Combinatorial Structure Research Articles

Lossless Approximate Pattern Matching: Automated Design of Efficient Search Schemes.

This study introduces a pioneering approach to automate the creation of search schemes for lossless approximate pattern matching. Search schemes are combinatorial structures that define a series of searches over a partitioned pattern. Each search specifies the processing order of these parts and the cumulative lower and upper bounds on the number of errors in each part of the pattern. Together, these searches ensure the identification of all approximate occurrences of a search pattern within a predefined limit of k errors. While existing literature offers designed schemes for up to k = 4 errors, designing search schemes for larger k values incurs escalating computational costs. Our method integrates a greedy algorithm and a novel Integer Linear Programming (ILP) formulation to design efficient search schemes for up to k = 7 errors. Comparative analyses demonstrate the superiority of our ILP-optimal schemes over alternative strategies in both theoretical and practical contexts. Additionally, we propose a dynamic scheme selection technique tailored to specific search patterns, further enhancing efficiency. Combined, this yields runtime reductions of up to 53% for higher k values. To facilitate search scheme generation, we present Hato, an open-source software tool (AGPL-3.0 license) employing the greedy algorithm and utilizing CPLEX for ILP solving. Furthermore, we introduce Columba 1.2, an open-source lossless read-mapper (AGPL-3.0 license) implemented in C++. Columba surpasses existing state-of-the-art tools by identifying all approximate occurrences of 100,000 Illumina reads (150 bp) in the human reference genome within 24 seconds (maximum edit distance of 4) and 75 seconds (maximum edit distance of 6) using a single CPU core. Notably, our study showcases Columba's capability to align 100,000 reads of length 50, with high error rates and up to an edit distance of 7, in a mere 2 hours and 15 minutes. This achievement is unmatched by other lossless aligners, which require over 3 hours for edit distance 5 alignments. Moreover, Columba exhibits a mapping rate four times higher than that of a lossy tool for this dataset.

Low-power coding method in data transmission systems

The object of the study is the Network-on-Chip (NoC) technology, which has become a popular choice for the on-chip communication architecture of modern System-on-Chip (SoC) devices. The subject matter of the article is methods of reducing dissipated power in NoC and SoC. The goal of the work is: development of a low-power coding method that allows for the efficient transmission or storage of information. The following tasks are solved in the article: analysis of classification methods for combinatorial structures, construction a system of typical representatives and analysis of their characteristics. The research methods are based on the use of set theory, system theory and combinatorics. The following results are obtained: analyzed factors that affect the dissipated power, considered principles of constructing energy-efficient codes. It is shown that switching activity significantly affects the total power and one of the effective methods for reducing switching activity during communication between devices or on-chip communication is the use of low-power coding methods. A method of hierarchical classification of unit distance codes and algorithms for solving step-by-step problems have been developed. The method is based on the invariant approach and construction of a system of different representatives. Estimates of their number have been obtained, characteristics have been determined, and catalogs of typical representatives have been formed. Conclusions. The article analyzes factors that affect dissipated power, and considers the principles of constructing energy-efficient codes. A method of hierarchical classification of single distance codes and algorithms for solving step-by-step problems have been developed, and catalogs of typical representatives have been formed. The application of the developed method will allow developers to analyze and select codes with the best properties and, as a result, obtain better results in terms of network delays, energy costs, and other design limitations for computer systems.

Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of <i>n</i> ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when <i>n</i> = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of <i>n</i> > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (<i>n</i> − 1).

Cross‐ratio degrees and triangulations

AbstractThe cross‐ratio degree problem counts configurations of points on with prescribed cross‐ratios. Cross‐ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well‐understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross‐ratio degrees indexed by triangulations of an ‐gon; these degrees are connected to the geometry of the real locus of , and to positive geometry.

An efficient exact method with polynomial time-complexity to achieve [formula omitted]-strong barrier coverage in heterogeneous wireless multimedia sensor networks

Barrier coverage in Wireless Sensor Networks (WSNs) plays a pivotal role in surveillance and security applications. It serves as a fundamental mechanism for identifying and detecting potential intruders who endeavor to infiltrate a sensor barrier. Achieving k-strong barrier coverage is a vital indicator of a WSN’s capability to detect unauthorized intrusions. This paper establishes efficient k-strong barrier coverage in hybrid wireless multimedia sensor networks, referred to as MMS-KSB. The primary goal is to identify a minimal number of mobile sensors to obtain k-barrier coverage. Exhibiting the combinatorial structure, previous research on building k-strong barrier has failed to provide a polynomial time solution for the considered problem and resorted to approximation algorithms. We, therefore, introduce a precise algorithm, named ExA-KSB, and provide theoretical analysis to substantiate that our proposed method achieves an exact solution with polynomial time complexity. Furthermore, we conduct comprehensive experiments to evaluate the efficacy of our algorithm by comparing it with existing approaches. Numerical results demonstrate that ExA-KSB surpasses previous algorithms, and offers superior solution quality with competitive computational efficiency.

Non-commutative probability insights into the double-scaling limit SYK model with constant perturbations: moments, cumulants and q-independence

Extending the results of Wu (2022 J. Phys. A 55 415207), we study the double-scaling limit Sachdev–Ye–Kitaev model with an additional diagonal matrix with a fixed number c of nonzero constant entries θ. This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of c. We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a q~ dependent mixture of independences within the moment formula. The parameter q~ , derived from the q-Ornstein–Uhlenbeck (q-OU) process, controls this transformation. It interpolates between classical independence ( q~=1 ) and Boolean independence ( q~=0 ). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.

Mini-Workshop: Combinatorial and Algebraic Structures in Rough Analysis and Related Fields

Recent years have seen an explosion of algebraic methods to study singular stochastic and rough dynamics. These include developments in geometric rough path theory based on the algebra of words, the introduction of decorated trees in regularity structures, and the recent approach to singular stochastic partial differential equations based on multi-indices. These developments have furthermore led to important links with numerical analysis, machine learning, stochastic quantisation, and the study of symmetries of physical systems. The aim of this mini-workshop was to bring together experts working on these fields using algebraic structures that appear in rough dynamics. The goal was to facilitate the exchange of ideas and encourage further connections to be established.

Construction of Coded Caching Scheme Based on Vector Space over Finite Field

Coded caching technology can better alleviate network traffic congestion. Since many of the centralized coded caching schemes now in use have high subpacketization, which makes scheme implementation more challenging, coded caching schemes with low subpacketization offer a wider range of practical applications. It has been demonstrated that the coded caching scheme can be achieved by creating a combinatorial structure named placement delivery array (PDA). In this work, we employ vector space over a finite field to obtain a class of PDA, calculate its parameters, and consequently achieve a coded caching scheme with low subpacketization. Subsequently, we acquire a new MN scheme and compare it with the new scheme developed in this study. The subpacketization \(F\) of the new scheme has significant advantages. Lastly, the number of users \(K\), cache fraction \(\frac{M}{N}\), and subpacketization \(F\) have advantages to some extent at the expense of partial transmission rate \(R\) when compared to the coded caching scheme in other articles.

Proceedings of the 13th edition of the conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings

Proceedings of the 13th edition of the conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings

Contextual and combinatorial structure in sperm whale vocalisations

Sperm whales (Physeter macrocephalus) are highly social mammals that communicate using sequences of clicks called codas. While a subset of codas have been shown to encode information about caller identity, almost everything else about the sperm whale communication system, including its structure and information-carrying capacity, remains unknown. We show that codas exhibit contextual and combinatorial structure. First, we report previously undescribed features of codas that are sensitive to the conversational context in which they occur, and systematically controlled and imitated across whales. We call these rubato and ornamentation. Second, we show that codas form a combinatorial coding system in which rubato and ornamentation combine with two context-independent features we call rhythm and tempo to produce a large inventory of distinguishable codas. Sperm whale vocalisations are more expressive and structured than previously believed, and built from a repertoire comprising nearly an order of magnitude more distinguishable codas. These results show context-sensitive and combinatorial vocalisation can appear in organisms with divergent evolutionary lineage and vocal apparatus.

Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions:c×⋯×c︸p(total distance):=∑♣1,⋯,♣p∈Zν♣1+⋯+♣p=total distance∏j=1pc(♣j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p=2[11], and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p=3. Next, we prove existence and uniqueness in the general case p≥2, which then covers the remaining cases p≥4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p.

Advances on similarity via transversal intersection of manifolds

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.

An Efficient Image Cryptosystem Utilizing Difference Matrix and Genetic Algorithm.

Aiming at addressing the security and efficiency challenges during image transmission, an efficient image cryptosystem utilizing difference matrix and genetic algorithm is proposed in this paper. A difference matrix is a typical combinatorial structure that exhibits properties of discretization and approximate uniformity. It can serve as a pseudo-random sequence, offering various scrambling techniques while occupying a small storage space. The genetic algorithm generates multiple ciphertext images with strong randomness through local crossover and mutation operations, then obtains high-quality ciphertext images through multiple iterations using the optimal preservation strategy. The whole encryption process is divided into three stages: first, the difference matrix is generated; second, it is utilized for initial encryption to ensure that the resulting ciphertext image has relatively good initial randomness; finally, multiple rounds of local genetic operations are used to optimize the output. The proposed cryptosystem is demonstrated to be effective and robust through simulation experiments and statistical analyses, highlighting its superiority over other existing algorithms.

Labeled sample compression schemes for complexes of oriented matroids

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admit a proper labeled sample compression scheme of size d. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture – one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Discrete model for discontinuous dynamic recrystallisation applied to grain structure evolution inside adiabatic shear bands

Discontinuous dynamic recrystallisation (DDRX) is a well-known phenomenon playing a significant role in the high-temperature processing of metals, including industrial forming and severe plastic deformations. The ongoing discussion on the Zener–Hollomon (Z–H) parameter as a descriptor of materials’ propensity to DDRX and a measure of microstructure homogeneity leaves more questions than answers and prevents its practical application. Most of the existing DDRX models are continuous, and so the geometry and topology of real grain microstructures cannot be considered. The present study uses a fully discrete representation of polycrystalline aluminium alloys as 2D/3D Voronoi space tessellations corresponding to EBSD maps. Such tessellations are geometric realisations of combinatorial structures referred to as polytopal cell complexes. Combining discrete models with FEM LS-Dyna simulations of shock-wave propagation in AA1050 and AW5083 aluminium alloys makes it possible to estimate for the first time the contribution of DDRX to the final material microstructure inside adiabatic shear bands. It is shown that the increase of the initial fraction of high-angle grain boundaries, caused by preliminary deformation, significantly increases the spatial homogeneity and decreases the clustering of DDRX grains. The obtained results contradict the conventional assumption that the microstructures obtained by severe plastic deformation under quasi-static and dynamic deformation conditions are similar due to the similar value of the Z–H parameter: competition between the two recrystallisation mechanisms leads to almost unpredictable final grain structures inside share bands that require further comprehensive experimental studies. This agrees with experimental evidence for high material sensitivity to the Z–H parameter.

On Constructions of GMGDs

Modified group divisible designs MGD ( k , λ , m , n ) are extensively studied because of an intriguing combinatorial structure that they possess and their applications. In this paper, we present a generalization of MGDs called GMGD ( k , λ 1 , λ 2 , m , n ) , and we provide some elementary results and constructions of some special cases of GMGDs. In addition, we show that the necessary conditions are sufficient for the existence of a GMGD ( 3 , λ , 2 λ , m , n ) for any positive integer λ , and a GMGD ( 3 , 2 , 3 , m , n ) . Though not a general result, the construction of a GMGD ( 3 , 3 , 2 , 2 , 6 ) given in the paper is worth mentioning in the abstract. Along with another example of a GMGD ( 3 , 3 , 2 , 2 , 4 ) , and n to t n construction, we have families of GMGD ( 3 , 3 λ , 2 λ , 2 , n ) s for n = 4 t or 6 t when t ≡ 0 , 1 ( mod 3 ) , for any positive integer λ .

Unlabeled sample compression schemes for oriented matroids

A long-standing sample compression conjecture asks to linearly bound the size of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC) dimension of an arbitrary class. In this paper, we explore the rich metric and combinatorial structure of oriented matroids (OMs) to construct proper unlabeled sample compression schemes for the classes of topes of OMs bounded by their VC-dimension. The result extends to the topes of affine OMs, as well as to the topes of the complexes of OMs that possess a corner peeling. The main tool that we use are the solutions of certain oriented matroid programs.

Research orientation and novelty discriminant for new metaheuristic algorithms

The rapid rate of generating a new metaheuristic algorithm almost every month is causing increasing concerns and disputes about their novelty. To stop the disputes and steer algorithm design in a healthy direction, this article presents a discriminant method of novelty and a research orientation for metaheuristic algorithms. The novelty discriminant is implemented by two novel mathematical definitions of homologous algorithms and root algorithms. The two definitions are developed to divide algorithms into two classes according to a discrepancy in whether the reproduction operator of an algorithm is a linear combination of existing operators. Root algorithms are strongly innovative because of the novelty of their reproduction operators. A homologous algorithm is recognized as a novel algorithm only when the practical value and academic significance of the new combinatorial structure of the algorithm’s reproduction operator is clearly highlighted. So a research orientation that the study of a homologous algorithm should focus on how a certain metaphor evokes a new combinatorial structure can be developed. Moreover, numerical experiments should be conducted to analyze the relationship between its search behavior and its new combinatorial structure. Further work can be directed towards studying the systematization of existing knowledge about search behaviors of metaheuristic algorithms.

Lifted algorithms for symmetric weighted first-order model sampling

Weighted model counting (WMC) is the task of computing the weighted sum of all satisfying assignments (i.e., models) of a propositional formula. Similarly, weighted model sampling (WMS) aims to randomly generate models with probability proportional to their respective weights. Both WMC and WMS are hard to solve exactly, falling under the #P-hard complexity class. However, it is known that the counting problem may sometimes be tractable, if the propositional formula can be compactly represented and expressed in first-order logic. In such cases, model counting problems can be solved in time polynomial in the domain size, and are known as domain-liftable. The following question then arises: Is it also the case for WMS? This paper addresses this question and answers it affirmatively. Specifically, we prove the domain-liftability under sampling for the two-variables fragment of first-order logic with counting quantifiers in this paper, by devising an efficient sampling algorithm for this fragment that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of cardinality constraints. To empirically validate our approach, we conduct experiments over various first-order formulas designed for the uniform generation of combinatorial structures and sampling in statistical-relational models. The results demonstrate that our algorithm outperforms a state-of-the-art WMS sampler by a substantial margin, confirming the theoretical results.