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$h$.     

Stratification associated with local b-functions

The local b$b$-function _bf,p(s)$b_{f,p}\left(s\right)$ of an n$n$-variate polynomial f∈C[x]$f\in \mathbb{C}\left[x\right]$ (x=(_x1,…,_xn)$x=(x_1,\dots ,x_n)$) at a point p∈Cⁿ$p\in {\mathbb{C}}^n$ is constant on each stratum of a stratification of Cⁿ${\mathbb{C}}^n$. We propose a new method for computing such a stratification and _bf,p(s)$b_{f,p}\left(s\right)$ on each stratum. In the existing method proposed in Oaku (1997b), a primary ideal decomposition of an ideal in C[x,s]$\mathbb{C}[x,s]$ is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in C[x]$\mathbb{C}\left[x\right]$ defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.     

Origami fold as algebraic graph rewriting

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,?)$(\mathbb{O},?)$, where O$\mathbb{O}$ is the set of abstract origamis and ?$?$ is a binary relation on O$\mathbb{O}$, that models fold  . An abstract origami is a structure (Π,∽,?)$(\Pi ,\backsim ,?)$, where Π$\Pi$ is a set of faces constituting an origami, and ∽$\backsim$ and ?$?$ are binary relations on Π$\Pi$, each representing adjacency and superposition relations between the faces.     

Algorithms for a class of infinite permutation groups

Motivated by the famous 3n+1$3n+1$ conjecture, we call a mapping from Z$\mathbb{Z}$ to Z$\mathbb{Z}$residue-class-wise affine   if there is a positive integer m$m$ such that it is affine on residue classes (mod m$m$). This article describes a collection of algorithms and methods for computation in permutation groups and monoids formed by residue-class-wise affine mappings.     

Conjunctive query evaluation by search-tree revisited

The most natural and perhaps most frequently used method for testing membership of an individual tuple in a conjunctive query is based on searching trees of partial solutions, or search-trees. We investigate the question of evaluating conjunctive queries with a time-bound guarantee that is measured as a function of the size of the optimal search-tree. We provide an algorithm that, given a database D$D$, a conjunctive query Q$Q$, and a tuple a$a$, tests whether Q(a)$Q\left(a\right)$ holds in D$D$ in time bounded by a polynomial in (sn)^logk(sn)^loglogn${\left(sn\right)}^{logk}{\left(sn\right)}^{loglogn}$ and n^r$n^r$, where n$n$ is the size of the domain of the database, k$k$ is the number of bound variables of the conjunctive query, s$s$ is the size of the optimal search-tree, and r$r$ is the maximum arity of the relations. In many cases of interest, this bound is significantly smaller than the n^O(k)$n^{O\left(k\right)}$ bound provided by the naive search-tree method. Moreover, our algorithm has the advantage of guaranteeing the bound for any given conjunctive query. In particular, it guarantees the bound for queries that admit an equivalent form that is much easier to evaluate, even when finding such a form is an NP-hard task. Concrete examples include the conjunctive queries that can be non-trivially folded into a conjunctive query of bounded size or bounded treewidth. All our results translate to the context of constraint-satisfaction problems via the well-publicized correspondence between both frameworks.     

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.     

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.     

The number of roots of a lacunary bivariate polynomial on a line

We prove that a polynomial f∈R[x,y]$f\in \mathbb{R}[x,y]$ with t$t$ non-zero terms, restricted to a real line y=ax+b$y=ax+b$, either has at most 6t−4$6t-4$ zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y−ax−b∈K[x,y]$y-ax-b\in K[x,y]$ divides a lacunary polynomial f∈K[x,y]$f\in K[x,y]$, where K$K$ is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f$f$, in the logarithm of the degree of f$f$, in the degree of the extension K/Q$K/\mathbb{Q}$ and in the logarithmic height of a$a$, b$b$ and f$f$.     

Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.     

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G=(V,E)$G=(V,E)$, with n$n$ nodes and m$m$ non-negatively real weighted edges, and a single source shortest paths tree (SPT) T$T$ of G$G$ rooted at an arbitrary node r$r$. If an edge in T$T$ is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge  , instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O(mlog²n)$O\left(m{log}^{2}n\right)$ time and O(m)$O\left(m\right)$ space algorithm to find a best swap edge for every edge of T$T$, thus improving for m=o(n²/log²n)$m=o(n^2/{log}^{2}n)$ the previously known O(n²)$O\left(n^2\right)$ time and space complexity algorithm.     

Modular Las Vegas algorithms for polynomial absolute factorization

Let f(X,Y)∈Z[X,Y]$f(X,Y)\in \mathbb{Z}[X,Y]$ be an irreducible polynomial over Q$\mathbb{Q}$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f$f$, or more precisely, of f$f$ modulo some prime integer p$p$. The same idea of choosing a p$p$ satisfying some prescribed properties together with LLL$LLL$ is used to provide a new strategy for absolute factorization of f(X,Y)$f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to construct the algebraic extension containing one absolute factor of a polynomial of degree up to 400.