Exact-size Sampling for Motzkin Trees in Linear Time via Boltzmann Samplers and Holonomic Specification

International audienceBoltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by Rémy's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average

Constructions par greffe, combinatoire analytique et génération analytique

Analytic combinatorics is a field which consist in applying methods from complex ana- lysis to combinatorial classes in order to obtain results on their asymptotic properties. We use for that specifications, which are a way to formalise the (often recursive) structure of the objects. In this thesis, we mainly devote ourselves to find new specifications for some combinatorial classes, in order to then apply more effective enumerative or random sampling methods. Indeed, for one combinatorial class several different specifications, based on different decompositions, may exist, making the classical methods - of asymptotic enu- meration or random sampling - more or less adapted. The first set of presented results focuses on Rémy’s algorithm and its underlying holonomic specification, based on a grafting operator. We develop a new and more efficient random sampler of binary trees and a random sampler of Motzkin trees based on the same principle. We then address some question relative to the study of subclasses of λ-terms. Finally, we present two other sets of results, on automatic specification of trees where occurrences of a given pattern are marked and on the asymptotic behaviour and the random sampling of digitally convex polyominoes. In every case, the new specifications give access to methods which could not be applied previously and lead to numerous new results.La combinatoire analytique est un domaine qui consiste à appliquer des méthodes issues de l’analyse complexe à des classes combinatoires afin d’obtenir des résultats sur leurs propriétés asymptotiques. On utilise pour cela des spécifications, qui sont une manière de formaliser la structure (souvent récursive) des objets. Dans cette thèse, nous nous attachons principalement à trouver des nouvelles spécifications pour certaines classes combinatoires, afin de pouvoir ensuite y appliquer des méthodes efficaces d’énumération ou de génération aléatoire. En effet, pour une même classe combinatoire il peut exister différentes spécifications, basées sur des décompositions différentes, rendant les méthodes classiques d’énumération asymptotique et de génération aléatoire plus ou moins adaptées. Le premier volet de résultats présentés concerne l’algorithme de Rémy et la spécification holonome qui y est sous-jacente, basée sur un opérateur de greffe. On y développe un nouvel algorithme, plus efficace, de génération aléatoire d’arbres binaires et un générateur aléatoire d’arbres de Motzkin basé sur le même principe. Nous abordons ensuite des questions relatives à l’étude de sous-classes de λ-termes. Enfin, nous présentons deux autres ensembles de résultats, sur la spécification automatique d’arbres où les occurrences d’un motif donné sont marquées et sur le comportement asymptotique et la génération aléatoire de polyominos digitalement convexes. Dans tous les cas, les nouvelles spécifications obtenues donnent accès à des méthodes qui ne pouvaient pas être utilisées jusque là et nous permettent d’obtenir de nombreux nouveaux résultats

Functional mechanisms underlying pleiotropic risk alleles at the 19p13.1 breast-ovarian cancer susceptibility locus

A locus at 19p13 is associated with breast cancer (BC) and ovarian cancer (OC) risk. Here we analyse 438 SNPs in this region in 46,451 BC and 15,438 OC cases, 15,252 BRCA1 mutation carriers and 73,444 controls and identify 13 candidate causal SNPs associated with serous OC (P=9.2 × 10-20), ER-negative BC (P=1.1 × 10-13), BRCA1-associated BC (P=7.7 × 10-16) and triple negative BC (P-diff=2 × 10-5). Genotype-gene expression associations are identified for candidate target genes ANKLE1 (P=2 × 10-3) and ABHD8 (P<2 × 10-3). Chromosome conformation capture identifies interactions between four candidate SNPs and ABHD8, and luciferase assays indicate six risk alleles increased transactivation of the ADHD8 promoter. Targeted deletion of a region containing risk SNP rs56069439 in a putative enhancer induces ANKLE1 downregulation; and mRNA stability assays indicate functional effects for an ANKLE1 3′-UTR SNP. Altogether, these data suggest that multiple SNPs at 19p13 regulate ABHD8 and perhaps ANKLE1 expression, and indicate common mechanisms underlying breast and ovarian cancer risk

The FANCM:p.Arg658* truncating variant is associated with risk of triple-negative breast cancer

Abstract: Breast cancer is a common disease partially caused by genetic risk factors. Germline pathogenic variants in DNA repair genes BRCA1, BRCA2, PALB2, ATM, and CHEK2 are associated with breast cancer risk. FANCM, which encodes for a DNA translocase, has been proposed as a breast cancer predisposition gene, with greater effects for the ER-negative and triple-negative breast cancer (TNBC) subtypes. We tested the three recurrent protein-truncating variants FANCM:p.Arg658*, p.Gln1701*, and p.Arg1931* for association with breast cancer risk in 67,112 cases, 53,766 controls, and 26,662 carriers of pathogenic variants of BRCA1 or BRCA2. These three variants were also studied functionally by measuring survival and chromosome fragility in FANCM−/− patient-derived immortalized fibroblasts treated with diepoxybutane or olaparib. We observed that FANCM:p.Arg658* was associated with increased risk of ER-negative disease and TNBC (OR = 2.44, P = 0.034 and OR = 3.79; P = 0.009, respectively). In a country-restricted analysis, we confirmed the associations detected for FANCM:p.Arg658* and found that also FANCM:p.Arg1931* was associated with ER-negative breast cancer risk (OR = 1.96; P = 0.006). The functional results indicated that all three variants were deleterious affecting cell survival and chromosome stability with FANCM:p.Arg658* causing more severe phenotypes. In conclusion, we confirmed that the two rare FANCM deleterious variants p.Arg658* and p.Arg1931* are risk factors for ER-negative and TNBC subtypes. Overall our data suggest that the effect of truncating variants on breast cancer risk may depend on their position in the gene. Cell sensitivity to olaparib exposure, identifies a possible therapeutic option to treat FANCM-associated tumors

Functional mechanisms underlying pleiotropic risk alleles at the 19p13.1 breast-ovarian cancer susceptibility locus

A locus at 19p13 is associated with breast cancer (BC) and ovarian cancer (OC) risk. Here we analyse 438 SNPs in this region in 46,451 BC and 15,438 OC cases, 15,252 BRCA1 mutation carriers and 73,444 controls and identify 13 candidate causal SNPs associated with serous OC (P = 9.2 x 10(-20)), ER-negative BC (P = 1.1 x 10(-13)), BRCA1-associated BC (P = 7.7 x 10(-16)) and triple negative BC (P-diff = 2 x 10(-5)). Genotype-gene expression associations are identified for candidate target genes ANKLE1 (P = 2 x 10(-3)) and ABHD8 (PPeer reviewe

Graft reconstruction, analytic combinatorics and analytical generation

La combinatoire analytique est un domaine qui consiste à appliquer des méthodes issues de l’analyse complexe à des classes combinatoires afin d’obtenir des résultats sur leurs propriétés asymptotiques. On utilise pour cela des spécifications, qui sont une manière de formaliser la structure (souvent récursive) des objets. Dans cette thèse, nous nous attachons principalement à trouver des nouvelles spécifications pour certaines classes combinatoires, afin de pouvoir ensuite y appliquer des méthodes efficaces d’énumération ou de génération aléatoire. En effet, pour une même classe combinatoire il peut exister différentes spécifications, basées sur des décompositions différentes, rendant les méthodes classiques d’énumération asymptotique et de génération aléatoire plus ou moins adaptées. Le premier volet de résultats présentés concerne l’algorithme de Rémy et la spécification holonome qui y est sous-jacente, basée sur un opérateur de greffe. On y développe un nouvel algorithme, plus efficace, de génération aléatoire d’arbres binaires et un générateur aléatoire d’arbres de Motzkin basé sur le même principe. Nous abordons ensuite des questions relatives à l’étude de sous-classes de λ-termes. Enfin, nous présentons deux autres ensembles de résultats, sur la spécification automatique d’arbres où les occurrences d’un motif donné sont marquées et sur le comportement asymptotique et la génération aléatoire de polyominos digitalement convexes. Dans tous les cas, les nouvelles spécifications obtenues donnent accès à des méthodes qui ne pouvaient pas être utilisées jusque là et nous permettent d’obtenir de nombreux nouveaux résultats.Analytic combinatorics is a field which consist in applying methods from complex ana- lysis to combinatorial classes in order to obtain results on their asymptotic properties. We use for that specifications, which are a way to formalise the (often recursive) structure of the objects. In this thesis, we mainly devote ourselves to find new specifications for some combinatorial classes, in order to then apply more effective enumerative or random sampling methods. Indeed, for one combinatorial class several different specifications, based on different decompositions, may exist, making the classical methods - of asymptotic enu- meration or random sampling - more or less adapted. The first set of presented results focuses on Rémy’s algorithm and its underlying holonomic specification, based on a grafting operator. We develop a new and more efficient random sampler of binary trees and a random sampler of Motzkin trees based on the same principle. We then address some question relative to the study of subclasses of λ-terms. Finally, we present two other sets of results, on automatic specification of trees where occurrences of a given pattern are marked and on the asymptotic behaviour and the random sampling of digitally convex polyominoes. In every case, the new specifications give access to methods which could not be applied previously and lead to numerous new results