Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings

We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretation of the number wall as an aperiodic plane tiling

A study and experiment plan for digital mobile communication via satellite

The viability of mobile communications is examined within the context of a frequency division multiple access, single channel per carrier satellite system emphasizing digital techniques to serve a large population of users. The intent is to provide the mobile users with a grade of service consistant with the requirements for remote, rural (perhaps emergency) voice communications, but which approaches toll quality speech. A traffic model is derived on which to base the determination of the required maximum number of satellite channels to provide the anticipated level of service. Various voice digitalization and digital modulation schemes are reviewed along with a general link analysis of the mobile system. Demand assignment multiple access considerations and analysis tradeoffs are presented. Finally, a completed configuration is described

Sparse approaches for the exact distribution of patterns in long state sequences generated by a Markov source

We present two novel approaches for the computation of the exact distribution of a pattern in a long sequence. Both approaches take into account the sparse structure of the problem and are two-part algorithms. The first approach relies on a partial recursion after a fast computation of the second largest eigenvalue of the transition matrix of a Markov chain embedding. The second approach uses fast Taylor expansions of an exact bivariate rational reconstruction of the distribution. We illustrate the interest of both approaches on a simple toy-example and two biological applications: the transcription factors of the Human Chromosome 5 and the PROSITE signatures of functional motifs in proteins. On these example our methods demonstrate their complementarity and their hability to extend the domain of feasibility for exact computations in pattern problems to a new level

Revisiting LFSMs

Linear Finite State Machines (LFSMs) are particular primitives widely used in information theory, coding theory and cryptography. Among those linear automata, a particular case of study is Linear Feedback Shift Registers (LFSRs) used in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSRs could be seen as particular LFSMs without inputs. In this paper, we first recall the description of LFSMs using traditional matrices representation. Then, we introduce a new matrices representation with polynomial fractional coefficients. This new representation leads to sparse representations and implementations. As direct applications, we focus our work on the Windmill LFSRs case, used for example in the E0 stream cipher and on other general applications that use this new representation. In a second part, a new design criterion called diffusion delay for LFSRs is introduced and well compared with existing related notions. This criterion represents the diffusion capacity of an LFSR. Thus, using the matrices representation, we present a new algorithm to randomly pick LFSRs with good properties (including the new one) and sparse descriptions dedicated to hardware and software designs. We present some examples of LFSRs generated using our algorithm to show the relevance of our approach.Comment: Submitted to IEEE-I

Personalized Purchase Prediction of Market Baskets with Wasserstein-Based Sequence Matching

Personalization in marketing aims at improving the shopping experience of customers by tailoring services to individuals. In order to achieve this, businesses must be able to make personalized predictions regarding the next purchase. That is, one must forecast the exact list of items that will comprise the next purchase, i.e., the so-called market basket. Despite its relevance to firm operations, this problem has received surprisingly little attention in prior research, largely due to its inherent complexity. In fact, state-of-the-art approaches are limited to intuitive decision rules for pattern extraction. However, the simplicity of the pre-coded rules impedes performance, since decision rules operate in an autoregressive fashion: the rules can only make inferences from past purchases of a single customer without taking into account the knowledge transfer that takes place between customers. In contrast, our research overcomes the limitations of pre-set rules by contributing a novel predictor of market baskets from sequential purchase histories: our predictions are based on similarity matching in order to identify similar purchase habits among the complete shopping histories of all customers. Our contributions are as follows: (1) We propose similarity matching based on subsequential dynamic time warping (SDTW) as a novel predictor of market baskets. Thereby, we can effectively identify cross-customer patterns. (2) We leverage the Wasserstein distance for measuring the similarity among embedded purchase histories. (3) We develop a fast approximation algorithm for computing a lower bound of the Wasserstein distance in our setting. An extensive series of computational experiments demonstrates the effectiveness of our approach. The accuracy of identifying the exact market baskets based on state-of-the-art decision rules from the literature is outperformed by a factor of 4.0.Comment: Accepted for oral presentation at 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD 2019

Fast algorithm for border bases of Artinian Gorenstein algebras

Given a multi-index sequence $$\sigma$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$\sigma$$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne l$I$\sigma of the Hankel operator $H$\sigma associated to $$\sigma$$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra $A$\sigma$$associated to the sequence$$\sigma yields the structure of the terms $\sigma\alpha$for all$$\alpha$ $\in$ N n$. This structure is explicitly given by a border basis of$A$\sigma$$, which is presented as a quotient of the polynomial ring$K[x 1 ,. .. , xn$] by the kernel$I$\sigma$$of the Hankel operator$H$\sigma$$. The algorithm provides generators of$I$\sigma$$constituting a border basis, pairwise orthogonal bases of$A$\sigma$$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm

Expressive Completeness of Existential Rule Languages for Ontology-based Query Answering

Existential rules, also known as data dependencies in Databases, have been recently rediscovered as a promising family of languages for Ontology-based Query Answering. In this paper, we prove that disjunctive embedded dependencies exactly capture the class of recursively enumerable ontologies in Ontology-based Conjunctive Query Answering (OCQA). Our expressive completeness result does not rely on any built-in linear order on the database. To establish the expressive completeness, we introduce a novel semantic definition for OCQA ontologies. We also show that neither the class of disjunctive tuple-generating dependencies nor the class of embedded dependencies is expressively complete for recursively enumerable OCQA ontologies.Comment: 10 pages; the full version of a paper to appear in IJCAI 2016. Changes (regarding to v1): a new reference has been added, and some typos have been correcte