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.     

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.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

A modified hidden Markov model

This paper considers two discrete time, finite state processes X$X$ and Y$Y$. In the usual hidden Markov model X$X$ modulates the values of Y$Y$. However, the values of Y$Y$ are then i.i.d. given X$X$. In this paper a new model is considered where the Markov chain X$X$ modulates the transition probabilities of the second, observed chain Y$Y$. This more realistically can represent problems arising in DNA sequencing. Algorithms for all related filters, smoothers and parameter estimations are derived. Versions of the Viterbi algorithms are obtained.     

Algorithms for near solutions to polynomial equations

Let F(x,y)$F(x,y)$ be a polynomial over a field K$K$ and m$m$ a nonnegative integer. We call a polynomial g$g$ over K$K$ an m$m$-near solution of F(x,y)$F(x,y)$ if there exists a c∈K$c\in K$ such that F(x,g)=cx^m$F(x,g)=c{x}^m$, and the number c$c$ is called an m$m$-value of F(x,y)$F(x,y)$ corresponding to g$g$. In particular, c$c$ can be 0. Hence, by viewing F(x,y)=0$F(x,y)=0$ as a polynomial equation over K[x]$K\left[x\right]$ with variable y$y$, every solution of the equation F(x,y)=0$F(x,y)=0$ in K[x]$K\left[x\right]$ is also an m$m$-near solution. We provide an algorithm that gives all m$m$-near solutions of a given polynomial F(x,y)$F(x,y)$ over K$K$, and this algorithm is polynomial time reducible to solving one variable equations over K$K$. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.     

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.     

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.     

On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P$P$ with integer coefficients in k$k$ variables over the standard simplex of R^k${\mathbb{R}}^k$, assuming that P$P$ is positive on the simplex. This bound depends only on the number of variables k$k$, the degree d$d$ and the bitsize τ$\tau$ of the coefficients of P$P$ and improves all the previous bounds for arbitrary polynomials which are positive over the simplex.     

On the parameterized complexity of multiple-interval graph problems

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k$k$-Independent Set, k$k$-Dominating Set, and k$k$-Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k$k$-Clique is in FPT, while k$k$-Independent Set and k$k$-Dominating Set are both W[1]-hard. We also prove that k$k$-Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k$k$-Multicolored Clique problem, a variant of k$k$-Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.     

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.     

Non-reducible descriptions for conditional Kolmogorov complexity

Assume that a program p$p$ on input a$a$ outputs b$b$. We are looking for a shorter program q$q$ having the same property (q(a)=b$q\left(a\right)=b$). In addition, we want q$q$ to be simple conditional to p$p$ (this means that the conditional Kolmogorov complexity K(q|p)$K\left(q\right|p)$ is negligible). In the present paper, we prove that sometimes there is no such program q$q$, even in the case when the complexity of p$p$ is much bigger than K(b|a)$K\left(b\right|a)$. We give three different constructions that use the game approach, probabilistic arguments and algebraic arguments, respectively.     

Local edge colouring of Yao-like subgraphs of Unit Disk Graphs

The focus of the present paper is on providing a local deterministic algorithm for colouring the edges of Yao-like   subgraphs of Unit Disk Graphs. These are geometric graphs such that for some positive integers l,k$l,k$ the following property holds at each node v$v$: if we partition the unit circle centered at v$v$ into 2k$2k$ equally sized wedges then each wedge can contain at most l$l$ points different from v$v$. We assume that the nodes are location aware, i.e. they know their Cartesian coordinates in the plane. The algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant (depending on k$k$ and l$l$, but not on the size of the graph) number of hops (measured in the original underlying Unit Disk Graph) away from it, and hence the algorithm terminates in a constant number of steps. The number of colours used is 2kl+1$2kl+1$ and this is optimal for local algorithms (since the maximal degree is 2kl$2kl$ and a colouring with 2kl$2kl$ colours can only be constructed by a global algorithm), thus showing that in this class of graphs the price for locality is only one additional colour.     

An equation-based Heap Sizing Rule

For garbage-collected applications, dynamically-allocated objects are contained in a heap. Programmer productivity improves significantly if there is a garbage collector to automatically deallocate objects that are no longer needed by the applications. However, there is a run-time performance overhead in garbage collection, and this cost is sensitive to heap size H$H$: a smaller H$H$ will trigger more collection, but a large H$H$ can cause page faults, as when H$H$ exceeds the size M$M$ of main memory allocated to the application.     

Multicriteria 0-1 knapsack problems with k-min objectives

This paper deals with the multicriteria 0-1 knapsack problem (KP) with k$k$-min objectives (MkMIN-KP) in which the first objective is of classical sum type and the remaining objectives are k$k$-min objective functions. The k$k$-min objectives are ordinal objectives, aiming at the maximization of the k$k$ th smallest objective coefficient in any feasible knapsack solution with at least k$k$ items in the knapsack. We develop efficient algorithms for the determination of the complete nondominated set of MkMIN-KP.     

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.