Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $\epsilon \in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $\epsilon$ and the case that it is at most $a_d \epsilon$, with query complexity and running time $c_d \epsilon^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns

Can fusion coefficients be calculated from the depth rule ?

The depth rule is a level truncation of tensor product coefficients expected to be sufficient for the evaluation of fusion coefficients. We reformulate the depth rule in a precise way, and show how, in principle, it can be used to calculate fusion coefficients. However, we argue that the computation of the depth itself, in terms of which the constraints on tensor product coefficients is formulated, is problematic. Indeed, the elements of the basis of states convenient for calculating tensor product coefficients do not have a well-defined depth! We proceed by showing how one can calculate the depth in an `approximate' way and derive accurate lower bounds for the minimum level at which a coupling appears. It turns out that this method yields exact results for $\widehat{su}(3)$ and constitutes an efficient and simple algorithm for computing $\widehat{su}(3)$ fusion coefficients.Comment: 27 page

Space-efficient detection of unusual words

Detecting all the strings that occur in a text more frequently or less frequently than expected according to an IID or a Markov model is a basic problem in string mining, yet current algorithms are based on data structures that are either space-inefficient or incur large slowdowns, and current implementations cannot scale to genomes or metagenomes in practice. In this paper we engineer an algorithm based on the suffix tree of a string to use just a small data structure built on the Burrows-Wheeler transform, and a stack of $O(\sigma^2\log^2 n)$ bits, where $n$ is the length of the string and $\sigma$ is the size of the alphabet. The size of the stack is $o(n)$ except for very large values of $\sigma$. We further improve the algorithm by removing its time dependency on $\sigma$, by reporting only a subset of the maximal repeats and of the minimal rare words of the string, and by detecting and scoring candidate under-represented strings that $\textit{do not occur}$ in the string. Our algorithms are practical and work directly on the BWT, thus they can be immediately applied to a number of existing datasets that are available in this form, returning this string mining problem to a manageable scale.Comment: arXiv admin note: text overlap with arXiv:1502.0637

Nonlinear optics and light localization in periodic photonic lattices

We review the recent developments in the field of photonic lattices emphasizing their unique properties for controlling linear and nonlinear propagation of light. We draw some important links between optical lattices and photonic crystals pointing towards practical applications in optical communications and computing, beam shaping, and bio-sensing.Comment: to appear in Journal of Nonlinear Optical Physics & Materials (JNOPM

Generating constrained random graphs using multiple edge switches

The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily possible to access all graphs obeying some given con- straints through a classical switching procedure calling on pairs of edges. We therefore propose to get round this issue by generalizing this classical approach through the use of higher-order edge switches. This method, which we denote by "k-edge switching", makes it possible to progres- sively improve the covered portion of a set of constrained graphs, thereby providing an increasing, asymptotically certain confidence on the statistical representativeness of the obtained sample.Comment: 15 page