A palindromization map for the free group

We define a self-map Pal:_F2→_F2$Pal:F_2\to F_2$ of the free group on two generators a,b$a,b$, using automorphisms of _F2$F_2$ that form a group isomorphic to the braid group _B3$B_3$. The map Pal$Pal$ restricts to de Luca’s right iterated palindromic closure on the submonoid generated by a,b$a,b$. We show that Pal$Pal$ is continuous for the profinite topology on _F2$F_2$; it is the unique continuous extension of de Luca’s right iterated palindromic closure to _F2$F_2$. The values of Pal$Pal$ are palindromes and coincide with the elements g∈_F2$g\in F_2$ such that abg$abg$ and bag$bag$ are conjugate.     

Choosing starting values for certain Newton–Raphson iterations

We aim at finding the best possible seed values when computing a^1/p$a^{1/p}$ using the Newton–Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f(a)$f(a)$ in the interval [_amin,_amax]$[a_{\min },a_{\max }]$, by building the sequence _xn$x_n$ defined by the Newton–Raphson iteration, the natural choice consists in choosing _x0$x_0$ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between _x0$x_0$ and f(a)$f(a)$. And yet, if we perform n$n$ iterations, what matters is to minimize the maximum possible distance between _xn$x_n$ and f(a)$f(a)$. In several examples, the value of the best starting point varies rather significantly with the number of iterations.     

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$h$.     

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.     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

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.     

Performance tradeoffs in structured peer to peer streaming

We consider the following basic question: a source node wishes to stream an ordered sequence of packets to a collection of receivers, which are in K$K$ clusters. A node may send a packet to another node in its own cluster in one time step and to a node in a different cluster in _Tc$T_c$ time steps (_Tc>1)$(T_c>1)$. Each cluster has two special nodes. We assume that the source and the special nodes in each cluster have a higher capacity and thus can send multiple packets at each step, while all other nodes can both send and receive a packet at each step. We construct two (intra-cluster) data communication schemes, one based on multi-trees (using a collection of d$d$-ary interior-disjoint trees) and the other based on hypercubes. The multi-tree scheme sustains streaming within a cluster with O(dlogN)$O(dlogN)$ maximum playback delay and O(dlogN)$O(dlogN)$ size buffers, while communicating with O(d)$O\left(d\right)$ neighbors, where N$N$ is the maximum size of any cluster. We also show that this protocol is optimal when d=2$d=2$ or 3. The hypercube scheme sustains streaming within a cluster, with O(log²(N))$O\left({log}^2\left(N\right)\right)$ maximum playback delay and O(1)$O\left(1\right)$ size buffers, while communicating with O(log(N))$O(log\left(N\right))$ neighbors, for arbitrary N$N$. In addition, we extend our multi-tree scheme to work when receivers depart and arrive over time. We also evaluate our dynamic schemes using simulations.     

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.     

Minimum cost source location problem with local 3-vertex-connectivity requirements

Let G=(V,E)$G=(V,E)$ be a simple undirected graph with a set V$V$ of vertices and a set E$E$ of edges. Each vertex v∈V$v\in V$ has a demand d(v)∈_Z+$d\left(v\right)\in Z_{+}$ and a cost c(v)∈_R+$c\left(v\right)\in R_{+}$, where _Z+$Z_{+}$ and _R+$R_{+}$ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G$G$ requires finding a set S$S$ of vertices minimizing _∑v∈Sc(v)${\sum }_{v\in S}c\left(v\right)$ such that there are at least d(v)$d\left(v\right)$ pairwise vertex-disjoint paths from S$S$ to v$v$ for each vertex v∈V−S$v\in V-S$. It is known that if there exists a vertex v∈V$v\in V$ with d(v)≥4$d\left(v\right)\ge 4$, then the problem is NP-hard even in the case where every vertex has a uniform cost. In this paper, we show that the problem can be solved in O(|V|⁴log²|V|)$O\left({\left|V\right|}^4{log}^2\left|V\right|\right)$ time if d(v)≤3$d\left(v\right)\le 3$ holds for each vertex v∈V$v\in V$.     

Adaptive timeliness of consensus in presence of crash and timing faults

The Δ$\mathrm{\Delta }$-timed uniform consensus is a stronger variant of the traditional consensus and it satisfies the following additional property: every correct process terminates its execution within a constant time Δ$\mathrm{\Delta }$ (Δ$\mathrm{\Delta }$-timeliness), and no two processes decide differently (uniformity). In this paper, we consider the Δ$\mathrm{\Delta }$-timed uniform consensus problem in presence of _fc$f_{\mathrm{c}}$ crash processes and _ft$f_{\mathrm{t}}$ timing-faulty processes, and propose a Δ$\mathrm{\Delta }$-timed uniform consensus algorithm. The proposed algorithm is adaptive in the following sense: it solves the Δ$\mathrm{\Delta }$-timed uniform consensus when at least _ft+1$f_{\mathrm{t}}+1$ correct processes exist in the system. If the system has less than _ft+1$f_{\mathrm{t}}+1$ correct processes, the algorithm cannot solve the Δ$\mathrm{\Delta }$-timed uniform consensus. However, as long as _ft+1$f_{\mathrm{t}}+1$ processes are non-crashed, the algorithm solves (non-timed) uniform consensus. We also investigate the maximum number of faulty processes that can be tolerated. We show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes requires that the system has at least _ft+1$f_{\mathrm{t}}+1$ correct processes. This impossibility result implies that the proposed algorithm attains the maximum resilience about the number of faulty processes. We also show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes cannot solve the (non-timed) uniform consensus when the system has less than _ft+1$f_{\mathrm{t}}+1$ non-crashed processes. This impossibility result implies that our algorithm attains the maximum adaptiveness.     

A bound for the Rosenfeld–Gröbner algorithm

We consider the Rosenfeld–Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n$n$ indeterminates. For a set of ordinary differential polynomials F$F$, let M(F)$M\left(F\right)$ be the sum of maximal orders of differential indeterminates occurring in F$F$. We propose a modification of the Rosenfeld–Gröbner algorithm, in which for every intermediate polynomial system F$F$, the bound M(F)?(n−1)!M(_F0)$M\left(F\right)?(n-1)!M\left(F_0\right)$ holds, where _F0$F_0$ is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal.     

Recognizing and parametrizing curves isomorphic to a line

Some time ago, Shpilrain and Yu reported an algorithm for deciding whether or not a polynomial p∈K[x,y]$p\in K[x,y]$ is a coordinate, or, equivalently, whether or not a plane curve p(x,y)=0$p(x,y)=0$ is isomorphic to a line. Here K$K$ is any constructible field of characteristic 0. In this paper, we show that their algorithm requires O(n²)$O\left(n^2\right)$ field operations, where n$n$ is the degree of a given polynomial. We also show how their algorithm can be used to find a polynomial parametrization of a plane curve p(x,y)=0$p(x,y)=0$ which is isomorphic to a line. This requires O(n²_log2n)$O\left(n^2{log}_{2}n\right)$ field operations.