Efficient generation of the ideals of a poset in Gray code order

Pruesse et Ruskey ont présenté un algorithme pour la génération de leur code Gray pour les idéaux d'un poset (ensemble partiellement ordonné) où deux idéaux adjacents diffèrent par un ou deux éléments. Leur algorithme fonctionne en temps amorti de O(n) par idéal. Squire a présenté une récurrence pour les idéaux d'un poset qui lui a permis de trouver un algorithme pour générer ces idéaux en temps amorti de O(log n) par idéal, mais pas en code Gray. Nous utilisons la récurrence de Squire pour trouver un code Gray pour les idéaux d'un poset, où deux idéaux adjacents diffèrent par un ou deux éléments. Dans le pire des cas, notre algorithme a la même complexité que celle de l'algorithme de Pruesse et Ruskey et dans les autres cas, sa complexité est meilleure que celle de leur algorithme et se rapproche de celle de l'algorithme de Squire. Squire a donné une condition pour obtenir cette complexité. Nous avons trouvé une condition moins restrictive que la sienne. Cette condition nous a permis d'améliorer la complexité de notre algorithme. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Poset, Extension linéaire, Cycle hamiltonien, Code Gray, Algorithme, Complexité

Efficient computation of rank probabilities in posets

As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be

Limiting distributions for additive functionals on Catalan trees

Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log n (the so-called shape functional) on uniformly distributed binary search trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution (\alpha = 1) and the normal distribution [case (b), and case (a) as $\alpha \downarrow 0$]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.Comment: 30 pages, 4 figures. Version 2 adds background information on singularity analysis and streamlines the presentatio

Les codes Gray pour les idéaux d'un poset et pour d'autres objets combinatoires

Pruesse et Ruskey ont trouvé un code Gray pour les idéaux d'un ensemble partiellement ordonné (poset) et un algorithme récursif pour les engendrer. Dans ce mémoire, un algorithme non-récursif qui engendre la même liste d'idéaux est présenté. De plus, plusieurs autres codes Gray classiques majoritairement reliés aux posets et leurs implantations\ud sont étudiés. Plus particulièrement, les codes Gray de Chase et de Ruskey pour les combinaisons, celui de Ruskey et Proskurowski pour les mots de Dyck et celui de Walsh pour les involutions sans point fixe sont étudiés. Le code Gray de Chase est présenté sous forme d'un programme FORTRAN. Vajnovszki et Walsh ont trouvé une implantation plus simple sans en donner une preuve formelle; une telle preuve est présentée dans ce mémoire. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Code Gray, Idéal, Ensemble partiellement ordonné (poset), Extension linéaire, Poset forêt, Algorithme, Non-récursif, Sans-boucle, Temps constant amorti (CAT)

Traversing combinatorial 0/1-polytopes via optimization

In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets and polytopes. Our method relies on a simple and versatile algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope \conv(X), where X\seq \{0,1\}^n. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem~$\min\{w\cdot x\mid x\in X\}$, and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is only by a factor of $\log n$ larger than the running time of the optimization algorithm. When $X$ encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope~\conv(X) along a Hamilton path corresponds to listing the combinatorial objects by local change operations, i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all ($c$-optimal) bases and independent sets in a matroid; ($c$-optimal) spanning trees, forests, matchings, maximum matchings, and $c$-optimal matchings in a general graph; vertex covers, minimum vertex covers, $c$-optimal vertex covers, stable sets, maximum stable sets and $c$-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, $c$-optimal antichains, and $c$-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope and order polytope, respectively. As another corollary from our framework, we obtain an \cO(t_{\upright{LP}} \log n) delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and~$b\in\mathbb{R}^m$, and t_{\upright{LP}} is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the 25-year old \cO(t_{\upright{LP}}\,n) delay algorithm due to Bussieck and L\"ubbecke