Showing papers on "Upper and lower bounds published in 1984"

Universal coding, information, prediction, and estimation

Abstract: A connection between universal codes and the problems of prediction and statistical estimation is established. A known lower bound for the mean length of universal codes is sharpened and generalized, and optimum universal codes constructed. The bound is defined to give the information in strings relative to the considered class of processes. The earlier derived minimum description length criterion for estimation of parameters, including their number, is given a fundamental information, theoretic justification by showing that its estimators achieve the information in the strings. It is also shown that one cannot do prediction in Gaussian autoregressive moving average (ARMA) processes below a bound, which is determined by the information in the data.

Parity, circuits and the polynomial time hierarchy

Abstract: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.

On H ∞ -optimal sensitivity theory for SISO feedback systems

Abstract: This paper deals with the design of feedback controllers which minimize the H^{\infty} -norm of the sensitivity function, suitably weighted. This approach to the theory of feedback design was introduced by Zames [1] and developed by Zames and Francis [2]. In this paper the theory of Sarason [3] is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm. The problem of achieving small sensitivity over a specified frequency band is studied, and the effect of nonminimum phase is elucidated. Finally, a method is introduced for handling plant poles and zeros on the imaginary axis.

Triangulating Simple Polygons and Equivalent Problems

Abstract: It' has long been known that the complexity of triangulation of simple polygons having an upper bound of 0 (n log n) but a lower bound higher than ~(n) has not been proved yet. We propose here an easily implemented route to the triangulation of simple polygons through the trapezoidization of simple polygons, which is currently done in O(n log n). Then the trapezoidized polygons are triangulated in O(n) time. Both of those steps can be performed on polygons with holes with the same complexity. We also show in this paper that a number of problems, such as the decomposition of simple polygons into convex, star, monotone, spiral, and trapezoidal polygons and the determination of edgevertex visibility, are linearly equivalent to the triangulation problem and therefore share the same lower bound. It is hoped that this will simplify the task of reducing the gap between the lower and upper bound for these problems.

Tight bounds on the complexity of parallel sorting

Abstract: In this paper, we prove tight upper and lower bounds on the number or processors, information transfer, wire area and time needed to sort N numbers in a bounded-degree fixed-connection network. Our most important new results are: 1) the construction of an O(N))-node bounded-degree network capable of sorting N numbers in O(log N) word steps, 2) a proof that any network capable of sorting N(7 log N)-bit numbers in T bit-steps requires area A where AT2≥ Ω(N2log2N), and 3) the construction of a “small-constant-factor” bounded-degree network that sorts N θ(log N)-bit numbers in T = θ(log N) bit steps with A = θ(N2) area.

On the sequential nature of unification

Abstract: The problem of unification of terms is log-space complete for P. In deriving this lower bound no use is made of the potentially concise representation of terms by directed acyclic graphs. In addition, the problem remains complete even if infinite substitutions are allowed. A consequence of this result is that parallelism cannot significantly improve on the best sequential solutions for unification. However, we show that for the problem of term matching, an important subcase of unification, there is a good parallel algorithm using O ( log 2 n ) time and n O(1) processors on a PRAM. For the O ( log 2 n ) parallel time upper bound we assume that the terms are represented by directed acyclic graphs; if the longer string representation is used we obtain an O ( log n ) parallel time bound.

New lower bound techniques for VLSI

Abstract: In this paper, we use crossing number and wire area arguments to find lower bounds on the layout area and maximum edge length of a variety of new and computationally useful networks. In particular, we describe

On the Structure of Armstrong Relations for Functional Dependencies

Abstract: An Armstrong relation for a set of functional dependencies (FDs) is a relation that satisfies each FD implied by the set but no FD that is not implied by it. The structure and size (number of tuples) of Armstrong relatsons are investigated. Upper and lower bounds on the size of minimal-sized Armstrong relations are derived, and upper and lower bounds on the number of distinct entries that must appear m an Armstrong relation are given. It is shown that the time complexity of finding an Armstrong relation, gwen a set of functional dependencies, is precisely exponential in the number of attributes. Also shown ,s the falsity of a natural conjecture which says that almost all relations obeying a given set of FDs are Armstrong relations for that set of FDs. Finally, Armstrong relations are used to generahze a result, obtained by Demetrovics using quite complicated methods, about the possible sets of keys for a relauon.

Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields

Abstract: An algorithm is described producing for each formula of the first order theory of algebraically closed fields an equivalent free of quantifiers one Denote by N a number of polynomials occuring in the formula, by d an upper bound on the degrees of polynomials, by n a number of variables, by a a number of quantifier alternations (in the prefix form) Then the algorithm works within the polynomial in the formula's size and in (Nd)n(2a+2) time Up to now a bound (Nd)no(n) was known ([5], [7], [15])

The self-equilibration of residuals and complementarya posteriori error estimates in the finite element method

Abstract: It is demonstrated that the residual in a compatible (displacement) finite element solution can be partitioned into local self-equilibrating systems on each element. An a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved.

Option Pricing Bounds in Discrete Time

Abstract: Upper and lower bounds are derived for call options traded at discrete intervals. These bounds are independent of assumptions on the stock price distribution other than a restriction satisfied by the stock being "non-negative beta." The development of the bounds relies on the single-price law and arbitrage arguments. Both single-period and multiperiod results are produced, and put option bounds follow by extension. The bounds exist as equilibrium values given a consensus on stock price distribution; they are also valid for empirical studies, being adjustable for dividends and commissions. FOLLOWING THE IMPETUS of the Black-Scholes pricing model, the majority of option analyses have been in continuous time. Such approaches have relied on the formation of perfect, riskless hedges involving the option, its underlying security, and a riskless asset; since the hedge must be continuously revised and maintained, while actual trading opportunities are discrete, this crucial assumption limits the accuracy and applicability of the Black-Scholes and related formulae. In fact, the Black-Scholes formula can only be considered as "a valid approximation to the discrete-time solution" (Merton [8, p. 663]), while the closeness of that approximation has not been ascertained in general.1 Earlier discrete models such as Boness [3] and Samuelson [12] have been succeeded by those of Brennan [5], Cox et al. [6], and Rubinstein [11], in which distributional restrictions are imposed on share returns; usually, the discrete formula converges to the Black-Scholes expression under appropriate limit conditions. Using the Rubinstein [11] approach, we derive upper and lower bounds for option prices with both a general price distribution and discrete trading opportunities. The bounds are functions of the share and exercise prices, the riskless rate of interest, the distribution of the share price change per period, and the number of periods of expiration of the option. The lower bound is tighter than that of Merton [7], derived from more general arbitrage considerations, while being less exact than the results already mentioned for restricted distributions.

An Efficient Algorithm for Computing Optimal s, S Policies

Abstract: This paper presents an algorithm to compute an optimal s, S policy under standard assumptions stationary data, well-behaved one-period costs, discrete demand, full backlogging, and the average-cost criterion. The method is iterative, starting with an arbitrary, given s, S policy and converging to an optimal policy in a finite number of iterations. Any of the available approximations can thus be used as an initial solution. Each iteration requires only modest computations. Also, a lower bound on the true optimal cost can be computed and used in a termination test. Empirical testing suggests very fast convergence.

On the Cramer- Rao bound for time delay and Doppler estimation (Corresp.)

Abstract: Using a theorem due to Whittle, simple derivations of the Cramer-Rao lower bound are presented for some delay estimation problems related to a single source, multiple sources, and multipath. The problem of Doppler estimation is briefly discussed.

Two exact algorithms for the distance‐constrained vehicle routing problem

Abstract: This paper considers a version of the vehicle routing problem in which all vehicles are identical and where the distance travelled by any vehicle may not exceed A prespecified upper bound. The problem is first formulated as an integer program which is solved by means of a constraint relaxation procedure. Two exact algorithms are developed: one based on Gomory cutting planes and one on branch and bound. Numerical results are reported. (Author/TRRL)

Tight lower and upper bounds for some distributed algorithms for a complete network of processors

Abstract: Distributed algorithms for complete asynchronous networks of processors (i.e., networks where each pair of processors is connected by a communication line) are discussed. The main result is O(nlogn) lower and upper bounds on the number of messages required by any algorithm in a given class of distributed algorithms for such networks. This class includes algorithms for problems like finding a leader or constructing a spanning tree (as far as we know, all known algorithms for those problems may require O(n2) messages when applied to complete networks). O(n2) bounds for other problems, like constructing a maximal matching or a Hamiltonian circuit are also given. In proving the lower bound we are counting the edges which carry messages during the executions of the algorithms (ignoring the actually number of messages carried by each edge). Interestingly, this number is shown to be of the same order of magnitude of the total number of messages needed by these algorithms. In the upper bounds, the length of any message is at most log2[4mlog2n] bits, where m is the maximum identity of a node in the network. One implication of our results is that finding a spanning tree in a complete network is easier than finding a minimum weight spanning tree in such a network, which may require O(n2) messages.

On characteristic exponents in turbulence

Abstract: Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Θ. The latter is particularly important because it yields an upper bound to the Hausdorff dimension of attracting sets. However, Ruelle's bound on the number has three deficiences: (i) it relies on some unproved conjectures about certain constants; (ii) it is valid only in dimensions ≧ 3 and not 2; (iii) it is valid only in the limit Θ → ∞. In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.

A golden ratio control policy for a multiple-access channel

Abstract: Consider n stations sharing a single communications channel. Each station has a buffer of length one. If the arrival rate of station i is r i , then 1-\Pi_{i}(1- r_{i}) is shown to be an upper bound (over all policies) on the throughput of the channel. Moreover, an optimal policy always exists and is stationary and periodic. The throughput of two policies, the random-policy, and the golden-ratio policy, are analyzed for a finite and infinite number of stations. The latter is shown to approach a limit which is within at least 98.4 percent of the upper bound.

Edge‐coloring of multigraphs: Recoloring technique

Abstract: New upper and lower bounds for the chromatic index of a finite muitigraph are obtained. A complete description is given of multigraphs satisfying χ′ > 1/8(9ρ + 6) where χ′ and ρ denote the chromatic index and the maximum vertex degree of a multigraph. Some new and old conjectures are discussed.

An exponential lower bound for one-time-only branching programs

Abstract: We d e f i n e b r a n c h i n g p rog rams f o l l o w i n ~ B o r o d i n e t a l . [2 7 . ~ a n c h ~ , ~ program_a r e a c y c l i c l a b e l l e d g r a p h s w i t h t he f o l l o w i n g p r o p e r t i e s : ( t ) T h e r e i s e x a c t l y one s o u r c e . ( i t ) E v e r y node h a s o u t d e g r e e a t mos t 2. ( i i i ) F o r e v e r y node v w i t h o u t d e g r e e 2, one o f t h e e d g e s l e a v i n g v i s l a b e l l e d by a B o o l e a n v a r i a b l e x i and t h e o t h e r by i t s comp l e m o n t xi • (iv) Every s~n¥ is labelled by 0 or I .

Balancing poset extensions

Abstract: We show that any finite partially ordered setP (not a total order) contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11. A consequence is that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant. Precisely: ifX is a totally ordered set about which we are given some partial information, and ife(X) is the number of total orderings ofX compatible with this information, then it is possible to sortX using no more thanC log2e (X) comparisons whereC is approximately 2.17.

Projections onto order simplexes

Abstract: Isotonic regression techniques are reinterpreted and extended to include upper and lower bounds on the ordered sequences in question. This amounts to solving the shortest distance problem for the order simplex $$S^n = \{ t \in R^n :0 \leqslant t_1 \leqslant t_2 \leqslant \cdots \leqslant t_n \leqslant 1\} $$ inR n . AnO(n) algorithm is presented for this problem, verified via the Kuhn-Tucker conditions, and explained geometrically in terms of the Lagrange multipliers. In this context, isotonic regression techniques are interpreted in terms of orthogonal projections onto faces of the order simplexS n . These projections provide a succinct characterization of the descent directions required for the design of methods for minimizing differentiable functions onS n . The latter problem arises in parameterized curve fitting.

The impact of synchronous communication on the problem of electing a leader in a ring

Abstract: We consider the problem of electing a leader in a synchronous ring of n processors. We obtain both positive and negative results. On the one hand, we show that if processor ID's are chosen from some countable set, then there is an algorithm which uses only O(n) messages in the worst case. On the other hand, we obtain two lower bound results: If the algorithm is restricted to use only comparisons of ID's, then we obtain an Ω(n log n) lower bound for the number of messages required in the worst case. Alternatively, there is a (very fast-growing) function f with the following property. If the number of rounds is required to be bounded by some t in the worst case, and ID's are chosen from any set having at least f(n,t) elements, then any algorithm requires Ω (n log n) messages in the worst case.

On the computation of the Cramer-Rao bound for ARMA parameter estimation

Abstract: The Cramer-Rao lower bound (CRLB) provides a useful tool for evaluating the performance of parameter estimation techniques. Several techniques for the computation of the asymptotic form of the CRLB for ARMA models are presented. It is shown that the asymptotic CRLB can be expressed as an explicit function of the model parameters.

Arbitrary adaptive pole placement for linear multivariable systems

Abstract: This paper presents a methodology for adaptively assigning the closed-loop poles of either continuous or discrete time linear mnltivariable systems. The scheme is direct in nature in that no parametrized model of the unknown plant needs to be explicitely identified. Rather the control parameters, and an "equivalent" plant parameterization are simultaneously estimated from input-output data using linear parameter estimation procedures. Implementation of the scheme requires only knowledge of the system controllability indexes, and an upper bound on the observabilily index.

On monotone formulae with restricted depth

Abstract: We prove a hierarchy theorem for the representation of monotone Boolean functions by monotone formulae with restricted depth. Specifically, we show that there are functions with πk-formula of size n for which every sk-formula has size exp ω(n1/(k−1)). A similar lower bound applies to concrete functions such as transitive closure and clique. We also show that any function with a formula of size n (and any depth) has a sk-formula of size exp o(n1/(k−1)). Thus our hierarchy theorem is the best possible.