arXiv subjects

Olivier Bodini

Publications and source records attributed to Olivier Bodini.

At least 19 recordsLinked to original sources

arXiv2026

Entropic Generation of Binary Words

The uniform generation of k Hamming weight binary words, equivalent to sampling k-subsets from n elements, relies on random bits, which can be expensive. We introduce a novel paradigm, random bit recycling, and use it to generate such binary words in linear time while consuming as few random bits as possible. The resulting algorithm is nearly optimal in terms of random bit consumption, meaning that it closely matches the Shannon entropic lower bound coming from information theory.

cs.DM↗

arXiv2026

Growing Binary Trees

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled structures and the analysis of polynomial iterates, we extend previous models of increasing trees with label repetitions by allowing growth branches to terminate. This mechanism enables a direct connection between dynamic evolutionary processes and classical unlabeled binary trees. We provide a combinatorial outlook for this model, linking our new approach to essential but traditionally complex parameters such as tree height, the maximum number of leaves at the deepest level (for a given tree size), and the overall tree profile. Our approach reveals structural links with Mandelbrot polynomials and coding theory. Furthermore, we leverage these structural insights to develop an efficient, iterative uniform random sampler for binary trees with a prescribed profile, achieving optimal complexity in both time and space and in random bit consumption.

math.CO↗

arXiv2026

The Distribution of the Deepest Leaves in Binary Trees

We study the extreme local structure of plane binary trees through the distribution of leaves at maximum depth. We first address two basic questions: (i) the asymptotic probability that exactly two leaves occur at the deepest level, and (ii) the asymptotic mean number of leaves at that level. These problems lead to generating functions coupled with the Catalan iteration $I_{k+1}(z)=1+zI_k(z)^2$ through quasi-logistic recurrences. We show that both associated series have dominant singularity $ρ=1/4$ and admit square-root singular expansions. The singular terms are obtained through a three-zone dominated-convergence analysis of the critical scaling regime of the truncation error. We then extend the framework to derive the full limiting distribution of the number of deepest leaves. Enumerating trees with exactly $2m$ deepest leaves yields a hierarchy of differential equations that reduces to successive polynomial integrations. Encoding these parameters into a bivariate generating function transforms the nonlinear dynamics back into the Catalan recurrence. Using continuous iteration theory and the Fatou coordinate associated with an Abel equation, we obtain a functional equation characterizing the distribution. Finally, singularity analysis implies a strict exponential tail: the probability of having $2m$ deepest leaves satisfies $κ[m]\sim 4^{-m+1}$. Numerical evaluation gives an average number of deepest leaves equal to $\hatκ\approx 2.8037$, while the probability of exactly two deepest leaves is $κ\approx 0.7009$.

math.CO↗

arXiv2024

Do Recommender Systems Promote Local Music? A Reproducibility Study Using Music Streaming Data

This paper examines the influence of recommender systems on local music representation, discussing prior findings from an empirical study on the LFM-2b public dataset. This prior study argued that different recommender systems exhibit algorithmic biases shifting music consumption either towards or against local content. However, LFM-2b users do not reflect the diverse audience of music streaming services. To assess the robustness of this study's conclusions, we conduct a comparative analysis using proprietary listening data from a global music streaming service, which we publicly release alongside this paper. We observe significant differences in local music consumption patterns between our dataset and LFM-2b, suggesting that caution should be exercised when drawing conclusions on local music based solely on LFM-2b. Moreover, we show that the algorithmic biases exhibited in the original work vary in our dataset, and that several unexplored model parameters can significantly influence these biases and affect the study's conclusion on both datasets. Finally, we discuss the complexity of accurately labeling local music, emphasizing the risk of misleading conclusions due to unreliable, biased, or incomplete labels. To encourage further research and ensure reproducibility, we have publicly shared our dataset and code.

cs.IR↗

arXiv2024

Optimal Generation of Strictly Increasing Binary Trees and Beyond

This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for weighted strictly increasing unary-binary increasing trees, optimizing randomness usage.

cs.DS↗

arXiv2021

Compaction for two models of logarithmic-depth trees: Analysis and Experiments

We are interested in the quantitative analysis of the compaction ratio for two classical families of trees: recursive trees and plane binary increasing trees. These families are typical representatives of tree models with a small depth. Once a tree of size $n$ is compacted by keeping only one occurrence of all fringe subtrees appearing in the tree the resulting graph contains only $O(n / \ln n)$ nodes. This result must be compared to classical results of compaction in the families of simply generated trees, where the analogous result states that the compacted structure is of size of order $n / \sqrt{\ln n}$. The result about the plane binary increasing trees has already been proved, but we propose a new and generic approach to get the result. Finally, an experimental study is presented, based on a prototype implementation of compacted binary search trees that are modeled by plane binary increasing trees.

math.CO↗

arXiv2021

Tuning as convex optimisation: a polynomial tuner for multi-parametric combinatorial samplers

Combinatorial samplers are algorithmic schemes devised for the approximate- and exact-size generation of large random combinatorial structures, such as context-free words, various tree-like data structures, maps, tilings, RNA molecules. They can be adapted to combinatorial specifications with additional parameters, allowing for a more flexible control over the output profile of parametrised combinatorial patterns. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain sub-patterns in generated strings. However, such a flexible control requires an additional and nontrivial tuning procedure. Using techniques of convex optimisation, we present an efficient tuning algorithm for multi-parametric combinatorial specifications. Our algorithm works in polynomial time in the system description length, the number of tuning parameters, the number of combinatorial classes in the specification, and the logarithm of the total target size. We demonstrate the effectiveness of our method on a series of practical examples, including rational, algebraic, and so-called Pólya specifications. We show how our method can be adapted to a broad range of less typical combinatorial constructions, including symmetric polynomials, labelled sets and cycles with cardinality lower bounds, simple increasing trees or substitutions. Finally, we discuss some practical aspects of our prototype tuner implementation and provide its benchmark results.

math.CO↗

arXiv2021

Asymptotic Distribution of Parameters in Trivalent Maps and Linear Lambda Terms

Structural properties of large random maps and lambda-terms may be gleaned by studying the limit distributions of various parameters of interest. In our work we focus on restricted classes of maps and their counterparts in the lambda-calculus, building on recent bijective connections between these two domains. In such cases, parameters in maps naturally correspond to parameters in lambda-terms and vice versa. By an interplay between lambda-terms and maps, we obtain various combinatorial specifications which allow us to access the distributions of pairs of related parameters such as: the number of bridges in rooted trivalent maps and of subterms in closed linear lambda-terms, the number of vertices of degree 1 in (1,3)-valent maps and of free variables in open linear lambda-terms etc. To analyse asymptotically these distributions, we introduce appropriate tools: a moment-pumping schema for differential equations and a composition schema inspired by Bender's theorem.

math.CO↗

arXiv2021

Generalized Tilings with Height Functions

In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected flip-accessibility graph can be determined, and the directed flip-accessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce.

math.CO↗

arXiv2019

On the Number of Increasing Trees with Label Repetitions

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.

math.CO↗

arXiv2019

The Combinatorics of Barrier Synchronization

In this paper we study the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.

cs.PL↗

arXiv2019

Ranked Schröder Trees

In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schröder tree model is not adapted to take into account the chronology between the branching nodes. In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there? In this paper, we enrich this model in two distinct ways in order to obtain two ranked tree models for phylogenetics, i.e. models coding chronology. For that purpose, we first develop a model of (strongly) increasing Schröder trees, symbolically described in the classical context of increasing labeling. Then we introduce a generalization for the labeling with some unusual order constraint in Analytic Combinatorics (namely the weakly increasing trees). Although these models are direct extensions of the Schröder tree model, it appears that they are also in one-to-one correspondence with several classical combinatorial objects. Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers.

cs.DS↗

arXiv2018

Statistical properties of lambda terms

We present a quantitative, statistical analysis of random lambda terms in the de Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the de Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

math.CO↗

arXiv2018

Beyond series-parallel concurrent systems: the case of arch processes

In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process with futures is more or less a directed acyclic graph. Since the quantitative study of such increasingly labeled graphs (directly related to processes) seems out of reach, we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of an uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.

cs.DM↗

arXiv2018

Asymptotic Distribution of Parameters in Random Maps

We consider random rooted maps without regard to their genus, with fixed large number of edges, and address the problem of limiting distributions for six different parameters: vertices, leaves, loops, root edges, root isthmus, and root vertex degree. Each of these leads to a different limiting distribution, varying from (discrete) geometric and Poisson distributions to different continuous ones: Beta, normal, uniform, and an unusual distribution whose moments are characterised by a recursive triangular array.

math.CO↗

arXiv2017

Polynomial tuning of multiparametric combinatorial samplers

Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tuning procedure. In this paper, we propose an efficient polynomial-time, with respect to the number of tuned parameters, tuning algorithm based on convex optimisation techniques. Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and Pólya structures including polyomino tilings with prescribed tile frequencies, planar trees with a given specific node degree distribution, and weighted partitions.

math.CO↗

arXiv2017

On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms

Uniquely closable skeletons of lambda terms are Motzkin-trees that predetermine the unique closed lambda term that can be obtained by labeling their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of closed lambda terms predetermine the unique simply-typed lambda term that can be obtained by labeling their leaves with de Bruijn indices. We derive, through a sequence of logic program transformations, efficient code for their combinatorial generation and study their statistical properties. As a result, we obtain context-free grammars describing closable and uniquely closable skeletons of lambda terms, opening the door for their in-depth study with tools from analytic combinatorics. Our empirical study of the more difficult case of (uniquely) typable terms reveals some interesting open problems about their density and asymptotic behavior. As a connection between the two classes of terms, we also show that uniquely typable closed lambda term skeletons of size $3n+1$ are in a bijection with binary trees of size $n$.

cs.PL↗

arXiv2017

Enumerating Lambda Terms by Weighted Length of Their De Bruijn Representation

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of 0-1-strings using the de Bruijn representation along with a weighting scheme. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$ and according to Tromp's weights) is $o(n^{-3/2} τ^{-n})$ for $τ\approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $Θ{n^{-3/2} ρ^{-n}}$ with some class dependent constant $ρ$, which in particular disproves the above mentioned conjecture. The methodology used is setting up the generating functions for the classes of lambda terms. These are infinitely nested radicals which are investigated then by a singularity analysis. We show further how some properties of random lambda terms can be analyzed and present a way to sample lambda terms uniformly at random in a very efficient way. This allows to generate terms of size more than one million within a reasonable time, which is significantly better than the samplers presented in the literature so far.

math.CO↗