The optimal information rate for graph access structures of nine participants

The information rate is an important metric of the performance of a secret-sharing scheme. In this paper we consider 272 non-isomorphic connected graph access structures with nine vertices and eight or nine edges, and either determine or bound the optimal information rate in each case. We obtain exact values for the optimal information rate for 231 cases and present a method that is able to derive information-theoretical upper bounds on the optimal information rate. Moreover, we apply some of the constructions to determine lower bounds on the information rate. Regarding information rate, we conclude with a full listing of the known optimal information rate (or bounds on the optimal information rate) for all 272 graphs access structures of nine participants.     

An updated version of the Kantorovich theorem for Newton's method

An affine invariant version of the Kantorovich theorem for Newton's method is presented. The result includes the Gragg-Tapia error bounds, as well as recent optimal and sharper upper bounds, new optimal and sharper lower bounds, and new inequalities showingq-quadratic convergence all in terms of the usual majorizing sequence. Closed form expressions for these bounds are given.     

Area-time lower-bound techniques with applications to sorting

Thearea-time complexity of VLSI computations is constrained by the flow and the storage of information in the two-dimensional chip. We study here the information exchanged across the boundary of the cells of asquare-tessellation of the layout. When the information exchange is due to thefunctional dependence between variables respectively input and output on opposite sides of a cell boundary, lower bounds are obtained on theAT ² measure (which subsume bisection bounds as a special case). When information exchange is due to thestorage saturation of the tessellation cells, a new type of lower bound is obtained on theAT measure. In the above arguments, information is essentially viewed as a fluid whose flow is uniquely constrained by the available bandwidth. However, in some computations, the flow is kept below capacity by the necessity to transform information before an output is produced. We call this mechanismcomputational friction and show that it implies lower bounds on theAT/logA measure. Regimes corresponding to each of the three mechanisms described above can appear by varying the problem parameters, as we shall illustrate by analyzing the problem of sortingn keys each ofk bits, for whichAT ²,AT, andAT/logA bounds are derived. Each bound is interesting, since it dominates the other two in a suitable range of key lengths and computations times.     

New bounds on D-ary optimal codes

We propose a simple method that, given a symbol distribution, yields upper and lower bounds on the average code length of a D-ary optimal code over that distribution. Thanks to its simplicity, the method permits deriving analytical bounds for families of parametric distributions. We demonstrate this by obtaining new bounds, much better than the existing ones, for Zipf and exponential distributions when D>2.     

Property Testing Lower Bounds via Communication Complexity

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of ??(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.     

Optimal left and right additive Schwarz preconditioning for minimal residual methods with Euclidean and energy norms

For the solution of non-symmetric or indefinite linear systems arising from discretizations of elliptic problems, two-level additive Schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem are independent of the mesh and the number of subdomains. These bounds are based on some kind of energy norm. However, in practice, iterative methods which minimize the Euclidean norm of the residual are used, despite the fact that the usual bounds are non-optimal, i.e., the quantities appearing in the bounds may depend on the mesh size; see [X.-C. Cai, J. Zou, Some observations on the l² convergence of the additive Schwarz preconditioned GMRES method, Numer. Linear Algebra Appl. 9 (2002) 379-397]. In this paper, iterative methods are presented which minimize the same energy norm in which the optimal Schwarz bounds are derived, thus maintaining the Schwarz optimality. As a consequence, bounds for the Euclidean norm minimization are also derived, thus providing a theoretical justification for the practical use of Euclidean norm minimization methods preconditioned with additive Schwarz. Both left and right preconditioners are considered, and relations between them are derived. Numerical experiments illustrate the theoretical developments.     

Lower Bounds for Dynamic Data Structures on Algebraic RAMs

In a seminal paper of 1989, Fredman and Saks proved lower bounds for some important data-structure problems in the cell probe model. This model assumes that data structures are stored in memory with a known word length. In this paper we consider random access machines (RAMs) that can add, subtract, compare, multiply and divide integer or real numbers, with no size limitation. These are referred to as algebraic RAMs . We prove new lower bounds for two important data-structure problems, union-findand dynamic prefix sums . To this end we apply the generalized Fredman—Saks techniqueintroduced by the authors in a previous paper. The generalized technique relies on a certain well-defined function, Output Variability , that characterizes in some sense the power of the computational model. Fredman and Saks' work implied bounds on output variability for the cell probe model; in this paper we provide the first bounds for algebraic RAMs, and show that they suffice for proving tight lower bounds for useful problems. An important feature of the problems we consider is that in a data structure of size n , the data stored are members of {0,\ldots,n} . This makes the derivation of lower bounds for such problems on a RAM without word-size limitations a particular challenge; previous RAM lower bounds we are aware of depend on the fact that the data for the computation can vary over a large domain.     

Tight end-to-end per-flow delay bounds in FIFO multiplexing sink-tree networks

Aggregate scheduling has been proposed as a solution for achieving scalability in large-size networks. However, in order to enable the provisioning of real-time services, such as video delivery or voice conversations, in aggregate scheduling networks, end-to-end delay bounds for single flows are required. In this paper, we derive per-flow end-to-end delay bounds in aggregate scheduling networks in which per-egress (or sink-tree) aggregation is in place, and flows traffic is aggregated according to a FIFO policy. The derivation process is based on Network Calculus, which is suitably extended to this purpose. We show that the bound is tight by deriving the scenario in which it is attained. A tight delay bound can be employed for a variety of purposes: for example, devising optimal aggregation criteria and rate provisioning policies based on pre-specified flow delay bounds.     

On the global avalanche characteristics between two Boolean functions and the higher order nonlinearity

The criterion for the global avalanche characteristics (GAC) of cryptographic functions is an important property. To measure the correlation between two arbitrary Boolean functions, we propose two new criteria called the sum-of-squares indicator and the absolute indicator of the cross-correlation between two Boolean functions. The two indicators generalize the GAC criterion. Based on the properties of the cross-correlation function, we deduce the rough lower and the rough upper bounds on the two indicators by hamming weights of two Boolean functions, and generalize some properties between the Walsh spectrum and the cross-correlation function. Furthermore, we give the tight upper and the tight lower bounds on the two indicators. Finally, we show some relationships between the upper bounds on the two indicators and the higher order nonlinearity.     

Decision algorithms for multiplayer noncooperative games of incomplete information

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines.In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S(n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S(n). Furthermore, S(n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S(n). This paper proves that this exponential blow-up can occur.We also show that time bounded games do not exhibit such hierarchy. A T(n) time bounded blindfold multiplayer game, as well as a T(n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T(n).     

Bandit problems with side observations

An extension of the traditional two-armed bandit problem is considered, in which the decision maker has access to some side information before deciding which arm to pull. At each time t, before making a selection, the decision maker is able to observe a random variable X/sub t/ that provides some information on the rewards to be obtained. The focus is on finding uniformly good rules (that minimize the growth rate of the inferior sampling time) and on quantifying how much the additional information helps. Various settings are considered and for each setting, lower bounds on the achievable inferior sampling time are developed and asymptotically optimal adaptive schemes achieving these lower bounds are constructed.     

An information statistics approach to data stream and communication complexity

We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.Our paradigm leads to two main results:(1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k>2, approximating the kth frequency moment in the data stream model requires essentially Ω(n1−2/k) space; this resolves a conjecture of Alon et al. (J. Comput. System Sci. 58(1) (1999) 137).(2) A lower bound for the L_p approximation problem in the general communication model; this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360-369). As a consequence, we show that for p>2, approximating the L_p norm to within a factor of n^ε in the data stream model with constant number of passes requires Ω(n1−4ε−2/p) space.     

New H ∞ Controller Design for Neural Networks with Interval Time-Varying Delays in State and Observation

This article addresses the H _∞ control problem of delayed neural networks, where the state input and observation output contain interval non-differentiable time-varying delays. Based on constructing a new set of Lyapunov–Krasovskii functionals, new delay-dependent sufficient criteria for H _∞ control are established in terms of linear matrix inequalities. The Lyapunov–Krasovskii functional is mainly based on the information of the lower and upper delay bounds, which allows us to avoid using additional free-weighting matrices and any assumption on the differentiability of the delay function. The obtained condition is less conservative because of the technique of designing state feedback controller. The H _∞ controller to be designed must satisfy some exponential stability constraints on the closed-loop poles. A numerical example is given to illustrate the effectiveness of our results.     

Cone-bounded nonlinearities and mean-square bounds—Smoothing and prediction

Mean-square performance bounds are derived for smoothing and prediction problems associated with the broad class of nonlinear dynamic systems which, when modeled by Ito differential equations, contain drift (·dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. Included in this paper are lower bounds on the error covariance attainable by any smoother or any predictor, including the optimum, and upper bounds on the performance of some simple, implementable predictors reminiscent of the designs which are optimal in the linear case. The lower bounds on smoothing and prediction performance are established using measure-transformation techniques to relate a version of the nonlinear problem to its linearization. The upper bound on prediction performance is constructed by a direct analysis of the estimation error. All the bounds hold for correlated system and observation noises. All are rigorously derived and independent of control or control law. In each case, the computational effort is comparable to that for the corresponding optimum linear smoothing or prediction problem. The bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.     

Computing (r, Q) policy for an inventory system with limited sharable resource

This paper deals with inventory systems with limited resource for a single item or multiple items under continuous review (r, Q) policies. For the single-item system with a stochastic demand and limited resource, it is shown that an existing algorithm can be applied to find an optimal (r, Q) policy that minimizes the expected system costs. For the multi-item system with stochastic demands and limited resource commonly shared among all items, an optimization problem is formulated for finding optimal (r, Q) policies for all items, which minimize the expected system costs. Bounds on the parameters (i.e., r and Q) of the optimal policies and bounds on the minimum expected system costs are obtained. Based on the bounds, an algorithm is developed for finding an optimal or near-optimal solution. A method is proposed for evaluating the quality of the solution. It is shown that the algorithm proposed in this paper finds a solution that is (i) optimal/near-optimal and/or (ii) significantly better than the optimal solution with unlimited resource.     

Online Search with Time-Varying Price Bounds

Online search is a basic online problem. The fact that its optimal deterministic/randomized solutions are given by simple formulas (however with difficult analysis) makes the problem attractive as a target to which other practical online problems can be transformed to find optimal solutions. However, since the upper/lower bounds of prices in available models are constant, natural online problems in which these bounds vary with time do not fit in the available models.We present two new models where the bounds of prices are not constant but vary with time in certain ways. The first model, where the upper and lower bounds of (logarithmic) prices have decay speed, arises from a problem in concurrent data structures, namely to maximize the (appropriately defined) freshness of data in concurrent objects. For this model we present an optimal deterministic algorithm with competitive ratio \(\sqrt{D}\), where D is the known duration of the game, and a nearly-optimal randomized algorithm with competitive ratio \(\frac{\ln D}{1+\ln2-\frac{2}{D}}\). We also prove that the lower bound of competitive ratios of randomized algorithms is asymptotically \(\frac{\ln D}{4}\).The second model is inspired by the fact that some applications do not utilize the decay speed of the lower bound of prices in the first model. In the second model, only the upper bound decreases arbitrarily with time and the lower bound is constant. Clearly, the lower bound of competitive ratios proved for the first model holds also against the stronger adversary in the second model. For the second model, we present an optimal randomized algorithm. Our numerical experiments on the freshness problem show that this new algorithm achieves much better/smaller competitive ratios than previous algorithms do, for instance 2.25 versus 3.77 for D=128.     

Computational algebraic algorithms for the reliability of generalized k-out-of-n and related systems

Identities and bounds for the reliability of coherent systems are analysed and computed using the techniques of commutative algebra. The techniques are applied to the analysis of some of the most relevant k-out-of-n class systems. The efficiency of the algebraic approach in obtaining exact identities, bounds and asymptotic formulas shows good performance when compared with results from the literature. The papers points to some new applications of these techniques that emphasize the connection of algebra and probability in this context.     

Learning the set covering machine by bound minimization and margin-sparsity trade-off

We investigate classifiers in the sample compression framework that can be specified by two distinct sources of information: a compression set and a message string of additional information. In the compression setting, a reconstruction function specifies a classifier when given this information. We examine how an efficient redistribution of this reconstruction information can lead to more general classifiers. In particular, we derive risk bounds that can provide an explicit control over the sparsity of the classifier and the magnitude of its separating margin and a capability to perform a margin-sparsity trade-off in favor of better classifiers. We show how an application to the set covering machine algorithm results in novel learning strategies. We also show that these risk bounds are tighter than their traditional counterparts such as VC-dimension and Rademacher complexity-based bounds that explicitly take into account the hypothesis class complexity. Finally, we show how these bounds are able to guide the model selection for the set covering machine algorithm enabling it to learn by bound minimization.     

List decoding for a multiple access hyperchannel

We obtain bounds on the rate of (optimal) list-decoding codes with a fixed list size L ≥ 1 for a q-ary multiple access hyperchannel (MAHC) with s ≥ 2 inputs and one output. By definition, an output signal of this channel is the set of symbols of a q-ary alphabet that occur in at least one of the s input signals. For example, in the case of a binary MAHC, where q = 2, an output signal takes values in the ternary alphabet {0, 1, {0, 1}}; namely, it equals 0 (1) if all the s input signals are 0 (1) and equals {0, 1} otherwise. Previously, upper and lower bounds on the code rate for a q-ary MAHC were studied for L ≥ 1 and q = 2, and also for the nonbinary case q ≥ 3 for L = 1 only, i.e., for so-called frameproof codes. Constructing upper and lower bounds on the rate for the general case of L ≥ 1 and q ≥ 2 in the present paper is based on a substantial development of methods that we designed earlier for the classical binary disjunctive multiple access channel.     

Differential approximation results for the traveling salesman and related problems

This paper deals with the problem of constructing a Hamiltonian cycle of optimal weight, called TSP. We show that TSP is 2/3-differential approximable and cannot be differential approximable greater than 649/650. Next, we demonstrate that, when dealing with edge-costs 1 and 2, the same algorithm idea improves this ratio to 3/4 and we obtain a differential non-approximation threshold equal to 741/742. Remark that the 3/4-differential approximation result has been recently proved by a way more specific to the 1-, 2-case and with another algorithm in the recent conference, Symposium on Fundamentals of Computation Theory, 2001. Based upon these results, we establish new bounds for standard ratio: 5/6 for MaxTSP[a,2a] and 7/8 for MaxTSP[1,2]. We also derive some approximation results on partition graph problems by paths.