Searcharxiv⌕ Search

arXiv subjects

Saúl A. Blanco

Publications and source records attributed to Saúl A. Blanco.

17 recordsLinked to original sources
arXiv2026

Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate $\mathrm{Sym}_n$. As the largest prefix reversal of length $n$ is always included in a triple, we give a complete solution of the problem when any of the two smallest or the two largest lengths but $n$ are included in a triple of prefix reversals. Moreover, some conditions implying a triple of prefix reversals does not generate $\mathrm{Sym}_n$ are considered. Computational results on the diameter and the girth of some cubic pancake graphs are presented, and conjectures for future research are formulated.

math.CO↗
arXiv2026

On the Laplacian spectral gap of generalized pancake graphs

The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\geq2$, the spectral gap $γ(P(m,n))$ of the normalized Laplacian satisfies $α_m/n\leqγ(P(m,n))\leq1/n$, where $α_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\geq2$, $γ(P(m,n))$ is $Θ_m(1/n)$ as $n\to\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\geq2$, we also establish that $γ(P(m,n))$ is $Θ_n(m^{-2})$ as $m\to\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.

math.CO↗
arXiv2025

On the spectra of prefix-reversal graphs

In this paper, we study spectral properties of prefix-reversal graphs. These graphs are obtained by connecting two elements of $C_m\wr S_n$ via prefix reversals. If $m=1,2$, the corresponding prefix-reversal graphs are the classic pancake and burnt pancake graphs. If $m>2$, then one can consider the directed and undirected versions of these graphs. We prove that the spectrum of the undirected prefix-reversal graph $\mathbb{P}_m(n)$ contains all even integers in the interval $[0,2n]\setminus\{2\lfloor n/2\rfloor\}$ and if $m\equiv0\pmod4$, we then show that the spectrum contains all even integers in $[0,2n]$. In the directed case, we show that the spectrum of the directed prefix-reversal graph $P(m,n)$ contains all integers in the interval $[0,n]\setminus\{\lfloor n/2\rfloor\}$. As a consequence, we show that in either case, the prefix-reversal graphs have a small spectral gap.

math.CO↗
arXiv2025

Multi-Agent Best Arm Identification in Stochastic Linear Bandits

We study the problem of collaborative best-arm identification in stochastic linear bandits under a fixed-budget scenario. In our learning model, we first consider multiple agents connected through a star network, interacting with a linear bandit instance in parallel. We then extend our analysis to arbitrary network topologies. The objective of the agents is to collaboratively identify the best arm of the given bandit instance with the help of a central server while minimizing the probability of error in best arm estimation. To this end, we propose two algorithms, MaLinBAI-Star and MaLinBAI-Gen for star networks and networks with arbitrary structure, respectively. Both algorithms utilize the technique of G-optimal design along with the successive elimination based strategy where agents share their knowledge through a central server at each communication round. We demonstrate, both theoretically and empirically, that our algorithms achieve exponentially decaying probability of error in the allocated time budget. Furthermore, experimental results on both synthetic and real-world data validate the effectiveness of our algorithms over the state-of-the art existing multi-agent algorithms.

cs.LG↗
arXiv2025

A categorical approach to additive combinatorics

Motivated by the definition of Freiman homomorphism, we explore the possibilities of formulating some basic notions and techniques of additive combinatorics in a categorical language. We show that additive sets and Freiman homomorphisms form a category and we study several limit and colimit constructions in this, and in an interesting subcategory of this category. Moreover, we study the additive structure of these (co)limit objects using additive doubling constant. We relate this category to that of finite sets and mappings, and that of abelian groups and group homomorphisms. We show that the Konyagin and Lev result on the existence of universal ambient groups is an instance of adjunction

math.CO↗
arXiv2024

Bounds on the genus for 2-cell embeddings of prefix-reversal graphs

In this paper, we provide bounds for the genus of the pancake graph $\mathbb{P}_n$, burnt pancake graph $\mathbb{BP}_n$, and undirected generalized pancake graph $\mathbb{P}_m(n)$. Our upper bound for $\mathbb{P}_n$ is sharper than the previously-known bound, and the other bounds presented are the first of their kind. Our proofs are constructive and rely on finding an appropriate rotation system (also referred to in the literature as Edmonds' permutation technique) where certain cycles in the graphs we consider become boundaries of regions of a 2-cell embedding. A key ingredient in the proof of our bounds for the genus $\mathbb{P}_n$ and $\mathbb{BP}_n$ is a labeling algorithm of their vertices that allows us to implement rotation systems to bound the number of regions of a 2-cell embedding of said graphs. All of our bounds are asymptotically tight; in particular, the genus of $\mathbb{P}_m(n)$ is $Θ(m^nnn!)$ for all $m\geq1$ and $n\geq2$.

math.CO↗
arXiv2024

Some integer values in the spectra of burnt pancake graphs

The burnt pancake graph, denoted by $\mathbb{BP}_n$, is formed by connecting signed permutations via prefix reversals. Here, we discuss some spectral properties of $\mathbb{BP}_n$. More precisely, we prove that the adjacency spectrum of $\mathbb{BP}_n$ contains all integer values in the set $\{0, 1, \ldots, n\}\setminus\{\left\lfloor n/2 \right\rfloor\}$.

math.CO↗
arXiv2023

Lengths of Cycles in Generalized Pancake Graphs

In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group $S(m,n)$, generated by prefix reversals. The generalized symmetric group $S(m,n)$ is the wreath product of the cyclic group of order $m$ and the symmetric group of order $n!$. Our main focus is the underlying \emph{undirected} graphs, denoted by $\mathbb{P}_m(n)$. In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, $\mathbb{P}_3(n)$ has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, $\mathbb{P}_4(n)$ has all the even-length cycles. We utilize these results as base cases and show that if $m>2$ is even, $\mathbb{P}_m(n)$ has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when $m>2$ is odd, $\mathbb{P}_m(n)$ has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of $\mathbb{P}_m(n)$ is $\min\{m,6\}$ if $m\geq3$, thus complementing the known results for $m=1,2.$

cs.DM↗
arXiv2023

A Novel Pseudo-Random Number Generator Based on Multi-Objective Optimization for Image-Cryptographic Applications

Pseudo-random number generators (PRNGs) play an important role to ensure the security and confidentiality of image cryptographic algorithms. Their primary function is to generate a sequence of numbers that possesses unpredictability and randomness, which is crucial for the algorithms to work effectively and provide the desired level of security. However, traditional PRNGs frequently encounter limitations like insufficient randomness, predictability, and vulnerability to cryptanalysis attacks. To overcome these limitations, we propose a novel method namely an elliptic curve genetic algorithm (ECGA) for the construction of an image-dependent pseudo-random number generator (IDPRNG) that merges elliptic curves (ECs) and a multi-objective genetic algorithm (MOGA). The ECGA consists of two primary stages. First, we generate an EC-based initial sequence of random numbers using pixels of a plain-image and parameters of an EC, that depart from traditional methods of population initialization. In our proposed approach, the image itself serves as the seed for the initial population in the genetic algorithm optimization, taking into account the image-dependent nature of cryptographic applications. This allows the PRNG to adapt its behavior to the unique characteristics of the input image, leading to enhanced security and improved resistance against differential attacks. Furthermore, the use of a good initial population reduces the number of generations required by a genetic algorithm, which results in decreased computational cost. In the second stage, we use well-known operations of a genetic algorithm to optimize the generated sequence by maximizing a multi-objective fitness function that is based on both the information entropy and the period of the PRNG. By combining elliptic curves and genetic algorithms, we enhance the randomness and security of the ECGA.

cs.CR↗
arXiv2023

An Algorithm to Enumerate Grid Signed Permutation Classes

In this paper, we present an algorithm that enumerates a certain class of signed permutations, referred to as grid signed permutation classes. In the case of permutations, the corresponding grid classes are of interest because they are equivalent to the permutation classes that can be enumerated by polynomials. Furthermore, we apply our results to genome rearrangements and establish that the number of signed permutations with fixed prefix reversal and reversal distance is given by polynomials that can be computed by our algorithm.

math.CO↗
arXiv2022

High-Accuracy Machine Learning Techniques for Functional Connectome Fingerprinting and Cognitive State Decoding

The human brain is a complex network comprised of functionally and anatomically interconnected brain regions. A growing number of studies have suggested that empirical estimates of brain networks may be useful for discovery of biomarkers of disease and cognitive state. A prerequisite for realizing this aim, however, is that brain networks also serve as reliable markers of an individual. Here, using Human Connectome Project data, we build upon recent studies examining brain-based fingerprints of individual subjects and cognitive states based on cognitively-demanding tasks that assess, for example, working memory, theory of mind, and motor function. Our approach achieves accuracy of up to 99\% for both identification of the subject of an fMRI scan, and for classification of the cognitive state of a previously-unseen subject in a scan. More broadly, we explore the accuracy and reliability of five different machine learning techniques on subject fingerprinting and cognitive state decoding objectives, using functional connectivity data from fMRI scans of a high number of subjects (865) across a number of cognitive states (8). These results represent an advance on existing techniques for functional connectivity-based brain fingerprinting and state decoding. Additionally, 16 different pre-processing pipelines are compared in order to characterize the effects of different aspects of the production of functional connectomes (FCs) on the accuracy of subject and task classification, and to identify possible confounds.

q-bio.NC↗
arXiv2020

Effects of Human vs. Automatic Feedback on Students' Understanding of AI Concepts and Programming Style

The use of automatic grading tools has become nearly ubiquitous in large undergraduate programming courses, and recent work has focused on improving the quality of automatically generated feedback. However, there is a relative lack of data directly comparing student outcomes when receiving computer-generated feedback and human-written feedback. This paper addresses this gap by splitting one 90-student class into two feedback groups and analyzing differences in the two cohorts' performance. The class is an intro to AI with programming HW assignments. One group of students received detailed computer-generated feedback on their programming assignments describing which parts of the algorithms' logic was missing; the other group additionally received human-written feedback describing how their programs' syntax relates to issues with their logic, and qualitative (style) recommendations for improving their code. Results on quizzes and exam questions suggest that human feedback helps students obtain a better conceptual understanding, but analyses found no difference between the groups' ability to collaborate on the final project. The course grade distribution revealed that students who received human-written feedback performed better overall; this effect was the most pronounced in the middle two quartiles of each group. These results suggest that feedback about the syntax-logic relation may be a primary mechanism by which human feedback improves student outcomes.

cs.HC↗
arXiv2019

On the number of pancake stacks requiring four flips to be sorted

Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$.

cs.DM↗
arXiv2019

Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.

cs.DM↗
arXiv2018

Some relations on prefix reversal generators of the symmetric and hyperoctahedral group

The pancake problem is concerned with sorting a permutation (a stack of pancakes of different diameter) using only prefix reversals (spatula flips). Although the problem description belies simplicity, an exact formula for the maximum number of flips needed to sort $n$ pancakes has been elusive. In this paper we present a different approach to the pancake problem, as a word problem on the symmetric group and hyperoctahedral group. Pancake flips are considered as generators and we study the relations satisfied by them. We completely describe the order of the product of any two of these generators, and provide some partial results on the order of the product of any three generators. Connections to the pancake graph of the hyperoctahedral group are also drawn.

cs.DM↗
arXiv2012

Bandwidth of the product of paths of the same length

In this note we give a numerical expression for the bandwidth $bw(P_{n}^{d})$ of the $d$-product of a path with $n$ edges, $P_{n}^{d}$. We prove that this bandwidth is given by the sum of certain multinomial coefficients. We also show that $bw(P_{n}^{d})$ is bounded above and below by the largest coefficient in the expansion of $(1+x+...+x^{n})^{k}$, with $k\in{d,d+1}$. Moreover, we compare the asymptotic behavior of $bw(P_{n}^{d})$ with the bandwidth of the labeling obtained by ordering the vertices of $P_{n}^{d}$ in lexicographic order.

math.CO↗