma Gamma Fig. 3. An incorrect encoding produced by Caseau s algorithm Habib and Nourine showed that constructing an optimal bit vector encoding for partially ordered sets is NP hard [HN94] They also showed in [HN94] and [HN96] that there exist some classes of lattices (distributive and simplicial lattices) where, for an optimal solution, all genes have to be different. For these classes of lattices, therefore, an optimal solution can be constructed in linear time. Partially ordered sets resulting from type hierarchies ....

Michel Habib and Lhouari Nourine. Tree structure for distributive lattices and its applications. Theoretical Computer Science, 165:391--405, 1996.

.... encoding addresses the need to expeditiously perform such operations as greatest lower bounds and comparability by constructing mappings from a given partial order to one which facilitates taxonomic operations (e.g. the lattice of bit vectors or logical terms) Recent research on encoding [3, 4, 8, 12, 15] has identified certain forms of taxonomies for which very efficient encoding schemes exist. In this paper, we provide a framework in which such techniques can be exploited when encoding a single taxonomy. Through modular decomposition [14, 18] we split taxonomies into suborders, each of which ....

....L is a distributive lattice, then we can preserve both meets and joins, although in general this is not possible. This arises from the fact that every distributive lattice is isomorphic to a lattice of sets [7] i.e. where meets and joins are computed by intersections and unions, respectively) In [12, 20], a detailed analysis of the properties of distributive and simplicial lattices related to encoding is given, and reference is made to an algorithm which can recognize distributive lattices in linear time. In their presentation, a decomposition based on the meet irreducible elements M(L) results ....

M. Habib and L. Nourine. Tree Structure for Distributive Lattices and its Applications. Research report, Universit 'e de Montpellier II, R.R. LIRMM 94036.

....the is a (or is a kind of) relation in Knowledge Representation Systems. In the future, in real applications, these hierarchies of objects will be very large. So, it will be very important to have efficient algorithmic techniques to represent them in memory and to compute object comparisons (see [HN] for a survey) The ultimate goal would be to represent any order in a very small sized data structure and to have a very efficient algorithm to compare any pair of elements. But this problem seems to be very difficult. In this paper, we expose an original method for representing any order as a ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. submitted to TCS.

....techniques will depend on the nature of these partial orders (e.g. whether they can change dynamically, whether certain properties such as distributivity or bounded width are satisfied) and the operations to be supported. Research on taxonomic encoding has explored a variety of possibilities (e.g. [2, 24, 34, 35, 43, 45, 61, 77, 78, 79, 93, 97, 101, 102, 104, 114]) In order to empower logical terms for encoding, we developed sparse terms [51] based on an analogy to sparse matrices. There are many similarities, but also some important differences, between sparse terms and terms in LIFE [4] as well as sorted feature structures [23, 118] Although ....

....of elements in Q corresponding to the meet crest in P . Here, Gamma1 must map each element in this set back to P . Note that if P and Q are both lattices, then must be meet preserving in the lattice theoretic sense. CHAPTER 4. THE FOUNDATIONS OF TAXONOMIC ENCODING 23 tree encoding scheme of [78], however, Q is a tree data structure, and maps elements of P to nodes of the tree. Operations in P are translated to operations on this data structure. In this chapter, it is our goal to develop a unified framework that separates the content (semantics) of the encoding map from its ....

[Article contains additional citation context not shown here]

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Technical Report R.R. LIRMM 94036, Universit'e de Montpellier II, Laboratoire d'Informatique, de Robotique et de Microelectronique de Montpellier, 1994.

....method 3 of Ait Kaci et al. 1] and the gene encoding technique of Caseau [2] which try to minimize the range of integers used to construct the sets, thus shortening the corresponding bit vectors. It is well known that finding an optimal bit vector encoding for partial ordered sets is NP hard [10] and that there exist classes of partial ordered sets (distributive and simplicial lattices) where an optimal encoding is as large as the number of types with only one supertype [10] Fortunately, type hierarchies can be encoded much more compactly than distributive lattices. In a previous paper, ....

....bit vectors. It is well known that finding an optimal bit vector encoding for partial ordered sets is NP hard [10] and that there exist classes of partial ordered sets (distributive and simplicial lattices) where an optimal encoding is as large as the number of types with only one supertype [10]. Fortunately, type hierarchies can be encoded much more compactly than distributive lattices. In a previous paper, we have developed a new and improved version of the Caseau approach [2] which we call Near Optimal Hierarchical Encoding (NHE) 12] This version generalizes Caseau s algorithm by ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Theoretical Computer Science, 165:391--405, 1996.

....as follows. In Sect. 2 we define the modele and the notations used, and we show the links existing between the P ossibly( Phi) problem and the ideals generation problem in order theory. The algorithm presented is fundmentally based on a particular spanning tree of the ideal lattice introduced in [10, 18] and called the Ideal Tree or the Consistent Cuts Tree of P . This tree and its properties are presented in Sect. 3. It has been already used in distributed system and ordered sets [16, 17] We think that the ideal tree is a pertinent structure to study the ideals of partial orders and thus global ....

....lattice The fundamental structure used by the algorithm in Sect. 4 is the ideal tree T (P ) associated to a linear extension of P . By using the labels of arcs, it gives an efficient way to generate and to encode ideals. The definition of this tree is given in the following theorem. Theorem 2. [10] Let L be any linear extension of P . There exists a unique spanning tree T (P ) of I(P ) whose root is the top element of I(P ) and whose sequence of labels of arcs e 1 ; Delta Delta Delta ; e k from the root to any ideal I is such that it satisfies e 1 L Delta Delta Delta L e k . Let us ....

[Article contains additional citation context not shown here]

M. Habib and L. Nourine. Tree structures for distributive lattice and its application. Research report, LIRMM, Montpellier, France, October 1993.

....elements of a partial order P . Then (GLB(x; y) x) y) furthermore GLB(x; y) Gamma1 ( x) y) The complexity of the algorithm computing GLB(x; y) depends on the data structure used for storing the code . To have good complexity, we use the tree data structure introduced in [8] for an optimal representation of a distributive lattice. This tree allows us to have a best time complexity to compute Gamma1 . Algorithm 1: Join preserving and meet preserving encodings Data : A partial order P = X; P ) Result : The meet preserving encoding : P 2 J(P ) the ....

....such that the union of labels from any node x to Root corresponds (x) sorted according to oe. The sons of each internal node are sorted according to oe Gamma1 Note that for a distributive lattice, every edge is labeled by one element of J(P ) Most operations on this tree are described in [8]. We use this tree to store the code : for a code (x) of an element x, we have to search the tree in a recursive way to verify if x is in the tree. If not, we can easily add it. Example 10. Figure 5 shows T for the encoding of the partial order of Figure 2, with oe = a b c d f i. ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. TCS, 165(2):391--405, october 1996.

....1 6 9 fg 9,10,14,17,18,16,12,13 5 Conclusions Notice that using the dual property one can derive easily an incremental algorithm to generate distributive lattice and therefore the down sets lattice of a partial order. An algorithm with such property can be found in [11]. Now consider a partition fL 1 ; L n ; L n 1 g of a distributive lattice L. Recall that if x 2 L i then i (x) is the unique element that cover x out of L i . The set of pairs (x; i (x) de ne a tree rooted in 1. This tree can be associated to L and it is a representation of L (see ....

....[11] Now consider a partition fL 1 ; L n ; L n 1 g of a distributive lattice L. Recall that if x 2 L i then i (x) is the unique element that cover x out of L i . The set of pairs (x; i (x) de ne a tree rooted in 1. This tree can be associated to L and it is a representation of L (see [11]) Moreover it is easy to generalize the algorithms presented above for extremal lattices and locally distributive lattices as de ned recently by Markowsky [13] ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Theor. Comp. Sci, 165(2):391-405, October 1996.

....encoding have been proposed in the literature. One of these encodings uses in an heuristic way the previous optimal encoding for a spanning tree of the order, see Agrawal et al. [1] and Talamo and Vocca [27] Other approaches are based on dimension approximation see Ellis [9] and Habib and Nourine [16, 15]. In the next section a new dimension parameter is introduced, namely the encoding dimension and we show how it can measure encoding into chain products. 3 Encoding dimension Let us start with the general definition of dimension of partially ordered sets. Definition 1. The k dimension of P , k ....

M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Research report, LIRMM, Montpellier, France, Avril 1994. to appear in Theoretical Computer Science.

No context found.

Michel Habib and Lhouari Nourine. Tree structure for distributive lattices and its applications. Theoretical Computer Science,165:391--405, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC