arXiv subjects

Florian Ingels

Publications and source records attributed to Florian Ingels.

8 recordsLinked to original sources

arXiv2026

Vigemers: on the number of $k$-mers sharing the same XOR-based minimizer

In bioinformatics, minimizers have become an inescapable method for handling $k$-mers (words of fixed size $k$) extracted from DNA or RNA sequencing, whether for sampling, storage, querying or partitioning. According to some fixed order on $m$-mers ($m<k$), the minimizer of a $k$-mer is defined as its smallest $m$-mer -- and acts as its fingerprint. Although minimizers are widely used for partitioning purposes, there is almost no theoretical work on the quality of the resulting partitions. For instance, it has been known for decades that the lexicographic order empirically leads to highly unbalanced partitions that are unusable in practice, but it was not until very recently that this observation was theoretically substantiated. The rejection of the lexicographic order has led the community to resort to (pseudo-)random orders using hash functions. In this work, we extend the theoretical results relating to the partitions obtained by the lexicographical order, departing from it to a (exponentially) large family of hash functions, namely where the $m$-mers are XORed against a fixed key. More precisely, provided a key $\gamma$ and a $m$-mer $w$, we investigate the function that counts how many $k$-mers admit $w$ as their minimizer (i.e. where $w\oplus\gamma$ is minimal among all $m$-mers of said $k$-mers). This number, denoted by $\pi_k^{\gamma}(w)$, represents the maximum size of the bucket associated with $w$, if all possible $k$-mers were to be seen and partitioned. We adapt the (lexicographical order) method of the literature to our framework and propose combinatorial equations that allow to compute, using dynamic programming, $\pi_k^{\gamma}(w)$ in $O(km^2)$ time and $O(km)$ space.

cs.DM↗

arXiv2025

Decomposing Words for Enhanced Compression: Exploring the Number of Runs in the Extended Burrows-Wheeler Transform

The Burrows-Wheeler Transform (BWT) is a fundamental component in many data structures for text indexing and compression, widely used in areas such as bioinformatics and information retrieval. The extended BWT (eBWT) generalizes the classical BWT to multisets of strings, providing a flexible framework that captures many BWT-like constructions. Several known variants of the BWT can be viewed as instances of the eBWT applied to specific decompositions of a word. A central property of the BWT, essential for its compressibility, is the number of maximal ranges of equal letters, named runs. In this article, we explore how different decompositions of a word impact the number of runs in the resulting eBWT. First, we show that the number of decompositions of a word is exponential, even under minimal constraints on the size of the subsets in the decomposition. Second, we present an infinite family of words for which the ratio of the number of runs between the worst and best decompositions is unbounded, under the same minimal constraints. These results illustrate the potential cost of decomposition choices in eBWT-based compression and underline the challenges in optimizing run-length encoding in generalized BWT frameworks.

cs.DS↗

arXiv2024

On the number of $k$-mers admitting a given lexicographical minimizer

The minimizer of a word of size $k$ (a $k$-mer) is defined as its smallest substring of size $m$ (with $m\leq k$), according to some ordering on $m$-mers. minimizers have been used in bioinformatics -- notably -- to partition sequencing datasets, binning together $k$-mers that share the same minimizer. It is folklore that using the lexicographical order lead to very unbalanced partitions, resulting in an abundant literature devoted to devising alternative orders for achieving better balanced partitions. To the best of our knowledge, the unbalanced-ness of lexicographical-based minimizer partitions has never been investigated from a theoretical point of view. In this article, we aim to fill this gap and determine, for a given minimizer, how many $k$-mers would admit the chosen minimizer -- i.e. what would be the size of the bucket associated to the chosen minimizer in the worst case, where all $k$-mers would be seen in the data. We show that this number can be computed in $O(km)$ space and $O(km^2)$ time. We further introduce approximations that can be computed in $O(k)$ space and $O(km)$ time. We also show on genomic datasets that the practical number of $k$-mers associated to a minimizer are closely correlated to the theoretical expected number. We introduce two conjectures that could help closely approximating the total number of $k$-mers sharing a minimizer. We believe that characterising the distribution of the number of $k$-mers per minimizer will help devise efficient lexicographic-based minimizer bucketting.

cs.DS↗

arXiv2023

Revisiting Tree Isomorphism: An Algorithmic Bric-\`a-Brac

The Aho, Hopcroft and Ullman (AHU) algorithm has been the state of the art since the 1970s for determining in linear time whether two unordered rooted trees are isomorphic or not. However, it has been criticized (by Campbell and Radford) for the way it is written, which requires several (re)readings to be understood, and does not facilitate its analysis. In this article, we propose a different, more intuitive formulation of the algorithm, as well as three propositions of implementation, two using sorting algorithms and one using prime multiplication. Although none of these three variants admits linear complexity, we show that in practice two of them are competitive with the original algorithm, while being straightforward to implement. Surprisingly, the algorithm that uses multiplications of prime numbers (which are also be generated during the execution) is competitive with the fastest variants using sorts, despite having a worst theoretical complexity. We also adapt our formulation of AHU to tackle to compression of trees in directed acyclic graphs (DAGs). This algorithm is also available in three versions, two with sorting and one with prime number multiplication. Our experiments are carried out on trees of size at most $10^6$, consistent with the actual datasets we are aware of, and done in Python with the library treex, dedicated to tree algorithms.

cs.DS↗

arXiv2023

Detection of Common Subtrees with Identical Label Distribution

Frequent pattern mining is a relevant method to analyse structured data, like sequences, trees or graphs. It consists in identifying characteristic substructures of a dataset. This paper deals with a new type of patterns for tree data: common subtrees with identical label distribution. Their detection is far from obvious since the underlying isomorphism problem is graph isomorphism complete. An elaborated search algorithm is developed and analysed from both theoretical and numerical perspectives. Based on this, the enumeration of patterns is performed through a new lossless compression scheme for trees, called DAG-RW, whose complexity is investigated as well. The method shows very good properties, both in terms of computation times and analysis of real datasets from the literature. Compared to other substructures like topological subtrees and labelled subtrees for which the isomorphism problem is linear, the patterns found provide a more parsimonious representation of the data.

cs.DS↗

arXiv2022

Enumeration of Irredundant Forests

Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general problem, the enumeration of sets of trees -- forests. Specifically, we mainly study irredundant forests, i.e., where no tree is a subtree of another. By compressing each such forest into a Directed Acyclic Graph (DAG), we develop a reverse search like method to enumerate DAGs compressing irredundant forests. Remarkably, we prove that these DAGs are in bijection with the row-Fishburn matrices, a well-studied class of combinatorial objects. In a second step, we derive our irredundant forest enumeration to provide algorithms for tackling related problems: (i) enumeration of forests in their classical sense (where redundancy is allowed); (ii) the enumeration of "subforests" of a forest, and (iii) the frequent "subforest" mining problem. All the methods presented in this article enumerate each item uniquely, up to isomorphism.

cs.DM↗

arXiv2021

Isomorphic unordered labeled trees up to substitution ciphering

Given two messages - as linear sequences of letters, it is immediate to determine whether one can be transformed into the other by simple substitution cipher of the letters. On the other hand, if the letters are carried as labels on nodes of topologically isomorphic unordered trees, determining if a substitution exists is referred to as marked tree isomorphism problem in the literature and has been show to be as hard as graph isomorphism. While the left-to-right direction provides the cipher of letters in the case of linear messages, if the messages are carried by unordered trees, the cipher is given by a tree isomorphism. The number of isomorphisms between two trees is roughly exponential in the size of the trees, which makes the problem of finding a cipher difficult by exhaustive search. This paper presents a method that aims to break the combinatorics of the isomorphisms search space. We show that in a linear time (in the size of the trees), we reduce the cardinality of this space by an exponential factor on average.

cs.DM↗

arXiv2020

The Weight Function in the Subtree Kernel is Decisive

Tree data are ubiquitous because they model a large variety of situations, e.g., the architecture of plants, the secondary structure of RNA, or the hierarchy of XML files. Nevertheless, the analysis of these non-Euclidean data is difficult per se. In this paper, we focus on the subtree kernel that is a convolution kernel for tree data introduced by Vishwanathan and Smola in the early 2000's. More precisely, we investigate the influence of the weight function from a theoretical perspective and in real data applications. We establish on a 2-classes stochastic model that the performance of the subtree kernel is improved when the weight of leaves vanishes, which motivates the definition of a new weight function, learned from the data and not fixed by the user as usually done. To this end, we define a unified framework for computing the subtree kernel from ordered or unordered trees, that is particularly suitable for tuning parameters. We show through eight real data classification problems the great efficiency of our approach, in particular for small datasets, which also states the high importance of the weight function. Finally, a visualization tool of the significant features is derived.

stat.ML↗