Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions

In a 2-dimensional (2D) pattern matching problem, the text is arranged as a matrix 𝖬[1..n, 1..n] and consists of N = n × n symbols drawn from alphabet set Σ of size σ. The query consists of a m × m square matrix 𝖯[1..m, 1..m] drawn from the same alphabet set Σ and the task is to find all the locations in 𝖬 where 𝖯 appears as a (contiguous) submatrix. The patterns can be of any size, but as long as they are square in shape data structures like suffix trees and suffix array exist [Raffaele Giancarlo, 1995; Dong Kyue Kim et al., 1998] for the task of efficient pattern matching. These are essentially 2D counterparts of classic suffix trees and arrays known for traditional 1-dimensional (1D) pattern matching. They work based on linearization of 2D suffixes which would preserve the prefix match property (i.e., every pattern match is a prefix of some suffix). The main limitation of the suffix trees and the suffix arrays (in 1D) was their space utilization of O(N log N) bits, where N is the size of the text. This was suboptimal compared to Nlog σ bits of space, which is information theoretic optimal for the text. With the advent of the field of succinct/compressed data structures, it was possible to develop compressed variants of suffix trees and array based on Burrows-Wheeler Tansform and LF-mapping (or Φ function) [Roberto Grossi and Jeffrey Scott Vitter, 2005; Paolo Ferragina and Giovanni Manzini, 2005; Kunihiko Sadakane, 2007]. These data structures indeed achieve O(N log σ) bits of space or better. This gives rise to the question: analogous to 1D case, can we design a succinct or compressed index for 2D pattern matching? Can there be a 2D compressed suffix tree? Are there analogues of Burrows-Wheeler Transform or LF-mapping? The problem has been acknowledged for over a decade now and there have been a few attempts at applying Φ function [Ankur Gupta, 2004] and achieving entropy based compression [Veli Mäkinen and Gonzalo Navarro, 2008]. However, achieving the complexity breakthrough akin to 1D case has yet to be found. In this paper, we still do not know how to answer suffix array queries in O(N log σ) bits of space - which would have led to efficient pattern matching. However, for the first time, we show an interesting result that it is indeed possible to compute inverse suffix array (ISA) queries in near compact space in O(polylog n) time. Our 2D succinct text index design is based on two 1D compressed suffix trees and it takes O(N log log N + N logσ) bits of space which is much smaller than its naive design that takes O(N log N) bits. Although the main problem is still evasive, this index gives a hope on the existence of a full 2D succinct index with all functionalities similar to that of 1D case.

The field of stringology studies algorithms and data structures used for processing strings efficiently. The goal of this thesis is to investigate 2-dimensional (2D) variants of some fundamental string problems, including \textit{Exact Pattern Matching} and \textit{Longest Common Substring}. In the 2D pattern matching problem, we are given a matrix $\M[1\dd n,1\dd n]$ that consists of $N = n \times n$ symbols drawn from an alphabet $\Sigma$ of size $\sigma$. The query consists of a $ m \times m$ square matrix $\PP[1\dd m, 1\dd m]$ drawn from the same alphabet, and the task is to find all the locations of $\PP$ in $\M$. For such square patterns, data structures such as suffix trees and suffix arrays exist for the task of efficient pattern matching. However, a suffix tree occupies $O(N \log N)$ bits, which is significantly more than that of the original text's size of $N\log \sigma$ bits. Therefore, the design of compressed data structures, that supports pattern matching queries efficiently and occupies space close to the original text's size, is imperative. In this thesis, we show an interesting result by designing a compact text index of size $O(N \log\log N + N \log\sigma)$ bits that at least supports efficient inverse suffix array queries. Although, the question of designing a compressed text index that would lead to efficient pattern matching is still evasive, this index gives a hope on the existence of a full 2D compressed text index with all functionalities similar to that of 1D case. On the other hand, the Longest Common 2D substring problem consists of two 2D strings (matrices), and the task is to report the size of the longest common 2D substring (submatrix) of these 2D strings. It is interesting to know if there exists a sublinear-time algorithm for solving this task. We answer this question positively by presenting a sublinear-time \textit{quantum} algorithm. In addition to this, we prove that any quantum algorithm requires at least $\tilde{\Omega}(N^{2/3})$ time to solve this problem.

Efficient pattern matching is fundamental for practical term rewrite engines. By preprocessing the given patterns into a finite deterministic automaton the matching patterns can be decided in a single traversal of the relevant parts of the input term. Most automaton-based techniques are restricted to linear patterns, where each variable occurs at most once, and require an additional post-processing step to check so-called variable consistency. However, we can show that interleaving the variable consistency and pattern matching phases can reduce the number of required steps to find a match all matches. Therefore, we take the existing adaptive pattern matching automata as introduced by Sekar et al and extend it these with consistency checks. We prove that the resulting deterministic pattern matching automaton is correct, and show that its evaluation depth is can be shorter than two-phase approaches.

Efficient pattern matching is fundamental for practical term rewrite engines. By preprocessing the given patterns into a finite deterministic automaton the matching patterns can be decided in a single traversal of the relevant parts of the input term. Most automaton-based techniques are restricted to linear patterns, where each variable occurs at most once, and require an additional post-processing step to check so-called variable consistency. However, we can show that interleaving the variable consistency and pattern matching phases can reduce the number of required steps to find all matches. Therefore, we take the existing adaptive pattern matching automata as introduced by Sekar et al and extend these with consistency checks. We prove that the resulting deterministic pattern matching automaton is correct, and show several examples where some reduction can be achieved.

Macaulay is a system for computing in algebraic geometry and cummutative algebra; it is capable of a variety of computations which are tedious or impossible to perform by hand. The primitive types in the system are polynomials, matrices, ideals, polynomial rings, modules, maps between rings, and complexes of modules. The system performs algebraic manipulation on objects of these types. The possible manipulations include the computation of standard (Gröbner) bases, modules of syzygies, finite free resolutions, Hilbert polynomials and functions. Using these basic operations, a variety of derived operations are possible, such as projections, ideal intersections, and the computation of coherent sheaf cohomology groups. The algorithm used for constructing standard (Gröbner) bases and syzygies is described in ([Buc76], [Zac78], [Sch80], [Bay82], [BaSt86a]).

Computer Algebra and Algebraic. Geometry—Achievements and Perspectives

Efficient pattern matching in compressed text via bit-parallel word-based encoding

This is a unique book which attempts to present the many and varied scientific models used to study matter along with an extensive collection of specific types of matter upon which these models can be brought to bear. The language and tone of the writing are both interesting and thoroughly scientific, and yet the author completely eschews the use of equations throughout the book. The topics include many from both physics and chemistry, and the book provides a good introduction to the way in which these two disciplines overlap to form the field of materials science. The order in which the topics are presented is interesting, with roughly the first half of the book putting forth several models invoked in the study of matter: forces or interactions; energy; thermodynamics and entropy; fractals and chaos; atoms, molecules and chemical bonding. The second half of the book then presents the applications of these models to ever more complex types of matter, beginning with electrons and fundamental particles, proceeding through atoms and molecules, and building up eventually to polymers, mixtures, solutions, surfaces and a host of others. The sheer number of examples presented is astounding, and they vary from the more esoteric (like the nucleation of freezing in water droplets by pseudomonad bacteria) to the commonplace (like silly putty). Many of the topics are of current interest, including electrorheological fluids, glass `xerogels', and liquid crystal films as information storage devices. The presentation is consistently interesting (with the help of figures and of cartoon illustrations by Andrew Slocombe), and is further enhanced by the copious chapter endnotes and bibliography. With so many examples about which a reader might like to learn more, these almost always provide an entry point into the relevant literature on the topic. The only thing unclear about this book is its intended audience. The complete avoidance of equations and the numerous everyday examples would tend to indicate that it is aimed at a very general audience. However, the breadth and depth of the topics covered, along with the unabashedly scientific tone of the language, seem better suited for a reader with a background in chemistry or physics. While perhaps not appropriate reading for an introductory-level course for undergraduates, it might be a good choice in an advanced seminar for either physics or chemistry majors with an interest in exploring areas of materials science. Perhaps the greatest strength of the book is its frequent demonstration of the power of using multiple models to examine the same phenomenon. The author is at his best when he reminds us that, `Each model we use provides us information on one aspect of a problem, but we can combine them so that with experience they form an integrated intellectual process'. Finally, as the title may suggest, a central theme throughout the book is the notion that science as a human endeavour is an ongoing, evolving process. The models we have today may not be the models of tomorrow. To quote from the book again, `... the number of possible states of matter is limited only by your ability to perceive them'. Because this book gives us a glimpse into the rapidly changing world of materials science and allows us to see both the variation between and the complementarity of the different models which have evolved, it is a worthy contribution to the literature.

This article shows the experience through Project Based Learning with STEM approach from the area of technology and computer science in vocational middle school students in the tenth and eleventh grades of a public school in Bogota, Colombia. The main objective is to influence students for a greater professional interest in STEM areas (Science, Technology, Engineering and Mathematics), while enhancing other competencies that occur through the STEM approach, such as critical thinking, creativity, communication, digital literacy, autonomy, collaboration, product design and manufacturing, problem-solving among others, using the Project Based Learning Didactic strategy. The results obtained show some motivation in the areas of Science, Mathematics and Technology.

This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed.KeywordsErgodic TheoremComputable FunctionKolmogorov ComplexityRandom StringRandom RealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

The International Conference "Computer Algebra" was held online via Zoom on June 28-29, 2021 and the conference web-site is http://www.ccas.ru/ca/conference. Co-organized by the Dorodnicyn Computing Center of Federal Research Center "Computer Science and Control" of Russian Academy of Science and the Peoples' Friendship University of Russia, the conference was devoted to inspiring discussions on computer algebra and related topics. Researchers from different countries presented talks on their latest research work. This is the fourth edition of the conference, and the previous three were in 2016, 2017 and 2019, respectively.

Generating interest in specialized areas of Computer Science (CS) is one of the goals of the department of Computer and Information Science at Spelman College as with most departments. Achieving this goal in a new, exciting, and innovative manner provided the inspiration to establish the Spelman College Computer Science Olympiad (SC CS Olympiad). The SC CS Olympiad is patterned after the Olympics athletic event. Students participate in the Olympiad as a part of a team as with the Olympics. In addition, there are several events for competition as there are in the Olympics. The events are designed to expose students to the interesting breadth of CS over several days. In this paper, the events are in the following categories: Cryptography, Robotics, Website Design, Hardware and Software Integration, and Programming. Teams use their CS knowledge and problem-solving skills to complete hands-on exercises in each area. Each teams receives points based on the quality of their results from the exercise. In this paper, we present the implementation, results, and future directions of the Spelman College Computer Science Olympiad.

DFA minimization is a central problem in algorithm design and is based on the notion of DFA equivalence: Two DFA's are equivalent if and only if they accept the same set of strings. In this paper, we propose a new notion of DFA equivalence (that we call weak-equivalence):We say that two DFA's are weakly equivalent if they both accept the same number of strings of length k for every k. The motivation for this problem is as follows. A large number of counting problems can be solved by encoding the combinatorial objects we want to count as strings over a finite alphabet. If the collection of encoded strings is accepted by a DFA, then standard algorithms from computational linear algebra can be used to solve the counting problem efficiently. When applying this approach to largwe-scale applications, the bottleneck is the space complexity since the computation involves a matrix of order k × k if k is the size of the underlying DFA M. This leads to the natural question: Is there a smaller DFA that is weakly equivalent to M? We present an algorithm of time complexity O(k2) to find a compact DFA equivalent to a given DFA. We illustrate, in the case of tiling problem, that our algorithm reduces a (strongly minimal) DFA by a factor close to 2.

Discrete mathematics has a central importance in the fundamental parts of the theory of computers. From the development of powerful algorithms to graph theory and optimization of networks discrete mathematics plays a role in the analysis and solution of computational problems. This paper aims to look at how discrete mathematics encompasses algorithms and graph theory within today’s real-world problems. As such, the study looks at how the concept of calculus in sets, relations, combinatory, and logic as fundamental blocks towards algorithm development or optimization forms the basis for growth within the field of computer science. Thus, discussing the recent changes and case studies the paper demonstrates that discrete mathematics is indeed a vital domain in computational research and its advancements’ applicability in such areas of study as data structures, network analysis, and computational complexity.