Showing papers on "Probabilistic analysis of algorithms published in 1982"

On the Analysis and Synthesis of VLSI Algorithms

Abstract: This correspondence is concerned with the development of algorithms for special-purpose VLSI arrays. The approach used in this correspondence is to identify algorithm transformations which modify favorably the index set and the data dependences, but perserve the ordering imposed on the index set by the data dependences. Conditions for the existance of such transformations are given for a class of algorithms. Also, a methodology is proposed for the synthesis of VLSI algorithms.

Problem solving techniques for the design of algorithms

Abstract: By studying the problem-solving techniques that people use to design algorithms we can learn something about building systems that automatically derive algorithms or assist human designers. In this paper we present a model of algorithm design based on our analysis of the protocols of two subjects designing three convex hull algorithms. The subjects work mainly in a data-flow problem space in which the objects are representations of partially specified algorithms. A small number of general-purpose operators construct and modify the representations; these operators are adapted to the current problem state by means-ends analysis. The problem space also includes knowledge-rich schemas such as divide and conquer that subjects incorporate into their algorithms. A particularly versatile problem-solving method in this problem space is symbolic execution, which can be used to refine, verify, or explain components of an algorithm. The subjects also work in a task-domain space about geometry. The interplay between problem solving in the two spaces makes possible the process of discovery. We have observed that the time a subject takes to design an algorithm is proportional to the number of components in the algorithm's data-flow representation. Finally, the details of the problem spaces provide a model for building a robust automated system.

Probabilistic analysis of some bin-packing problems

Abstract: We analyze the average-case behaviour of the Next-Fit algorithm for bin-packing, and obtain closed-form expressions for distributions of interest. Our analysis is based on a novel technique of partitioning the interval (0, 1) suitably and then formulating the problem as a matrix-differential equation. We compare our analytic results with previously known simulation results and show that there is an excellent agreement between the two. We also explain a certain empirically observed anomaly in the behaviour of the algorithm. Finally we establish that asymptotically perfect packing is possible when input items are drawn from a monotonically decreasing density function.

Robustness of probabilistic computational complexity classes under definitional perturbations

Abstract: Various types of probabilistic algorithms play an increasingly important role in computer science, especially in computational complexity theory. Probabilistic polynomial time complexity classes are defined and compared to each other, emphasizing some structural relationships to the known complexity classes P, NP, PSPACE. The classes R and ZPP, corresponding to the so-called Las Vegas polynomial time bounded algorithms, are given special attention. It is shown that many definitions, that arise naturally from different types of algorithms, are equivalent in defining the same class R (or ZPP, resp.). These robustness results justify finally the tractability of the above probabilistic classes.

Specification tests for probabilistic choice models

Abstract: Probabilistic choice models, such as logit and probit models, are highly sensitive to a variety of specification errors, including the use of incorrect functional forms for the systematic component of the utility function, incorrect specification of the probability distribution of the random component of the utility function, and incorrect specification of the choice set. Specification errors can cause large forecasting errors, so it is of considerable importance to have means of testing models for the presence of these errors. A number of tests based on the likelihood ratio statistic have been developed. These tests and available information on their power are summarized in this paper. The likelihood ratio test can entail considerable computational difficulty, owing to the need to evaluate the likelihood function for both the null and alternative hypotheses. Substantial gains in computational efficiency can be achieved through the use of a test that requires evaluating the likelihood function only for the null hypothesis. A lagrangian multiplier test that has this property is described, and numerical examples of its computational properties are given. An important disadvantage of conventional specification tests is that they do not permit comparisons of models that belong to different parametric families in order to determine which model best explains the available data. Thus, these tests cannot be used to compare models whose utility functions have substantially different functional forms or models that are based on different behavioural paradigms. Several methods for dealing with this problem, including the construction of hybrid models and the Cox test of separate families of hypotheses, are described. (Author/TRRL)

Wafer-scale integration of systolic arrays

Abstract: VLSI technologists are fast developing wafer-scale integration. Rather than partitioning a silicon wafer into chips as is usually done, the idea behind wafer-scale integration is to assemble an entire system (or network of chips) on a single wafer, thus avoiding the costs and performance loss associated with individual packaging of chips. A major problem with assembling a large system of microprocessors on a single wafer, however, is that some of the processors, or cells, on the wafer are likely to be defective. In the paper, we describe practical procedures for integrating "around" such faults. The procedures are designed to minimize the length of the longest wire in the system, thus minimizing the communication time between cells. Although the underlying network problems are NP-complete, we prove that the procedures are reliable by assuming a probabilistic model of cell failure. We also discuss applications of the work to problems in VLSI layout theory, graph theory, fault-tolerant systems, planar geometry, and the probabilistic analysis of algorithms.

A family of solution algorithms for nonlinear structural analysis based on relaxation equations

Abstract: A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko-Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by-product of the algorithms is that they provide a systematic and economical means for computing the stepsize at each iteration stage when a Newton-like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.

Probabilistic Analysis of Deposit Liquefaction

Abstract: A probabilistic methodology is presented for liquefaction analysis of horizontally layered sand deposits, subjected to vertically propagating seismic S waves. Three different deposit models are developed. The models are effectively one-dimensional and differ in that they include or neglect pore pressure diffusion and stiffness reduction due to pore pressure buildup. Conclusions are drawn about sensitivity of the results to different assumptions about the mechanical behavior of the deposit, the effect of vertical and horizontal variations of soil properties, and about the importance of statistical variability of the response spectrum at bedrock given peak acceleration. An approximate procedure is proposed for the calculation of the probability of almost complete layer liquefaction, for given input motion and given layer average SPT results. Predictions from the approximate procedure are in good agreement with historical data.

On the Power of Probabilistic Choice in Synchronous Parallel Computations

Abstract: This paper introduces probabilistic choice to synchronous parallel machine models; in particular parallel RAMs The power of probabilistic choice in parallel computations is illustrated by parallelizing some known probabilistic sequential algorithms We characterize the computational complexity of time, space, and processor bounded probabilistic prallel RAMs in terms of the computational complexity of probabilistic sequential RAMs We show that parallelism uniformly speeds up time bounded probabilistic sequential RAM computations by nearly a quadratic factor We also show that probabilistic choice can be, eliminated from parallel computations by introducing nonuniformity

Probabilistic analysis of combinatorial algorithms: A bibliography with selected annotations

Abstract: Probabilistic methods in evaluation of performance efficiency of combinatorial optimization algorithms are of continuously growing interest, and rapidly increasing effort of researchers in this field is observed. The present paper is a bibliography which contains 70 references dealing with probabilistic evaluation of time complexity and performance accuracy of deterministic algorithms for combinatorial decision and optimization problems. Some entries of the bibliography, mainly those having appreared in journals and nonperiodical issues of limited distribution are shortly annotated (18 references). Basic notions and definitions facilitating better understanding and plain presentation of different results are given.

Probabilistic Analysis of the LPT Processor Scheduling Heuristic

Abstract: The average performance of the LPT processor scheduling algorithm is analyzed, under the assumption that task times are drawn from a uniform distribution on (0,1]. The ratio of the expected length of the schedule to the expected length of a preemptive schedule is shown to be bounded by 1 + 0(m2 /n2 ), where m is the number of processors, and η is the number of tasks.

Multicommodity Network Flows with Probabilistic Losses

Abstract: This paper considers the problem of maximizing the expected value of multicommodity flows in a network in which the arcs experience probabilistic loss rates. Consideration of probabilistic losses are relevant, particularly, in communication and transportation networks. An arc-chain formulation of the problem and an efficient algorithm for computing an optimal solution are provided. The algorithm involves a modified column generation technique for identifying a constrained chain. Computational experience with the algorithm is included.

Probabilistic simulations (Preliminary Version)

Abstract: The results of this paper concern the question of how fast machines with one type of storage media can simulate machines with a different type of storage media. Most work on this question has focused on the question of how fast one deterministic machine can simulate another. In this paper we shall look at the question of how fast a probabilistic machine can simulate another. This approach should be of interest in its own right, in view of the great attention that probabilistic algorithms have recently attracted.

Algorithms for replace-add based paracomputers.

Abstract: The space between and surrounding aligned window openings of a vehicle cab and a camper carried thereby is sealed by an endless flexible member of sheet material of a length to surround the aligned window openings and of a width greater than the space therebetween. An endless resilient member is carried by the edge of each flexible member with each resilient member being of a length greater than the peripheral distance around the window opening adjacent thereto and adapted to snap therethrough and then retain a position outwardly thereof.

Probabilistic finite state system identification

Abstract: This paper explores the problem of finding behavioral models of finite state systems, given observed data of systems which are presumed to lie in this category. Following an informal introduction, a formal definition of an identification problem is given along with the form that a solution to such a problem may be expected to lake. The most general type of such problem involving finite state systems is described and several possible approaches to developing a solution to this general problem are discussed. Several classes of more specialized classes of systems and their identification problems are developed. For these, a set of algorithms are described which may be used to solve them. The complexity of these algorithms is discussed.

Deterministic Versus Probabilistic Factorization of Integral Polynomials

Abstract: We have shown that the Cantor-Zassenhaus probabilistic step to find a factor over GF(p) of a polynomial being the product of equal degree factors takes O(n3L2(p)Lβ(p)2) units of time. This is, using classical arithmetic, the cost in Rabin's algorithm to find a root of an irreducible factor of degree d in the extension field GF(pd). The constants involved in Rabin's algorithm seem to be higher than in the simple Cantor-Zassenhaus test, which is the most promising candidate of a probabilistic algorithmic to be compared with the deterministic Berlekamp-Hensel algorithm. A careful analysis, backed up by measurements using current technology in computer algebra, demonstrates that the time has not yet come to beat Berlekamp-Hensel. This is mainly due to the cost of exponentiation both in Berlekamp's Q-matrix and in the probabilistic test of Cantor-Zassenhaus which makes both algorithms for large primes intractable. In contradistinction : the restriction of the Berlekamp-Hensel algorithm to small primes is computationally its greatest strength.

Probabilistic analysis of the solution of the knapsack problem

Abstract: In this paper the value of the optimal solution of a random instance of the Knapsack problem is analyzed. With respect to this value, the performance of a simple greedy heuristic for the solution of this problem is evaluated. The results are compared with the performance of other greedy heuristics.

Analysis of Covariance Algorithms

Abstract: Computational algorithms specific to the analysis of covariance are discussed for the treatment of both balanced and unbalanced data. The use of covariance algorithms in the solution of missing data problems is also considered.

Probabilistic Analysis of Algorithms

Abstract: This paper is a brief introduction to the field of probabilistic analysis of algorithms; it is not a comprehensive survey. The first part of the paper examines three important probabilistic algorithms that together illustrate many of the important points of the field, and the second part then generalizes from those examples to provide a more systematic view.

Tracking performance of stochastic gradient algorithms for nonstationary processes

Abstract: The solution to the linear mean-square estimation problem on the basis of a priori knowledge of the relevant correlations is well known. However, in many applications a priori knowledge of the correlations is not available and the correlations may be changing with time. In this case, adaptive estimation is indicated. In this research, probabilistic analysis of learning characteristics of the class of stochastic-gradient-descent algorithms for adapting the parameters of a non-recursive linear filter in order to identify a time-varying system from noisy measurements is presented. Characteristics analyzed include stability, rate of convergence, steady state average and RMS misadjustments, average- and RMS-optimum adaptation step sizes, and sensitivity. A measure of the degree of nonstationarity for the system to be identified is proposed and its effects together with the effects of gradient-averaging, adaptation step size, and signal-to-noise ratio, on the learning characteristics are studied. The tracking performance of this class of algorithms is studied for the case in which the unknown system evolves according to (i) a first-order Markov random process and (ii) a periodic process. Simulation results that verify the accuracy of theoretical performance predictions are presented.