On Hoffman's celebrated cycling LP example

We answer two questions that naturally arise while dealing with Hoffman's celebrated 50-year-old linear program to be solved by the primal simplex method, where an angle θ$\theta$ and a scaling factor ω$\omega$ are adjustable parameters. In particular, we determine what conditions have to be imposed on ω$\omega$ for classical cycling to occur with θ=2π/5$\theta =2\pi /5$, and what on θ$\theta$ with ω=±tan(θ)$\omega =\pm \tan (\theta )$. The first answer reveals that the sufficient condition widely spread over the literature is false, so fixing it turns this example into a correct example of classical cycling. Some progress towards necessary and sufficient conditions for cycling to occur in this example is also reported.     

On semimeasures predicting Martin-Löf random sequences

Solomonoff’s central result on induction is that the prediction of a universal semimeasure M$M$ converges rapidly and with probability 1 to the true sequence generating predictor μ$\mu$, if the latter is computable. Hence, M$M$ is eligible as a universal sequence predictor in the case of unknown μ$\mu$. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Löf) random sequences remained open. Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of M$M$ itself. We show that there are universal semimeasures M$M$ which do not converge to μ$\mu$ on all μ$\mu$-random sequences, i.e. we give a partial negative answer to the open problem. We also provide a positive answer for some non-universal semimeasures. We define the incomputable measure D$D$ as a mixture over all computable measures and the enumerable semimeasure W$W$ as a mixture over all enumerable nearly measures. We show that W$W$ converges to D$D$ and D$D$ to μ$\mu$ on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role.     

Learning multiple languages in groups

We consider a variant of Gold’s learning paradigm where a learner receives as input n$n$ different languages (in the form of one text where all input languages are interleaved). Our goal is to explore the situation when a more “coarse” classification of input languages is possible, whereas more refined classification is not. More specifically, we answer the following question: under which conditions, a learner, being fed n$n$ different languages, can produce m$m$ grammars covering all input languages, but cannot produce k$k$ grammars covering input languages for any k>m$k>m$. We also consider a variant of this task, where each of the output grammars may not cover more than r$r$ input languages. Our main results indicate that the major factor affecting classification capabilities is the difference n−m$n-m$ between the number n$n$ of input languages and the number m$m$ of output grammars. We also explore the relationship between classification capabilities for smaller and larger groups of input languages. For the variant of our model with the upper bound on the number of languages allowed to be represented by one output grammar, for classes consisting of disjoint languages, we found complete picture of relationship between classification capabilities for different parameters n$n$ (the number of input languages), m$m$ (number of output grammars), and r$r$ (bound on the number of languages represented by each output grammar). This picture includes a combinatorial characterization of classification capabilities for the parameters n,m,r$n,m,r$ of certain types.     

From Nerode’s congruence to suffix automata with mismatches

In this paper we focus on the minimal deterministic finite automaton _Sk$S_k$ that recognizes the set of suffixes of a word w$w$ up to k$k$ errors. As first result we give a characterization of the Nerode’s right-invariant congruence that is associated with _Sk$S_k$. This result generalizes the classical characterization described in [A. Blumer, J. Blumer, D. Haussler, A. Ehrenfeucht, M. Chen, J. Seiferas, The smallest automaton recognizing the subwords of a text, Theoretical Computer Science, 40, 1985, 31–55]. As second result we present an algorithm that makes use of _Sk$S_k$ to accept in an efficient way the language of all suffixes of w$w$ up to k$k$ errors in every window of size r$r$ of a text, where r$r$ is the repetition index of w$w$. Moreover, we give some experimental results on some well-known words, like prefixes of Fibonacci and Thue-Morse words. Finally, we state a conjecture and an open problem on the size and the construction of the suffix automaton with mismatches.     

An example of relations on the Ext-quiver for the Suzuki group Sz(8) in characteristic 2

Let G$G$ be the smallest Suzuki group Sz(8) and let F$F$ be an algebraically closed field of characteristic 2. The basic algebra of the group algebra of G$G$ over F$F$ is described by its Ext-quiver and a certain set of relations.     

Isotopic triangulation of a real algebraic surface

We present a new algorithm for computing the topology of a real algebraic surface S$S$ in a ball B$B$, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to S$S$, and even a simplicial complex isotopic to S$S$ by exploiting properties of the contour curve of S$S$. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of S$S$, by resultant computation. Using Thom’s isotopy lemma, we show how to deduce the topology of S$S$ from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper.     

Drawing slicing graphs with face areas

We consider orthogonal drawings of a plane graph G$G$ with specified face areas. For a natural number k$k$, a k$k$-gonal drawing of G$G$ is an orthogonal drawing such that the boundary of G$G$ is drawn as a rectangle and each inner face is drawn as a polygon with at most k$k$ corners whose area is equal to the specified value. In this paper, we show that every slicing graph G$G$ with a slicing tree T$T$ and a set of specified face areas admits a 10-gonal drawing D$D$ such that the boundary of each slicing subgraph that appears in T$T$ is also drawn as a polygon with at most 10 corners. Such a drawing D$D$ can be found in linear time.     

Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s$s$ and t$t$ in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t$s-t$ path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s$s$ and t$t$ for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k$k$ pairwise vertex/edge disjoint properly edge-colored s−t$s-t$ paths/trails in a c$c$-edge-colored graph G^c$G^c$ is NP-complete even for k=2$k=2$ and c=Ω(n²)$c=\Omega \left(n^2\right)$, where n$n$ denotes the number of vertices in G^c$G^c$. Moreover, we prove that these problems remain NP-complete for c$c$-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n)$c=\Omega \left(n\right)$. We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.     

A four-stage algorithm for updating a Burrows–Wheeler transform

We present a four-stage algorithm that updates the Burrows–Wheeler Transform of a text T$T$, when this text is modified. The Burrows–Wheeler Transform is used by many text compression applications and some self-index data structures. It operates by reordering the letters of a text T$T$ to obtain a new text bwt(T)$bwt\left(T\right)$ which can be better compressed.     

Quasi-quadratic elliptic curve point counting using rigid cohomology

Let E$E$ be a nonsupersingular elliptic curve over the finite field with pⁿ$p^n$ elements. We present a deterministic algorithm that computes the zeta function and hence the number of points of such a curve E$E$ in time quasi-quadratic in n$n$. An older algorithm having the same time complexity uses the canonical lift of E$E$, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.     

Simple,space-efficient,and fairness improved FCFS mutual exclusion algorithms

Let n$n$ be the number of threads that can compete for a shared resource R$R$. The mutual exclusion problem involves coordinating these n$n$ concurrent threads in accessing R$R$ in a mutually exclusive way. This paper addresses two basic questions related to the First-Come-First-Served (FCFS) mutual exclusion algorithms that use only read–write operations: one is regarding the lower bound on the shared space requirement and the other is about fairness.     

Claw finding algorithms using quantum walk

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. Given two functions, f$f$ and g$g$, with domain sizes N$N$ and M$M$(N≤M)$(N\le M)$, respectively, and the same range, the goal of the problem is to find x$x$ and y$y$ such that f(x)=g(y)$f\left(x\right)=g\left(y\right)$. This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm that uses quantum walk to solve this problem. Our algorithm can be slightly modified to solve the more general problem of finding a tuple consisting of elements in the two function domains that has a prespecified property. It can also be generalized to find a claw of k$k$ functions for any constant integer k>1$k>1$, where the domain sizes of the functions may be different.     

Rational two-parameter families of spheres and rational offset surfaces

The present paper investigates two-parameter families of spheres in R³${\mathbb{R}}^3$ and their corresponding two-dimensional surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$. Considering a rational surface Φ$\Phi$ in R⁴${\mathbb{R}}^4$, the envelope surface Ψ$\Psi$ of the corresponding family of spheres in R³${\mathbb{R}}^3$ is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface Ψ$\Psi$ and its offset surfaces admit rational parameterizations if and only if Φ$\Phi$ is a rational sub-variety of a rational isotropic hyper-surface in R⁴${\mathbb{R}}^4$. The close relation between the envelope surfaces Ψ$\Psi$ and rational offset surfaces in R³${\mathbb{R}}^3$ is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$ whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.     

Efficient enumeration of words in regular languages

The cross-section enumeration problem is to list all words of length n$n$ in a regular language L$L$ in lexicographical order. The enumeration problem is to list the first m$m$ words in L$L$ according to radix order. We present an algorithm for the cross-section enumeration problem that is linear in n+t$n+t$, where t$t$ is the output size. We provide a detailed analysis of the asymptotic running time of our algorithm and that of known algorithms for both enumeration problems. We discuss some shortcomings of the enumeration algorithm found in the Grail computation package. In the practical domain, we modify Mäkinen’s enumeration algorithm to get an algorithm that is usually the most efficient in practice. We performed an extensive performance analysis of the new and previously known enumeration and cross-section enumeration algorithms and found when each algorithm is preferable.     

Transposition invariant words

We define an operation called transposition on words of fixed length. This operation arises naturally when the letters of a word are considered as entries of a matrix. Words that are invariant with respect to transposition are of special interest. It turns out that transposition invariant words have a simple interpretation by means of elementary group theory. This leads us to investigate some properties of the ring of integers modulo n$n$ and primitive roots. In particular, we show that there are infinitely many prime numbers p$p$ with a primitive root dividing p+1$p+1$ and infinitely many prime numbers p$p$ without a primitive root dividing p+1$p+1$. We also consider the orbit of a word under transposition.