arXiv subjects

Bastien Cazaux

Publications and source records attributed to Bastien Cazaux.

10 recordsLinked to original sources

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.

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arXiv2022

Algorithms and Complexity on Indexing Founder Graphs

We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: E.g. Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (MSAs). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free non-elastic founder graph from a gapless MSA, and (parameterized) near-linear time algorithms to construct semi-repeat-free (repeat-free, respectively) elastic founder graphs from general MSAs. Finally, we show that repeat-free elastic founder graphs admit a reduction to Wheeler graphs in polynomial time.

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arXiv2021

Greedy-reduction from Shortest Linear Superstring to Shortest Circular Superstring

A superstring of a set of strings correspond to a string which contains all the other strings as substrings. The problem of finding the Shortest Linear Superstring is a well-know and well-studied problem in stringology. We present here a variant of this problem, the Shortest Circular Superstring problem where the sought superstring is a circular string. We show a strong link between these two problems and prove that the Shortest Circular Superstring problem is NP-complete. Moreover, we propose a new conjecture on the approximation ratio of the Shortest Circular Superstring problem.

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arXiv2021

A Linear Time Algorithm for Constructing Hierarchical Overlap Graphs

The hierarchical overlap graph (HOG) is a graph that encodes overlaps from a given set P of n strings, as the overlap graph does. A best known algorithm constructs HOG in O(||P|| log n) time and O(||P||) space, where ||P|| is the sum of lengths of the strings in P. In this paper we present a new algorithm to construct HOG in O(||P||) time and space. Hence, the construction time and space of HOG are better than those of the overlap graph, which are O(||P|| + n^2).

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arXiv2020

Linear Time Construction of Indexable Founder Block Graphs

We introduce a compact pangenome representation based on an optimal segmentation concept that aims to reconstruct founder sequences from a multiple sequence alignment (MSA). Such founder sequences have the feature that each row of the MSA is a recombination of the founders. Several linear time dynamic programming algorithms have been previously devised to optimize segmentations that induce founder blocks that then can be concatenated into a set of founder sequences. All possible concatenation orders can be expressed as a founder block graph. We observe a key property of such graphs: if the node labels (founder segments) do not repeat in the paths of the graph, such graphs can be indexed for efficient string matching. We call such graphs segment repeat-free founder block graphs. We give a linear time algorithm to construct a segment repeat-free founder block graph given an MSA. The algorithm combines techniques from the founder segmentation algorithms (Cazaux et al. SPIRE 2019) and fully-functional bidirectional Burrows-Wheeler index (Belazzougui and Cunial, CPM 2019). We derive a succinct index structure to support queries of arbitrary length in the paths of the graph. Experiments on an MSA of SAR-CoV-2 strains are reported. An MSA of size $410\times 29811$ is compacted in one minute into a segment repeat-free founder block graph of 3900 nodes and 4440 edges. The maximum length and total length of node labels is 12 and 34968, respectively. The index on the graph takes only $3\%$ of the size of the MSA.

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arXiv2019

Minimum Segmentation for Pan-genomic Founder Reconstruction in Linear Time

Given a threshold $L$ and a set $\mathcal{R} = \{R_1, \ldots, R_m\}$ of $m$ haplotype sequences, each having length $n$, the minimum segmentation problem for founder reconstruction is to partition the sequences into disjoint segments $\mathcal{R}[i_1{+}1,i_2], \mathcal{R}[i_2{+}1, i_3], \ldots, \mathcal{R}[i_{r-1}{+}1, i_r]$, where $0 = i_1 < \cdots < i_r = n$ and $\mathcal{R}[i_{j-1}{+}1, i_j]$ is the set $\{R_1[i_{j-1}{+}1, i_j], \ldots, R_m[i_{j-1}{+}1, i_j]\}$, such that the length of each segment, $i_j - i_{j-1}$, is at least $L$ and $K = \max_j\{ |\mathcal{R}[i_{j-1}{+}1, i_j]| \}$ is minimized. The distinct substrings in the segments $\mathcal{R}[i_{j-1}{+}1, i_j]$ represent founder blocks that can be concatenated to form $K$ founder sequences representing the original $\mathcal{R}$ such that crossovers happen only at segment boundaries. We give an optimal $O(mn)$ time algorithm to solve the problem, improving over earlier $O(mn^2)$. This improvement enables to exploit the algorithm on a pan-genomic setting of haplotypes being complete human chromosomes, with a goal of finding a representative set of references that can be indexed for read alignment and variant calling.

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arXiv2018

Strong link between BWT and XBW via Aho-Corasick automaton and applications to Run-Length Encoding

The boom of genomic sequencing makes compression of set of sequences inescapable. This underlies the need for multi-string indexing data structures that helps compressing the data. The most prominent example of such data structures is the Burrows-Wheeler Transform (BWT), a reversible permutation of a text that improves its compressibility. A similar data structure, the eXtended Burrows-Wheeler Transform (XBW), is able to index a tree labelled with alphabet symbols. A link between a multi-string BWT and the Aho-Corasick automaton has already been found and led to a way to build a XBW from a multi-string BWT. We exhibit a stronger link between a multi-string BWT and a XBW by using the order of the concatenation in the multi-string. This bijective link has several applications: first, it allows to build one data structure from the other; second, it enables one to compute an ordering of the input strings that optimises a Run-Length measure (i.e., the compressibility) of the BWT or of the XBW.

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arXiv2018

Hierarchical Overlap Graph

Given a set of finite words, the Overlap Graph (OG) is a complete weighted digraph where each word is a node and where the weight of an arc equals the length of the longest overlap of one word onto the other (Overlap is an asymmetric notion). The OG serves to assemble DNA fragments or to compute shortest superstrings which are a compressed representation of the input. The OG requires a space is quadratic in the number of words, which limits its scalability. The Hierarchical Overlap Graph (HOG) is an alternative graph that also encodes all maximal overlaps, but uses a space that is linear in the sum of the lengths of the input words. We propose the first algorithm to build the HOG in linear space for words of equal length.

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arXiv2017

The Compressed Overlap Index

For analysing text algorithms, for computing superstrings, or for testing random number generators, one needs to compute all overlaps between any pairs of words in a given set. The positions of overlaps of a word onto itself, or of two words, are needed to compute the absence probability of a word in a random text, or the numbers of common words shared by two random texts. In all these contexts, one needs to compute or to query overlaps between pairs of words in a given set. For this sake, we designed COvI, a compressed overlap index that supports multiple queries on overlaps: like computing the correlation of two words, or listing pairs of words whose longest overlap is maximal among all possible pairs. COvI stores overlaps in a hierarchical and non-redundant manner. We propose an implementation that can handle datasets of millions of words and still answer queries efficiently. Comparison with a baseline solution - called FullAC - relying on the Aho-Corasick automaton shows that COvI provides significant advantages. For similar construction times, COvI requires half the memory FullAC, and still solves complex queries much faster.

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arXiv2016

Read Mapping on de Bruijn graph

Background Next Generation Sequencing (NGS) has dramatically enhanced our ability to sequence genomes, but not to assemble them. In practice, many published genome sequences remain in the state of a large set of contigs. Each contig describes the sequence found along some path of the assembly graph, however, the set of contigs does not record all the sequence information contained in that graph. Although many subsequent analyses can be performed with the set of contigs, one may ask whether mapping reads on the contigs is as informative as mapping them on the paths of the assembly graph. Currently, one lacks practical tools to perform mapping on such graphs. Results Here, we propose a formal definition of mapping on a de Bruijn graph, analyse the problem complexity which turns out to be NP-complete, and provide a practical solution.We propose a pipeline called GGMAP (Greedy Graph MAPping). Its novelty is a procedure to map reads on branching paths of the graph, for which we designed a heuristic algorithm called BGREAT (de Bruijn Graph REAd mapping Tool). For the sake of efficiency, BGREAT rewrites a read sequence as a succession of unitigs sequences. GGMAP can map millions of reads per CPU hour on a de Bruijn graph built from a large set of human genomic reads. Surprisingly, results show that up to 22% more reads can be mapped on the graph but not on the contig set. Conclusions Although mapping reads on a de Bruijn graph is complex task, our proposal offers a practical solution combining efficiency with an improved mapping capacity compared to assembly-based mapping even for complex eukaryotic data. Availability: github.com/Malfoy/BGREAT Keywords: Read mapping; De bruijn graphs; NGS; NP-completeness

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