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$h$.     

Compatibility of unrooted phylogenetic trees is FPT

A collection of _T1,_T2,…,_Tk$T_1,T_2,\dots ,T_k$ of unrooted, leaf labelled (phylogenetic) trees, all with different leaf sets, is said to be compatible   if there exists a tree T$T$ such that each tree _Ti$T_i$ can be obtained from T$T$ by deleting leaves and contracting edges. Determining compatibility is NP-hard, and the fastest algorithm to date has worst case complexity of around Ω(n^k)$\mathrm{\Omega }(n^k)$ time, n$n$ being the number of leaves. Here, we present an O(nf(k))$\mathrm{O}(\mathit{nf}(k))$ algorithm, proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable   (FPT) with respect to the number k$k$ of trees.     

Fast algorithms for some dominating induced matching problems

We describe O(n)$O(n)$ time algorithms for finding the minimum weighted dominating induced matching of chordal, dually chordal, biconvex, and claw-free graphs. For the first three classes, we prove tight O(n)$O(n)$ bounds on the maximum number of edges that a graph having a dominating induced matching may contain. By applying these bounds, and employing existing O(n+m)$O(n+m)$ time algorithms we show that they can be reduced to O(n)$O(n)$ time. For claw-free graphs, we describe a variation of the existing algorithm for solving the unweighted version of the problem, which decreases its complexity from O(n²)$O(n^2)$ to O(n)$O(n)$, while additionally solving the weighted version. The same algorithm can be easily modified to count the number of DIM's of the given graph.     

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.     

On miniaturized problems in parameterized complexity theory

We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t]$\mathrm{W}[t]$ lies in W[t]$\mathrm{W}[t]$, too. In particular, the miniaturization of the dominating set problem is in W[2]$\mathrm{W}[2]$. Furthermore, we investigate the relation between f(k)·n^o(k)$f(k)·n^{\mathrm{o}(k)}$ time and subexponential time algorithms for the dominating set problem and for the clique problem.     

Faster two-dimensional pattern matching with rotations

The most efficient currently known algorithms for two-dimensional pattern matching with rotations have a worst case time complexity of O(n²m³)$\mathrm{O}(n^2{m}^3)$, where the size of the text is n×n$n\times n$ and the size of the pattern is m×m$m\times m$. In this paper we present a new algorithm for the problem whose running time is O(n²m²)$\mathrm{O}(n^2{m}^2)$.     

A self-stabilizing algorithm to maximal 2-packing with improved complexity

In self-stabilization, each node has a local view of the distributed network system, in a finite amount of time the system converges to a global setup with desired property, in this case establishing a 2-packing set. Using a graph G=(V,E)$G=(V,E)$ to represent the network, a subset S⊆V$S\subseteq V$ is a 2-packing if ∀i∈V:|N[i]∩S|?1$\forall i\in V:|N[i]\cap S|?1$. In this paper, we first propose an ID-based, constant space, self-stabilizing algorithm that stabilizes to a maximal 2-packing in an arbitrary graph. We show that the algorithm stabilizes in O(mn)$O(mn)$ moves under any scheduler (such as a distributed daemon). Secondly, we show that the algorithm stabilizes in O(n²)$O(n^2)$ rounds under a synchronous daemon where every privileged node moves at each round.     

Rate-adaptive weighted fair queueing for energy-aware scheduling

In this paper we study energy-aware scheduling that trades energy consumption against a traditional performance measure of delay. We use the power-rate function f(x)=c+x^α$f(x)=c+x^{\alpha }$ for x>0$x>0$ and f(0)=0$f(0)=0$ to model the power consumption, where c>0$c>0$ represents the base power. We give a definition of a rate-adaptive version of the Weighted Fair Queueing scheduling algorithm, and prove its energy consumption is within a bounded factor of the best possible when the algorithm guarantees the classic end-to-end delay for every connection.     

Computing phylogenetic roots with bounded degrees and errors is NP-complete

In this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E)$G=(V,E)$, we wish to find a tree T such that (1) the degree of each internal node of T   is at least 3 and at most Δ$\Delta$, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E   and {{u,v}:u,v$\{\{u,v\}:u,v$ are leaves of T   and _dT(u,v)≤k}$d_T(u,v)\le k\}$, is as small as possible, where _dT(u,v)$d_T(u,v)$ denotes the distance between u$u$ and v$v$ in tree T  . We show that this problem is NP-hard for all fixed constants Δ,k≥3$\Delta ,k\ge 3$.     

A note on scaling the Linpack benchmark

Dimensional analysis yields a new scaling formula for the Linpack benchmark. The computational power r(_p0,_q0)$r(p_0,q_0)$ on a set of processors decomposed into a (_p0,_q0)$(p_0,q_0)$ grid determines the computational power r(p,q)$r(p,q)$ on a set of processors decomposed into a (p,q)$(p,q)$ grid by the formula r(p,q)=(p/_p0)^α(q/_q0)^βr(_p0,_q0)$r(p,q)=(p/p_0{)}^{\alpha }(q/q_0{)}^{\beta }r(p_0,q_0)$. The two scaling parameters α$\alpha$ and β$\beta$ measure interprocessor communication overhead required by the algorithm. A machine that scales perfectly corresponds to α=β=1$\alpha =\beta =1$; a machine that scales not at all corresponds to α=β=0$\alpha =\beta =0$. We have determined the two scaling parameters by imposing a fixed-time constraint on the problem size such that the execution time remains constant as the number of processors changes. Results for a collection of machines confirm that the formula suggested by dimensional analysis is correct. Machines with the same values for these parameters are self-similar. They scale the same way even though the details of their specific hardware and software may be quite different.     

Optimum sweeps of simple polygons with two guards

A polygon P admits a sweep if two mobile guards can detect an unpredictable, moving target inside P  , no matter how fast the target moves. Two guards move on the polygon boundary and are required to always be mutually visible. The objective of this study is to find an optimum sweep such that the sum of the distances travelled by the two guards in the sweep is minimized. We present an O(n²)$O(n^2)$ time and O(n)$O(n)$ space algorithm for optimizing this metric, where n   is the number of vertices of the given polygon. Our result is obtained by reducing this problem to finding a shortest path between two nodes in a graph of size O(n)$O(n)$.     

On mimicking networks representing minimum terminal cuts

Given a capacitated undirected graph G=(V,E)$G=(V,E)$ with a set of terminals K⊂V$K\subset V$, a mimicking network   is a smaller graph H=(_VH,_EH)$H=(V_H,E_H)$ which contains the set of terminals K   and for every bipartition [U,K−U]$[U,K-U]$ of the terminals, the cost of the minimum cut separating U   from K−U$K-U$ in G is exactly equal to the cost of the minimum cut separating U   from K−U$K-U$ in H.     

A simple optimal representation for balanced parentheses

We consider succinct, or highly space-efficient, representations of a (static) string consisting of n   pairs of balanced parentheses, which support natural operations such as finding the matching parenthesis for a given parenthesis, or finding the pair of parentheses that most tightly enclose a given pair. This problem was considered by Jacobson [Space-efficient static trees and graphs, in: Proc. of the 30th FOCS, 1989, pp. 549–554] and Munro and Raman [Succinct representation of balanced parentheses and static trees, SIAM J. Comput. 31 (2001) 762–776] who gave O(n)$\mathrm{O}(n)$-bit and 2n+o(n)$2n+\mathrm{o}(n)$-bit representations, respectively, that supported the above operations in O(1)$\mathrm{O}(1)$ time on the RAM model of computation. This data structure is a fundamental tool in succinct representations, and has applications in representing suffix trees, ordinal trees, planar graphs and permutations.     

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4ⁿ)$O(4^n)$ constraints whose coefficients belong to {−1,0,1}$\{-1,0,1\}$. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(_kmax2ⁿ)$O(k_{\mathit{max}}{2}^n)$, where k_max is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.