arXiv subjects

Carine Pivoteau

Publications and source records attributed to Carine Pivoteau.

10 recordsLinked to original sources

arXiv2026

Leap generators for composition schemes

Leap generators have been introduced in [Duchon et al.'04] for exact-size random generation of structures in a class of the form $\mathcal{C}=\mathrm{Seq}(\mathcal{B})$ (sequence construction), in the supercritical case. We extend these generators to supercritical composition schemes $\mathcal{C}=\mathcal{A}\circ\mathcal{B}$. Compared to the sequence construction, the obtained exact-size random generator for $\mathcal{C}$ still has linear time complexity (under conditions on the sampling complexity in $\mathcal{A}$ and $\mathcal{B}$), but perfect uniformity of the distribution is lost in general. However the distribution on $\mathcal{C}_n$, called leap distribution, is asymptotically uniform, the total variation distance from the uniform distribution being $(c+o(1))n^{-1/2}$ for an explicit constant $c$. These generators are simple to implement and can be applied to several classes of walks and trees, in particular Pólya trees. Leap generators can also be given for certain critical composition schemes, those relating planar map families, where this time the total variation distance to the uniform distribution is $\sim c\,n^{-1/3}$ for an explicit constant $c$.

math.CO↗

arXiv2025

Effective Asymptotics of Combinatorial Systems

Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that allows one to derive precise asymptotic expansions starting from systems of combinatorial equations. In the situation where the combinatorial system involves only cartesian products and disjoint unions, the generating functions satisfy polynomial systems with positivity constraints for which many results and algorithms are known. We extend these results to the general situation. This produces an almost complete algorithmic chain going from combinatorial systems to asymptotic expansions. Thus, it is possible to compute asymptotic expansions of all generating functions produced by the symbolic method of Flajolet and Sedgewick when they have algebraic-logarithmic singularities (which can be decided), under the assumption that Schanuel's conjecture from number theory holds. That conjecture is not needed for systems that do not involve the constructions of sets and cycles.

math.CO↗

arXiv2025

Branch Prediction Analysis of Morris-Pratt and Knuth-Morris-Pratt Algorithms

We analyze the classical Morris-Pratt and Knuth-Morris-Pratt pattern matching algorithms through the lens of computer architecture, investigating the impact of incorporating a simple branch prediction mechanism into the model of computation. Assuming a fixed pattern and a random text, we derive precise estimates of the number of mispredictions these algorithms produce using local predictors. Our approach is based on automata theory and Markov chains, providing a foundation for the theoretical analysis of other text algorithms and more advanced branch prediction strategies.

cs.DS↗

arXiv2024

Record-biased permutations and their permuton limit

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call \emph{record-biased permutations}. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterize. This model was introduced in our earlier work [N. Auger, M. Bouvel, C. Nicaud, C. Pivoteau, \emph{Analysis of Algorithms for Permutations Biased by Their Number of Records}, AofA 2016], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

math.PR↗

arXiv2019

On the Worst-Case Complexity of TimSort

TimSort is an intriguing sorting algorithm designed in 2002 for Python, whose worst-case complexity was announced, but not proved until our recent preprint. In fact, there are two slightly different versions of TimSort that are currently implemented in Python and in Java respectively. We propose a pedagogical and insightful proof that the Python version runs in $\mathcal{O}(n\log n)$. The approach we use in the analysis also applies to the Java version, although not without very involved technical details. As a byproduct of our study, we uncover a bug in the Java implementation that can cause the sorting method to fail during the execution. We also give a proof that Python's TimSort running time is in $\mathcal{O}(n + n\log ρ)$, where $ρ$ is the number of runs (i.e. maximal monotonic sequences), which is quite a natural parameter here and part of the explanation for the good behavior of TimSort on partially sorted inputs.

cs.DS↗

arXiv2016

An algorithm computing combinatorial specifications of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.

math.CO↗

arXiv2016

Analysis of Algorithms for Permutations Biased by Their Number of Records

The topic of the article is the parametric study of the complexity of algorithms on arrays of pairwise distinct integers. We introduce a model that takes into account the non-uniformness of data, which we call the Ewens-like distribution of parameter $θ$ for records on permutations: the weight $θ^r$ of a permutation depends on its number $r$ of records. We show that this model is meaningful for the notion of presortedness, while still being mathematically tractable. Our results describe the expected value of several classical permutation statistics in this model, and give the expected running time of three algorithms: the Insertion Sort, and two variants of the Min-Max search.

cs.DM↗

arXiv2012

Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations

We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the constructible classes of Flajolet and Sedgewick with Joyal's species theory. We extend the implicit species theorem to structures of size zero. A quadratic iterative Newton method is shown to solve well-founded systems combinatorially. From there, truncations of the corresponding generating series are obtained in quasi-optimal complexity. This iteration transfers to a numerical scheme that converges unconditionally to the values of the generating series inside their disk of convergence. These results provide important subroutines in random generation. Finally, the approach is extended to combinatorial differential systems.

math.CO↗

arXiv2012

Combinatorial specification of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positiveand algebraic. It also yields a uniform random sampler of permutations in C. The method presentedis fully algorithmic.

math.CO↗

arXiv2009

Random sampling of plane partitions

This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for $(a\times b)$-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

math.CO↗