Showing papers on "Upper and lower bounds published in 1997"

Systems with finite communication bandwidth constraints. I. State estimation problems

Abstract: In this paper, we investigate a state estimation problem involving finite communication capacity constraints. Unlike classical estimation problems where the observation is a continuous process corrupted by additive noises, there is a constraint that the observations must be coded and transmitted over a digital communication channel with finite capacity. This problem is formulated mathematically, and some convergence properties are defined. Moreover, the concept of a finitely recursive coder-estimator sequence is introduced. A new upper bound for the average estimation error is derived for a large class of random variables. Convergence properties of some coder-estimator algorithms are analyzed. Various conditions connecting the communication data rate with the rate of change of the underlying dynamics are established for the existence of stable and asymptotically convergent coder-estimator schemes.

Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations

Abstract: We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.

Optimal eavesdropping in quantum cryptography. i. information bound and optimal strategy

Abstract: We consider the Bennett-Brassard cryptographic scheme, which uses two conjugate quantum bases. An eavesdropper who attempts to obtain information on qubits sent in one of the bases causes a disturbance to qubits sent in the other basis. We derive an upper bound to the accessible information in one basis, for a given error rate in the conjugate basis. Independently fixing the error rates in the conjugate bases, we show that both bounds can be attained simultaneously by an optimal eavesdropping probe. The probe interaction and its subsequent measurement are described explicitly. These results are combined to give an expression for the optimal information an eavesdropper can obtain for a given average disturbance when her interaction and measurements are performed signal by signal. Finally, the relation between quantum cryptography and violations of Bell's inequalities is discussed.

Piecewise Linear Quadratic Optimal Control

Abstract: The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds

Abstract: Certain types of routing, scheduling, and resource-allocation problems in a distributed setting can be modeled as edge-coloring problems We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point-to-point model of computation Our algorithms compute an edge coloring of a graph $G$ with $n$ nodes and maximum degree $\Delta$ with at most $16 \Delta + O(\log^{1+ \delta} n)$ colors with high probability (arbitrarily close to 1) for any fixed $\delta > 0$; they run in polylogarithmic time The upper bound on the number of colors improves upon the $(2 \Delta - 1)$-coloring achievable by a simple reduction to vertex coloring To analyze the performance of our algorithms, we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables The Chernoff--Hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities However, they assume stochastic independence among certain random variables, which may not always hold Our results extend the Chernoff--Hoeffding bounds to certain types of random variables which are not stochastically independent We believe that these results are of independent interest and merit further study

Multifractal formalism for functions part I: results valid for all functions

Abstract: The multifractal formalism for functions relates some functional norms of a signal to its "Holder spectrum" (which is the dimension of the set of points where the signal has a given Holder regularity). This formalism was initially introduced by Frisch and Parisi in order to numerically determine the spectrum of fully turbulent fluids; it was later extended by Arneodo, Bacry, and Muzy using wavelet techniques and has since been used by many physicists. Until now, it has only been supported by heuristic arguments and verified for a few specific examples. Our purpose is to investigate the mathematical validity of these formulas; in particular, we obtain the following results: The multifractal formalism yields for any function an upper bound of its spectrum. We introduce a "case study," the self-similar functions; we prove that these functions have a concave spectrum (increasing and then decreasing) and that the different formulas allow us to determine either the whole increasing part of their spectrum or a p...

Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic

Abstract: A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems:(a) resolution(b) a subsystem of LK corresponding to the bounded arithmetic theory (α)(c) linear equational calculus(d) cutting planes.(2) New proofs of the exponential lower bounds (for new formulas)(a) for resolution ([15])(b) for the cutting planes proof system with coefficients written in unary ([4]).(3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Crossing Numbers and Hard Erdös Problems in Discrete Geometry

Abstract: We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution

Abstract: Recent traffic measurements in corporate local-area networks (LANs), variable-bit-rate video sources, ISDN control-channels, and other communication systems, have indicated traffic behaviour of self-similar nature. This paper first discusses some definitions and properties of (second-order) self-similarity and gives simpler criteria for it. It then gives a model of self-similar traffic suitable for queuing system analysis of an asynchronous transfer mode (ATM) queue. A lower bound to the overflow probability of a finite ATM buffer is obtained, as also a lower bound to the cell loss probability. Finally, the stationary distribution of the cell delay in an infinite ATM buffer is obtained.

Sensitivity analysis in discrete Bayesian networks

Abstract: This paper presents an efficient computational method for performing sensitivity analysis in discrete Bayesian networks. The method exploits the structure of conditional probabilities of a target node given the evidence. First, the set of parameters which is relevant to the calculation of the conditional probabilities of the target node is identified. Next, this set is reduced by removing those combinations of the parameters which either contradict the available evidence or are incompatible. Finally, using the canonical components associated with the resulting subset of parameters, the desired conditional probabilities are obtained. In this way, an important saving in the calculations is achieved. The proposed method can also be used to compute exact upper and lower bounds for the conditional probabilities, hence a sensitivity analysis can be easily performed. Examples are used to illustrate the proposed methodology.

Extended Ziv-Zakai lower bound for vector parameter estimation

Abstract: The Bayesian Ziv-Zakai bound on the mean square error (MSE) in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than MSE. The extended bound is evaluated for some representative problems in time-delay and bearing estimation. The resulting bounds have simple closed-form expressions, and closely predict the simulated performance of the maximum-likelihood estimator in all regions of operation.

An Exact Algorithm for the Vehicle Routing Problem with Backhauls

Abstract: The Vehicle Routing Problem with Backhauls is an extension of the capacitated Vehicle Routing Problem where the customers' set is partitioned into two subsets. The first is the set of Linehaul, or Delivery, customers, while the second is the set of Backhaul, or Pickup, customers. The problem is known to be NP-hard in the strong sense and finds many practical applications in distribution planning. In this paper we consider, in a unified framework, both the symmetric and the asymmetric versions of the vehicle routing problem with backhauls, for which we present a new integer linear programming model and a Lagrangian lower bound which is strengthened in a cutting plane fashion. The Lagrangian lower bound is then combined, according to-the additive approach, with a lower bound obtained by dropping the capacity constraints, thus obtaining an effective overall bounding procedure. A branch-and-bound algorithm, reduction procedures and dominance criteria are also described. Computational tests on symmetric and as...

Approximating total flow time on parallel machines

Abstract: We consider the problem of optimizing the total flow time of a stream of jobs that are released over time in a multiprocessor setting. This problem is NP-hard even when there are only two machines and preemption is allowed. Although the total (or average) flow time is widely accepted as a good measurement of the overall quality of service, no approximation algorithms were known for this basic scheduling problem. This paper contains two main results. We first prove that when preemption is allowed, Shortest Remaining Processing Time (SRPT) is an O(log(min{nm,P})) approximation algorithm for the total flow time, where n is the number of jobs, m is the number of machines, and P is the ratio between the maximum and the minimum processing time of a job. We also provide an @W(log(nm+P)) lower bound on the (worst case) competitive ratio of any randomized algorithm for the on-line problem in which jobs are known at their release times. Thus, we show that up to a constant factor SRPT is an optimal on-line algorithm. Our second main result addresses the non-preemptive case. We present a general technique that allows to transform any preemptive solution into a non-preemptive solution at the expense of an O(nm) factor in the approximation ratio of the total flow time. Combining this technique with our previous result yields an O(nmlognm) approximation algorithm for this case. We also show an @W(n^1^3^-^@e) lower bound on the approximability of this problem (assuming P NP).

Crossing Numbers and Hard Erd} os Problems in Discrete Geometry

Abstract: We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances amongn points, the minimum number of distinct distances among n points. \A statement about curves is not interesting unless it is already interesting in the case of a circle." (H. Steinhaus)

Scattering asymptotics for Riemann surfaces

Abstract: In this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends. We also give Weyl asymptotics for the relative scattering phase of such a surface. The proofs are based on trace formulae analogous to those of the Euclidean odd-dimensional scattering. The main technical ingredient is a new proof of the Poisson formula (Theorem 5.7) which is applicable in the Euclidean case as well. Our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries. The previous general lower bounds or asymptotics were either nonoptimal ([25], [58], [9]), one-dimensional or radial ([65], [67] and [54], [41]1) or they required some degeneracy of the

The Elmore delay as a bound for RC trees with generalized input signals

Abstract: The Elmore delay is an extremely popular timing-performance metric which is used at all levels of electronic circuit design automation, particularly for resistor-capacitor (RC) tree analysis. The widespread usage of this metric is mainly attributable to it being a delay measure that is a simple analytical function of the circuit parameters. The only drawback to this delay metric is the uncertainty of its accuracy and the restriction to it being an estimate only for the step response delay. In this paper, we prove that the Elmore delay measure is an absolute upper bound on the actual 50% delay of an RC tree response. Moreover, we prove that this bound holds for input signals other than steps and that the actual delay asymptotically approaches the Elmore delay as the input signal rise time increases. A lower bound on the delay is also developed using the Elmore delay and the second moment of the impulse response. The utility of this bound is for understanding the accuracy and the limitations of the Elmore metric as we use it as a performance metric for design automation.

Lower bounds of tower type for Szemerédi's uniformity lemma

Abstract: It is known that the size of the partition obtained in Szemeredi's Uniformity Lemma can be bounded above by a number given by a tower of 2s of height proportional to $\epsilon^{-5}$ , where $\epsilon$ is the desired accuracy. In this paper, we first show that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to $ \log{(1/ \epsilon)} $ ). We then give a different construction which improves the bound, even for certain weaker versions of the statement.

Slope stability analysis by the upper bound approach: fundamentals and methods

Abstract: A new method for stability analysis in soils and rocks is presented, based on the upper bound theorem of classical plasticity. The sliding mass is divided into a small number of discrete blocks, with linear interfaces between blocks and either linear or curved bases to individual blocks. By equating the work done by external loads and body forces to the energy dissipated in shearing, either a safety factor or a disturbance factor may be calculated. The rigorous theoretical background is established, from which it may be demonstrated that for several well-defined classical slope problems the equations for the multi-block solution reduce to the published closed-form solutions. Powerful optimization routines are provided in the computer program EMU to search for the critical failure mechanism giving the lowest factor of safety. Several examples are given to demonstrate that, for problems where the exact answers are known, the new method produces accurate values of safety factor and predictions of failure mechanism. Applications to practical problems have shown that the new method is as simple as the conventional limit equilibrium methods for practitioners.

Better bounds for online scheduling

Abstract: We study a classical problem in online scheduling. A sequence of jobs must be scheduled on m identical parallel machines. As each job arrives, its processing time is known. The goal is to minimize the makespan. Bartal et al. (J. Comput. System Sci., 51 (1995), pp. 359{366) gave a deterministic online algorithm that is 1.986-competitive. Karger, Phillips, and Torng (J. Algorithms, 20 (1996), pp. 400{430) generalized the algorithm and proved an upper bound of 1.945. The best lower bound currently known on the competitive ratio that can be achieved by deterministic online algorithms is equal to 1.837. In this paper we present an improved deterministic online scheduling algorithm that is 1.923-competitive; for all m 2. The algorithm is based on a new scheduling strategy, i.e., it is not a generalization of the approach by Bartal et al. Also, the algorithm has a simple structure. Furthermore, we develop a better lower bound. We prove that, for general m, no deterministic online scheduling algorithm can be better than 1.852-competitive.

Regularity of weak solutions of the compressible isentropic Navier-Stokes equations

Abstract: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations is proven for small time in dimension N = 2 or 3 under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to τ > 1 when N = 2 and p(φ) = aφτ. Moreover,weak solutions in T2turn out to be smooth as long as the density remains bounded in L∞( T2).

Effective properties of nonlinear composites

Abstract: These lectures describe several procedures commonly used or recently developed to predict the overall behavior of nonlinear composites from the behavior of their individual constituents and from statistical information about their microstructure. Secant methods are discussed in section 2. A modified method based on the second-order moment of the strain field is proposed and compared with the classical secant method in specific situations, composites with large or small contrast and power-law materials. The incremental method is presented in section 3. It appears much stiffer than the two secant methods. Its predictions for isotropic two-phase power-law composites even violate a rigorous upper bound when the nonlinearity is strong. A variational procedure leading to rigorous upper bounds for the effective potential of the composite is presented in section 4. Specific forms for voided or rigidly reinforced power-law composites are given first. Then a general upper bound applying to a general class of nonlinear composites is derived. The variational procedure coincides with the secant approach based on second-order moments and with the variational procedure of Ponte Castnneda. These different schemes are applied in section 5 to predict the overall behavior of metal-matrix composites. A simplified model based on the variational procedure is proposed. Its predictions compared well with simulations performed by the Finite Element Method.

Relativistic one-electron hamiltonians 'for electrons only' and the variational treatment of the dirac equation

Abstract: A relation χ = Xϕ between the upper and lower components of the Dirac spinor ψ =( ϕ , χ ), with X satisfying a nonlinear operator equation, guarantees that the expectation value 〈 D 〉 ψ of the Dirac operator is bounded from below by the exact electronic ground state energy. Unfortunately X can, except for special cases, not be constructed in closed form. It is relatively easy to satisfy the condition χ = Xϕ if ϕ has the correct behaviour near a point nucleus, i.e. if it goes in the spherical average as r ν with ν slightly smaller than 0. To satisfy χ = Xϕ for trial functions regular at the position of the nuclei is possible in principle, but very difficult in practice. If one satisfies the condition χ = Xϕ only approximatively , one may still get a lower bound, i.e. a variationally stable approach, but the lower bound is below its exact counterpart, i.e. the exact electronic ground state. Examples of variationally stable approaches are analyzed, in particular the regularized stationary direct perturbation theory (SDPT), the so-called regular approximation (RA), and the Douglas-Kroll-Hess transformation (DKH), of which a few new features are revealed. Finally the possibility of a minimax principle for the Dirac equation is discussed.

Stability of 2-D discrete systems described by the Fornasini-Marchesini second model

Abstract: Based on the Fornasini-Marchesini second local state-space (LSS) model, criteria that sufficiently guarantee the asymptotic stability of 2-D discrete systems are given. A sufficient condition for a 2-D nonlinear discrete system to be free of overflow oscillations is then shown in the case when a 2-D discrete system is employed by saturation arithmetic. Finally, an upper bound on parameter variations which guarantees the asymptotic stability of a perturbed 2-D discrete system is considered. It is shown that the upper bound stated here is less conservative than the existing ones.

A Barankin-type lower bound on the estimation error of a hybrid parameter vector

Abstract: The Barankin (1949) bound is a realizable lower bound on the mean-square error (MSE) of any unbiased estimator of a (nonrandom) parameter vector. We present a Barankin-type bound which is useful in problems where there is a prior knowledge on some of the parameters to be estimated. That is, the parameter vector is a hybrid vector in the sense that some of its entries are deterministic while other are random variables. We present a simple expression for a positive-definite matrix which provides bounds on the covariance of any unbiased estimator of the nonrandom parameters and an estimator of the random parameters, simultaneously. We show that the Barankin bound for deterministic parameters estimation and the Bobrovsky-Zakai (1976) bound for random parameters estimation are special cases of our proposed bound.

Lower bounds for cutting planes proofs with small coefficients

Abstract: We consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

Bayesian Algorithms for One-Dimensional GlobalOptimization

Abstract: In this paper Bayesian analysis and Wiener process are used in order to build an algorithm to solve the problem of global optimization The paper is divided in two main parts In the first part an already known algorithm is considered: a new (Bayesian) stopping rule is added to it and some results are given, such as an upper bound for the number of iterations under the new stopping rule In the second part a new algorithm is introduced in which the Bayesian approach is exploited not only in the choice of the Wiener model but also in the estimation of the parameter \sigma^2 of the Wiener process, whose value appears to be quite crucial Some results about this algorithm are also given

Tight Bounds on the Information Rate of Secret SharingSchemes

Abstract: A secret sharing scheme is a protocol by means of which a dealer distributes a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P, non-qualified to know s, cannot determine anything about the value of the secret. In this paper we provide a general technique to prove upper bounds on the information rate of secret sharing schemes. The information rate is the ratio between the size of the secret and the size of the largest share given to any participant. Most of the recent upper bounds on the information rate obtained in the literature can be seen as corollaries of our result. Moreover, we prove that for any integer d there exists a d-regular graph for which any secret sharing scheme has information rate upper bounded by 2/(d+1). This improves on van Dijk‘s result dik and matches the corresponding lower bound proved by Stinson in [22].