Ranked Schröder Trees

Bodini, Olivier; Genitrini, Antoine; Naima, Mehdi

doi:10.1137/1.9781611975505.2

Cited by 6 publications

(9 citation statements)

References 12 publications

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“…It is also used as a basic building block in scheduling problems [3], as well as, e.g., in software testing [4]. In other contexts, it appears as the core problem for the generation of complex structures: we can for example cite phylogenetics [5] and bioinformatics [6].…”

Section: Introductionmentioning

confidence: 99%

Lexicographic Unranking of Combinations Revisited

Genitrini¹,

Pépin²

2021

Algorithms

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In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

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Section: Introductionmentioning

confidence: 99%

Lexicographic Unranking of Combinations Revisited

Genitrini¹,

Pépin²

2021

Algorithms

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“…This example is of interest as it indicates where the change of asymptotic behavior occurs when one varies the allowed node arities, going from binary trees with label repetitions (see [4]) to weakly increasing Schröder trees (see [5]) (with arbitrary arity of nodes).…”

Section: Strictly Monotonic Treesmentioning

confidence: 99%

“…In the study of permutations, increasing trees found usage in illuminating the behavior of local order patterns in permutations (see, for example, [6]). In biology, increasing trees find application as models of phylogenetic trees which, apart from encoding the relations between species, also encode temporal information in a way such that it encodes the history of some evolutionary process (see [5]). For a detailed and generic analysis of families increasing trees, see [1].…”

Section: Introductionmentioning

confidence: 99%

“…Firstly, we can interpret φ i , i ≥ 2, as the number of possible colors of a node of degree i. In this context the objects of study will correspond to the aforementioned strictly monotonic trees (these also include the so-called weakly increasing trees of [4,5]). Alternatively, the coefficients φ i , i ≥ 2, can be seen as the number of trees with i leaves belonging to some class of plane rooted unlabeled trees (in the sequel, we will refer to elements of such classes as tree-shapes).…”

Section: Introductionmentioning

confidence: 99%

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Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

Bodini

Genitrini

Naima

et al. 2020

Computer Science – Theory and Applications

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There exists a wealth of literature concerning families of increasing trees, particularly suitable for representing the evolution of either data structures in computer science, or probabilistic urns in mathematics, but are also adapted to model evolutionary trees in biology. The classical notion of increasing trees corresponds to labeled trees such that, along paths from the root to any leaf, node labels are strictly increasing; in addition nodes have distinct labels. In this paper we introduce new families of increasingly labeled trees relaxing the constraint of unicity of each label. Such models are especially useful to characterize processes evolving in discrete time whose nodes evolve simultaneously. In particular, we obtain growth processes for biology much more adequate than the previous increasing models. The families of monotonic trees we introduce are much more delicate to deal with, since they are not decomposable in the sense of Analytic Combinatorics. New tools are required to study the quantitative statistics of such families. In this paper, we first present a way to combinatorially specify such families through evolution processes, then, we study the tree enumerations.

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“…Such a model may serve to describe population evolution processes where each individual can give birth to two descendants independently of the other individuals. In another paper [4] we have presented an increasing model with repetitions of Schröder trees that encodes the chronology in phylogenetic trees.…”

Section: Introductionmentioning

confidence: 99%

On the number of increasing trees with label repetitions

Bodini

Genitrini

Gittenberger

et al. 2020

Discrete Mathematics

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We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree.In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument.The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents.

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Ranked Schröder Trees

Cited by 6 publications

References 12 publications

Lexicographic Unranking of Combinations Revisited

Lexicographic Unranking of Combinations Revisited

Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

On the number of increasing trees with label repetitions

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