Approximately optimal trees for group key management with batch updates

We investigate the group key management problem for broadcasting applications. Previous work showed that, in handling key updates, batch rekeying can be more cost effective than individual rekeying. One model for batch rekeying is to assume that every user has probability p$p$ of being replaced by a new user during a batch period with the total number of users unchanged. Under this model, it was recently shown that an optimal key tree can be constructed in linear time when p$p$ is a constant and in O(n⁴)$O\left(n^4\right)$ time when p→0$p\to 0$. In this paper, we investigate more efficient algorithms for the case p→0$p\to 0$, i.e., when membership changes are sparse. We design an O(n)$O\left(n\right)$ heuristic algorithm for the sparse case and show that it produces a nearly 2-approximation to the optimal key tree. Simulation results show that its performance is even better in practice. We also design a refined heuristic algorithm and show that it achieves an approximation ratio of 1+?$1+?$ for any fixed ?>0$?>0$ and n$n$, as p→0$p\to 0$. Finally, we give another approximation algorithm for any p∈(0,0.693)$p\in (0,0.693)$ which is shown to be quite good by our simulations.     

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.     

A note on the number of squares in a word

Fraenkel and Simpson [A.S. Fraenkel, J. Simpson, How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112–120] proved that the number of squares in a word of length n$n$ is bounded by 2n$2n$. In this note we improve this bound to 2n−Θ(logn)$2n-\Theta (logn)$. Based on the numerical evidence, the conjectured bound is n$n$.     

Expanders and time-restricted branching programs

The replication number   of a branching program is the minimum number R$R$ such that along every accepting computation at most R$R$ variables are tested more than once; the sets of variables re-tested along different computations may be different. For every branching program, this number lies between 0$0$ (read-once programs) and the total number n$n$ of variables (general branching programs). The best results so far were exponential lower bounds on the size of branching programs with R=o(n/logn)$R=o(n/logn)$. We improve this to R≤?n$R\le ?n$ for a constant ?>0$?>0$. This also gives an alternative and simpler proof of an exponential lower bound for (1+?)n$(1+?)n$ time branching programs for a constant ?>0$?>0$. We prove these lower bounds for quadratic functions of Ramanujan graphs.     

Practical perfect hashing in nearly optimal space

A hash function is a mapping from a key universe U   to a range of integers, i.e., h:U?{0,1,…,m−1}$h:U?\{\mathrm{0,1},\dots ,m-1\}$, where m is the range's size. A perfect hash function   for some set S⊆U$S\subseteq U$ is a hash function that is one-to-one on S  , where m≥|S|$m\ge \left|S\right|$. A minimal perfect hash function   for some set S⊆U$S\subseteq U$ is a perfect hash function with a range of minimum size, i.e., m=|S|$m=\left|S\right|$. This paper presents a construction for (minimal) perfect hash functions that combines theoretical analysis, practical performance, expected linear construction time and nearly optimal space consumption for the data structure. For n keys and m=n   the space consumption ranges from 2.62n+o(n)$2.62n+o(n)$ to 3.3n+o(n)$3.3n+o(n)$ bits, and for m=1.23n$m=1.23n$ it ranges from 1.95n+o(n)$1.95n+o(n)$ to 2.7n+o(n)$2.7n+o(n)$ bits. This is within a small constant factor from the theoretical lower bounds of 1.44n$1.44n$ bits for m=n   and 0.89n$0.89n$ bits for m=1.23n$m=1.23n$. We combine several theoretical results into a practical solution that has turned perfect hashing into a very compact data structure to solve the membership problem when the key set S is static and known in advance. By taking into account the memory hierarchy we can construct (minimal) perfect hash functions for over a billion keys in 46 min using a commodity PC. An open source implementation of the algorithms is available at http://cmph.sf.net under the GNU Lesser General Public License (LGPL).     

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.     

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.     

First steps to the runtime complexity analysis of ant colony optimization

The paper presents results on the runtime complexity of two ant colony optimization (ACO) algorithms: ant system, the oldest ACO variant, and GBAS, the first ACO variant for which theoretical convergence results have been established. In both cases, as the class of test problems under consideration, a slight generalization of the well-known OneMax test function has been chosen. The techniques used for the runtime analysis of the two algorithms differ: in the case of GBAS, the expected runtime until the optimal solution is reached is studied by a direct bound estimation approach inspired by comparable results for the (1+1)$(1+1)$ evolutionary algorithm (EA). A runtime bound of order O(mlogm)$O(m\log m)$, where m   is the problem instance size, is obtained. In the case of ant system, the original discrete stochastic process is approximated by a suitable continuous deterministic process. The validity of the approximation is shown by means of a rigid convergence theorem exploiting a classical result from mathematical learning theory. Using this approximation, it is demonstrated that for the considered OneMax-type problems, a runtime of order O(mlog(1/ε))$O(m\log (1/\epsilon ))$ until reaching an expected relative   solution quality of 1-ε$1-\epsilon$, and a runtime of O(mlogm)$O(m\log m)$ until reaching the optimal   solution with high probability can be predicted. Our results are the first to show competitiveness in runtime complexity with (1+1$1+1$) EA on OneMax for a proper ACO algorithm.     

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.     

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.     

State-complexity hierarchies of uniform languages of alphabet-size length

We study the state complexity of certain simple languages. If A$A$ is an alphabet of k$k$ letters, then a k$k$-language   is a nonempty set of words of length k$k$, that is, a uniform language of length k$k$. We show that the minimal state complexity of a k$k$-language is k+2$k+2$, and the maximal, (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. We prove constructively that, for every i$i$ between the minimal and maximal bounds, there is a language of state complexity i$i$. We introduce a class of automata accepting sets of words that are permutations of A$A$; these languages define a complete hierarchy of complexities between k²−k+3$k^2-k+3$ and 2^k+1$2^k+1$. The languages of another class of automata, based on k$k$-ary trees, define a complete hierarchy of complexities between 2^k+1$2^k+1$ and (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. This provides new examples of uniform languages of maximal complexity.     

On hypercube packings,blocking sets and a covering problem

Consider sets S of hypercubes of side 2 in the discrete n-dimensional torus of side 4 with the property that every possible hypercube of side 2 has a nonempty intersection with some hypercube in S. The problem of minimizing the size of S is studied in two settings, depending on whether intersections between hypercubes in S are allowed or not. If intersections are not allowed, then one is asking for the smallest size of a non-extensible packing S  ; this size is denoted by f(n)$f(n)$. If intersections are allowed, then the structure S is called a blocking set. The smallest size of a blocking set S   is denoted by h(n)$h(n)$. By computer-aided techniques, it is shown that f(5)=12$f(5)=12$, f(6)=16$f(6)=16$, h(6)=15$h(6)=15$ and h(7)≤23$h(7)\le 23$. Also, non-extensible packings as well as blocking sets of certain small sizes are classified for n≤6$n\le 6$. There is a direct connection between these problems and a covering problem originating from the football pools.     

On the privacy offered by (k, δ)-anonymity

The widespread deployment of technologies with tracking capabilities, like GPS, GSM, RFID and on-line social networks, allows mass collection of spatio-temporal data about their users. As a consequence, several methods aimed at anonymizing spatio-temporal data before their publication have been proposed in recent years. Such methods are based on a number of underlying privacy models. Among these models, (k,δ)-anonymity$(k,\delta )\mathrm{-}\mathrm{anonymity}$ claims to extend the widely used k  -anonymity concept by exploiting the spatial uncertainty δ≥0$\delta \ge 0$ in the trajectory recording process. In this paper, we prove that, for any δ>0$\delta >0$ (that is, whenever there is actual uncertainty), (k,δ)-anonymity$(k,\delta )\mathrm{-}\mathrm{anonymity}$ does not offer trajectory k-anonymity, that is, it does not hide an original trajectory in a set of k   indistinguishable anonymized trajectories. Hence, the methods based on (k,δ)-anonymity$(k,\delta )\mathrm{-}\mathrm{anonymity}$, like Never Walk Alone (NWA) and Wait For Me (W4M) can offer trajectory k  -anonymity only when δ=0$\delta =0$ (no uncertainty). Thus, the idea of exploiting the recording uncertainty δ$\delta$ to achieve trajectory k  -anonymity with information loss inversely proportional to δ$\delta$ turns out to be flawed.     

Fractional Adams–Moulton methods

In the simulation of dynamical systems exhibiting an ultraslow decay, differential equations of fractional order have been successfully proposed. In this paper we consider the problem of numerically solving fractional differential equations by means of a generalization of k  -step Adams–Moulton multistep methods. Our investigation is focused on stability properties and we determine intervals for the fractional order for which methods are at least A(π/2)$A(\pi /2)$-stable. Moreover we prove the A-stable character of k  -step methods for k=0$k=0$ and k=1$k=1$.     

Nonstandard numerical methods for a mathematical model for influenza disease

In this paper we construct and develop two competitive implicit finite difference scheme for a deterministic mathematical model associated with the evolution of influenza A disease in human population. Qualitative dynamics of the model is determined by the basic reproduction number, _R0$R_0$. Numerical schemes developed here are based on nonstandard finite difference methods. Our aim is to transfer essential properties of the continuous model to the discrete schemes and to obtain unconditional stable schemes. The proposed numerical schemes have two fixed points which are identical to the critical points of the continuous model and it is shown that they have the same stability properties. Numerical simulations with different initial conditions, parameters values and time step sizes are developed for different values of parameter _R0$R_0$, convergence to the disease free equilibrium point when _R0<1$R_0<1$ and to the endemic equilibrium point when _R0>1$R_0>1$ are obtained independent of the time step size. These numerical integration schemes are useful since can reproduce the dynamics of original differential equations.     

On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion _I1$I_1$ and _I2$I_2$ corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald q$q$-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0$q=0$ in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions _In$I_n$ (n=1,2,…$n=1\text{,}2\text{,}\dots$), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0$q\to 0$, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.