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.     

Borderedness-preserving homomorphisms

This paper aims at investigating properties of homomorphisms which preserve the bordered words. The bordered words are classified into _d1$d_1$-words and _d2$d_2$-words, where the length of the proper border of _d2$d_2$-word is greater than half of it. Some characterizations of _d1$d_1$-word-preserving homomorphism are studied. Some relationships among d-primitivity-preserving homomorphisms, _d1$d_1$-word-preserving homomorphisms, and _d2$d_2$-word-preserving homomorphisms are investigated. We show that every d-primitivity-preserving homomorphism is _d1$d_1$-word-preserving and every _d1$d_1$-word-preserving homomorphism is _d2$d_2$-word-preserving.     

Set-theoretic generators of rational space curves

We show how to calculate three low degree set-theoretic generators (i.e., algebraic surfaces) for all rational space curves of low degree (degree ≤6$\le 6$) as well as for all higher degree rational space curves where at least one element of their μ$\mu$-basis has degree 1 from a μ$\mu$-basis of the parametrization. In addition to having low degree, at least two of these surface generators are always ruled surfaces. Whenever possible we also show how to compute two set-theoretic complete intersection generators for these rational space curves from a μ$\mu$-basis of their parametrization.     

The complexity of the zero-sum 3-flows

A zero-sum k-flow for a graph G   is a vector in the null-space of the 0,1$0,1$-incidence matrix of G   such that its entries belong to {±1,?,±(k−1)}$\{\pm 1,?,\pm (k-1)\}$. Akbari et al. (2009) [5] conjectured that if G is a graph with a zero-sum flow, then G   admits a zero-sum 6-flow. (2,3)$(2,3)$-semiregular graphs are an important family in studying zero-sum flows. Akbari et al. (2009) [5] proved that if Zero-Sum Conjecture is true for any (2,3)$(2,3)$-semiregular graph, then it is true for any graph. In this paper, we show that there is a polynomial time algorithm to determine whether a given (2,3)$(2,3)$-graph G   has a zero-sum 3-flow. In fact, we show that, there is a polynomial time algorithm to determine whether a given (2,4)$(2,4)$-graph G with n   vertices has a zero-sum 3-flow, where the number of vertices of degree four is O(log?n)$O(\log ?n)$. Furthermore, we show that it is NP-complete to determine whether a given (3,4)$(3,4)$-semiregular graph has a zero-sum 3-flow.     

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.     

A rounding algorithm for approximating minimum Manhattan networks

For a set T$T$ of n$n$ points (terminals) in the plane, a Manhattan network   on T$T$ is a network N(T)=(V,E)$N\left(T\right)=(V,E)$ with the property that its edges are horizontal or vertical segments connecting points in V⊇T$V\supseteq T$ and for every pair of terminals, the network N(T)$N\left(T\right)$ contains a shortest _l1$l_1$-path between them. A minimum Manhattan network   on T$T$ is a Manhattan network of minimum possible length. The problem of finding minimum Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan [J. Gudmundsson, C. Levcopoulos, G. Narasimhan, Approximating a minimum Manhattan network, Nordic Journal of Computing 8 (2001) 219–232. Proc. APPROX’99, 1999, pp. 28–37] and its complexity status is unknown. Several approximation algorithms (with factors 8, 4, and 3) have been proposed; recently Kato, Imai, and Asano [R. Kato, K. Imai, T. Asano, An improved algorithm for the minimum Manhattan network problem, ISAAC’02, in: LNCS, vol. 2518, 2002, pp. 344–356] have given a factor 2-approximation algorithm, however their correctness proof is incomplete. In this paper, we propose a rounding 2-approximation algorithm based on an LP-formulation of the minimum Manhattan network problem.     

Simplifying the weft hierarchy

We give simple, self-contained proofs of the basic hardness results for the classes W[t]$W[t]$ of the weft hierarchy. We extend these proofs to higher levels of the hierarchy and illuminate the distinctions among its classes. The anti-monotone collapse at W[1,s]$W[1,s]$ and the normalization of weft-t$t$ formulas arise as by-products of the proofs.     

An approximation algorithm to the k-Steiner Forest problem

Given a graph G$G$, an integer k$k$, and a demand set D={(_s1,_t1),…,(_sl,_tl)}$D=\{(s_1,t_1),\dots ,(s_l,t_l)\}$, the k$k$-Steiner Forest problem finds a forest in graph G$G$ to connect at least k$k$ demands in D$D$ such that the cost of the forest is minimized. This problem was proposed by Hajiaghayi and Jain in SODA’06. Thereafter, using a Lagrangian relaxation technique, Segev et al. gave the first approximation algorithm to this problem in ESA’06, with performance ratio O(n^2/3logl)$O(n^{2/3}logl)$. We give a simpler and faster approximation algorithm to this problem with performance ratio O(n^2/3logk)$O(n^{2/3}logk)$ via greedy approach, improving the previously best known ratio in the literature.     

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.     

Thread vulnerability in parallel applications

Continuously reducing transistor sizes and aggressive low power operating modes employed by modern architectures tend to increase transient error rates. Concurrently, multicore machines are dominating the architectural spectrum today in various application domains. These two trends require a fresh look at resiliency of multithreaded applications against transient errors from a software perspective. In this paper, we propose and evaluate a new metric called the Thread Vulnerability Factor (TVF$TVF$). A distinguishing characteristic of TVF$TVF$ is that its calculation for a given thread (which is typically one of the threads of a multithreaded application) does not depend on its code alone, but also on the codes of the threads that share resources and data with that thread. As a result, we decompose TVF$TVF$ of a thread into two complementary parts: local and remote. While the former captures the TVF$TVF$ induced by the code of the target thread, the latter represents the vulnerability impact of the threads that interact with the target thread. We quantify the local and remote TVF$TVF$ values for three architectural components (register file, ALUs, and caches) using a set of ten multithreaded applications from the Parsec and Splash-2 benchmark suites. Our experimental evaluation shows that TVF$TVF$ values tend to increase as the number of cores increases, which means the system becomes more vulnerable as the core count rises. We further discuss how TVF$TVF$ metric can be employed to explore performance–reliability tradeoffs in multicores. Reliability-based analysis of compiler optimizations and redundancy-based fault tolerance are also mentioned as potential usages of our TVF$TVF$ metric.     

Hamiltonian circuit and linear array embeddings in faulty k-ary n-cubes

In this paper, we investigate the fault-tolerant capabilities of the k-ary n-cubes for even integer k with respect to the hamiltonian and hamiltonian-connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F   be a faulty set with nodes and/or links, and let k?3$k?3$ be an odd integer. When |F|?2n-2$\left|F\right|?2n-2$, we show that there exists a hamiltonian cycle in a wounded k-ary n  -cube. In addition, when |F|?2n-3$\left|F\right|?2n-3$, we prove that, for two arbitrary nodes, there exists a hamiltonian path connecting these two nodes in a wounded k-ary n-cube. Since the k-ary n  -cube is regular of degree 2n$2n$, the degrees of fault-tolerance 2n-3$2n-3$ and 2n-2$2n-2$ respectively, are optimal in the worst case.     

A new variance-based global sensitivity analysis technique

A new set of variance-based sensitivity indices, called W$W$-indices, is proposed. Similar to the Sobol’s indices, both main and total effect indices are defined. The W$W$-main effect indices measure the average reduction of model output variance when the ranges of a set of inputs are reduced, and the total effect indices quantify the average residual variance when the ranges of the remaining inputs are reduced. Geometrical interpretations show that the W$W$-indices gather the full information of the variance ratio function, whereas, Sobol’s indices only reflect the marginal information. Then the double-loop-repeated-set Monte Carlo (MC) (denoted as DLRS MC) procedure, the double-loop-single-set MC (denoted as DLSS MC) procedure and the model emulation procedure are introduced for estimating the W$W$-indices. It is shown that the DLRS MC procedure is suitable for computing all the W$W$-indices despite its highly computational cost. The DLSS MC procedure is computationally efficient, however, it is only applicable for computing low order indices. The model emulation is able to estimate all the W$W$-indices with low computational cost as long as the model behavior is correctly captured by the emulator. The Ishigami function, a modified Sobol’s function and two engineering models are utilized for comparing the W$W$- and Sobol’s indices and verifying the efficiency and convergence of the three numerical methods. Results show that, for even an additive model, the W$W$-total effect index of one input may be significantly larger than its W$W$-main effect index. This indicates that there may exist interaction effects among the inputs of an additive model when their distribution ranges are reduced.     

Optimal coverage of an infrastructure network using sensors with distance-decaying sensing quality

Motivated by recent applications of wireless sensor networks in monitoring infrastructure networks, we address the problem of optimal coverage of infrastructure networks using sensors whose sensing performance decays with distance. We show that this problem can be formulated as a continuous p-median problem on networks. The literature has addressed the discrete p-median problem   on networks and in continuum domains, and the continuous p$p$-median problem in continuum domains extensively. However, in-depth analysis of the continuous p$p$-median problem on networks has been lacking. With the sensing performance model that decays with distance, each sensor covers a region equivalent to its Voronoi partition on the network in terms of the shortest path distance metric. Using Voronoi partitions, we define a directional partial derivative of the coverage metric with respect to a sensor’s location. We then propose a gradient descent algorithm to obtain a locally optimal solution with guaranteed convergence. The quality of an optimal solution depends on the choice of the initial configuration of sensors. We obtain an initial configuration using two approaches: by solving the discrete p$p$-median problem on a lumped   network and by random sampling. We consider two methods of random sampling: uniform sampling and D²$D^2$-sampling. The first approach with the initial solution of the discrete p$p$-median problem leads to the best coverage performance for large networks, but at the cost of high running time. We also observe that the gradient descent on the initial solution with the D²$D^2$-sampling method yields a solution that is within at most 7% of the previous solution and with much shorter running time.     

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.     

Non-associative Gröbner bases,finitely-presented Lie rings and the Engel condition,II

We give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n$n$-Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t$t$-generator Lie rings that satisfy the n$n$-Engel condition, for (t,n)=(2,3),(3,3),(4,3),(2,4)$(t,n)=(2,3),(3,3),(4,3),(2,4)$.     

Exploiting Universum data in AdaBoost using gradient descent

Recently, Universum data that does not belong to any class of the training data, has been applied for training better classifiers. In this paper, we address a novel boosting algorithm called U$\mathfrak{U}$AdaBoost that can improve the classification performance of AdaBoost with Universum data. U$\mathfrak{U}$AdaBoost chooses a function by minimizing the loss for labeled data and Universum data. The cost function is minimized by a greedy, stagewise, functional gradient procedure. Each training stage of U$\mathfrak{U}$AdaBoost is fast and efficient. The standard AdaBoost weights labeled samples during training iterations while U$\mathfrak{U}$AdaBoost gives an explicit weighting scheme for Universum samples as well. In addition, this paper describes the practical conditions for the effectiveness of Universum learning. These conditions are based on the analysis of the distribution of ensemble predictions over training samples. Experiments on handwritten digits classification and gender classification problems are presented. As exhibited by our experimental results, the proposed method can obtain superior performances over the standard AdaBoost by selecting proper Universum data.     

Separation of synchronous and asynchronous communication via testing

One of the early results concerning the asynchronous π$\pi$-calculus which significantly contributed to its popularity is the capability of encoding the output prefix of the (choiceless) π$\pi$-calculus in a natural and elegant way. Encodings of this kind were proposed by Honda and Tokoro, by Nestmann and (independently) by Boudol. We investigate whether the above encodings preserve De Nicola and Hennessy’s testing semantics. In this sense, it turns out that, under some general conditions, no encoding of the output prefix is able to preserve the must testing. This negative result is due to (a) the non-atomicity of the sequences of steps which are necessary in the asynchronous π$\pi$-calculus to mimic synchronous communication, and (b) testing semantics’ sensitivity to divergence.