Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

Towards optimal simulations of formulas by bounded-width programs

We show that, over an arbitrary ring, for any fixed >0, all balanced algebraic formulas of sizes are computed by algebraic straight-line programs that employ a constant number of registers and have lengthO (s ¹⁺). In particular, in the special case where the ring isGF(2), we obtain a technique for simulating balanced Boolean formulas of sizes by bounded-width branching programs of lengthO(s ¹⁺), for any fixed >0. This is an asymptotic improvement in efficiency over previous simulations in both the Boolean and algebraic settings.     

Strongly adaptive token distribution

The token distribution (TD) problem, an abstract static variant of load balancing, is defined as follows: letM be a (parallel processor) network with setP of processors. Initially, each processorP P has a certain amountl(P) of tokens. The goal of a TD algorithm, run onM, is to distribute the tokens evenly among the processors. In this paper we introduce the notion of strongly adaptive TD algorithms, i.e., algorithms whose running times come close to the best possible runtime, the off-line complexity of the TD problem, for each individual initial token distributionl. Until now, only weakly adaptive algorithms have been considered, where the running time is measured in terms of the maximum initial load max{l(P)P P}.We design an almost optimal, strongly adaptive algorithm on mesh-connected networks of arbitrary dimension extended by a single 1-bit bus. This result shows that an on-line TD algorithm can come close to the optimal (off-line) bound for each individual initial load. Furthermore, we exactly characterize the off-line complexity of arbitrary initial token distributions on arbitrary networks. As an intermediate result, we design almost optimal weakly adaptive algorithms for TD on mesh-connected networks of arbitrary dimension.This research was partially supported by DFG-Forschergruppe Effiziente Nutzung massiv paralleler Systeme, Teilprojekt 4, by the ESPRIT Basic Research Action No. 7141 (ALCOM II), and by the Volkswagen-stiftung. A preliminary version was presented at the 20th ICALP, 1993, see [9].     

An Interval Linear Tolerance Problem

This paper is concerned with a tolerance problem for an interval linear system A x = b requiring inner estimation of the admissible solution set {x ⁿ | (A A)(Ax b)} formed by vectors x for which the product Ax remains within b for any possible A A. Methods for verifying the emptiness and nonemptiness of admissible solution sets are developed. Formulas for the dimensions of the interval solution of a tolerance problem with known center are derived.     

A combinatorial property of EOL languages

Let be an alphabet and II its nontrivial binary partition. Each word over can uniquely be decomposed in subwords (called blocks) consisting of letters of II _i only,i {1,2}. LetK ^*.K has a long block property (with respect to II), abbreviated asLB-property, if there exists a functionf:N ⁺ N ⁺ such that for everyw K and every positive integerm the number of blocks of length at mostm inw is bounded byf(m). K has a clustered block property (with respect to II), abbreviated asCB-property, if there exists a positive integern _o and a growing functiong:N ⁺ N ⁺ such that for everyw K and for every positive integerm the blocks of length at mostm can be covered by at mostn _o segments of length at mostg(m).It is proved that aCB-property always implies aLB-property but not necessarily other way around. It is proved that an EOL language has aLB-property if and only if it has aCB-property.     

Randomness in interactive proofs

This paper initiates a study of the quantitative aspects of randomness in interactive proofs. Our main result, which applies to the equivalent form of IP known as Arthur-Merlin (AM) games, is a randomness-efficient technique for decreasing the error probability. Given an AM proof system forL which achieves error probability 1/3 at the cost of Arthur sendingl(n) random bits per round, and given a polynomialk=k(n), we show how to construct an AM proof system forL which, in the same number of rounds as the original proof system, achieves error 2 ^–k(n) at the cost of Arthur sending onlyO(l+k) random bits per round.Underlying the transformation is a novel sampling method for approximating the average value of an arbitrary functionf:{0,1} ^l [0,1]. The method evaluates the function onO(^–2 log ^–1) sample points generated using onlyO(l + log ^–1) coin tosses to get an estimate which with probability at least 1- is within of the average value of the function.     

Multi-time scale zero-sum differential games with perfect state measurements

We obtain necessary and sufficient conditions for the existence of approximate saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters ₁, and ₂, and the terminology approximate saddle-point solution is used to refer to saddle-point policies that do not depend on ₁ and ₂, while providing cost levels withinO(₁) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames-slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in theH -optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.Research supported in part by the National Science Foundation under Grant ECS 91-13153, and in part by the U.S. Department of Energy under Grant DE-FG-02-88-ER-13939. This paper is dedicated to the memory of John Breakwell.     

PAC Learning Axis-aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples

We describe a polynomial-time algorithm for learning axis-aligned rectangles in Q ^d with respect to product distributions from multiple-instance examples in the PAC model. Here, each example consists of n elements of Q^d together with a label indicating whether any of the n points is in the rectangle to be learned. We assume that there is an unknown product distribution D over Q ^d such that all instances are independently drawn according to D. The accuracy of a hypothesis is measured by the probability that it would incorrectly predict whether one of n more points drawn from D was in the rectangle to be learned. Our algorithm achieves accuracy with probability 1- in O (d⁵ n¹²/²⁰ log² nd/ time.     

Bounding Perturbations in Zeros of Nonlinear Systems

Algorithms for determining computationally rigorous componentwise bounds on the solutions x R ⁿ of equations F(x, t) = 0 R ^m containing parameters t R ^l due to small perturbations in t when m n and when F is at most twice continuously differentiable in x and in t are described. Numerical results which illustrate the behaviour of the algorithms are presented.     

A Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness

We study the scheduling situation where n tasks, subjected to release dates and due dates, have to be scheduled on m parallel processors. We show that, when tasks have unit processing times and either require 1 or m processors simultaneously, the minimum maximal tardiness can be computed in polynomial time. Two algorithms are described. The first one is based on a linear programming formulation of the problem while the second one is a combinatorial algorithm. The complexity status of this tall/small task scheduling problem P|r _i ,p _i =1, size _i {1,m}|T _max was unknown before, even for two processors.     

A deterministic test for permutation polynomials

A new deterministic algorithm is presented for testing whether a given polynomial of degreen over a finite field ofq elements is a permutation polynomial. The algorithm has computing time (nq)^6/7+, and gives a positive answer to a question of Lidl and Mullen.     

Infinitesimal Generators on the Quantum Group SUq(2) in the Classical and Anti-Classical Limit

In [5] we found an explicit formula for the infinitesimal generators of white noises on the quantum group SU _q (2) in the case q (-1, 1). In [7] we compared the case q (-1, 1) with the classical case q = 1 which corresponds to the infinitesimal generators of Lévy processes on SU(2). We pointed out that our formula is the perfect analogue of Hunt's formula which describes the classical infinitesimal generators. In the first part of these notes we find Hunt's formula for SU _q (2) in the anti-classical case q = -1. This completes our theory of infinitesimal generators on SU _q (2) for all q [-1, 1].In the second part of these notes we show that SU _q (2) is a 'good' quantization of SU(2). We show that not only states on the algebra of functions on SU(2), but even infinitesimal generators may be approximated in a suitable sense by states and infinitesimals generators on SU _q (2) for q (-1, 1). Simultaneously, we show that a similar result is true for SU _-1(2).     

Computing Hough transforms on hypercube multicomputers

Efficient algorithms to compute the Hough transform on MIMD and SIMD hypercube multicomputer are developed. Our algorithms can compute p angles of the Hough transform of an N × N image, p N, in 0(p + log N) time on both MIMD and SIMD hypercubes. These algorithms require 0(N ²) processors. We also consider the computation of the Hough transform on MIMD hypercubes with a fixed number of processors. Experimental results on an NCUBE/7 hypercube are presented.This research was supported by the National Science Foundation under grants DCR84-20935 and 86-17374. All correspondence should be mailed to Sanjay Ranka.     

Dynamical Properties of Timed Automata

Timed automata are an important model for specifying and analyzing real-time systems. The main analysis performed on timed automata is the reachability analysis. In this paper we show that the standard approach for performing reachability analysis is not correct when the clocks drift even by a very small amount. Our formulation of the reachability problem for timed automata is as follows: we define the set R ^*(T,Z ₀)=_>0 Reach(T, Z ₀ where T is obtained from timed automaton T by allowing an drift in the clocks. R ^*(T,Z ₀) is the set of states which can be reached in the timed automatonT from the initial states in Z₀ when the clocks drift by an infinitesimally small amount. We present an algorithm for computing R ^*(T,Z ₀)and provide a proof of its correctness. We show that R ^*(T,Z ₀)is robust with respect to various types of modeling errors. To prove the correctness of our algorithm, we need to understand the dynamics of timed automata—in particular, the structure of the limit cycles of timed automata.     

Parallel integer sorting using small operations

We consider the problem of sortingn integers in the range [0,n ^c -1], wherec is a constant. It has been shown by Rajasekaran and Sen [14] that this problem can be solved optimally inO(logn) steps on an EREW PRAM withO(n) n -bit operations, for any constant >O. Though the number of operations is optimal, each operation is very large. In this paper, we show thatn integers in the range [0,n ^c -1] can be sorted inO(logn) time withO(nlogn)O(1)-bit operations andO(n) O(logn)-bit operations. The model used is a non-standard variant of an EREW PRAMtthat permits processors to have word-sizes ofO(1)-bits and (logn)-bits. Clearly, the speed of the proposed algorithm is optimal. Considering that the input to the problem consists ofO (n logn) bits, the proposed algorithm performs an optimal amount of work, measured at the bit level.This work was partially supported by The Northeast Parallel Architectures Center (NPAC) at Syracuse University, Syracuse, NY 13244 and The Rome Air Development Center, under contract F30602-88-D-0027.     

A Problem of Control of Random Processes

A control problem for a random process depending additively on the fractional Brownian motion with the Hurst parameter H (1/2, 1) is analyzed.     

Simulating Shared Memory in Real Time: On the Computation Power of Reconfigurable Architectures

We consider randomized simulations of shared memory on a distributed memory machine (DMM) where thenprocessors and thenmemory modules of the DMM are connected via a reconfigurable architecture. We first present a randomized simulation of a CRCW PRAM on a reconfigurable DMM having a complete reconfigurable interconnection. It guarantees delay (log *n), with high probability. Next we study a reconfigurable mesh DMM (RM-DMM). Here thenprocessors andnmodules are connected via ann×nreconfigurable mesh. It was already known that ann×mreconfigurable mesh can simulate in constant time ann-processor CRCW PRAM with shared memory of sizem. In this paper we present a randomized step by step simulation of a CRCW PRAM with arbitrarily large shared memory on an RM-DMM. It guarantees constant delay with high probability, i.e., it simulates in real time. Finally we prove a lower bound showing that sizeΩ(n²) for the reconfigurable mesh is necessary for real time simulations.     

Self-organizing neural networks for the analysis and representation of data: Some financial cases

Many recent papers have dealt with the application of feedforward neural networks in financial data processing. This powerful neural model can implement very complex nonlinear mappings, but when outputs are not available or clustering of patterns is required, the use of unsupervised models such as self-organizing maps is more suitable. The present work shows the capabilities of self-organizing feature maps for the analysis and representation of financial data and for aid in financial decision-making. For this purpose, we analyse the Spanish banking crisis of 1977–1985 and the Spanish economic situation in 1990 and 1991, making use of this unsupervised model. Emphasis is placed on the analysis of the synaptic weights, fundamental for delimiting regions on the map, such as bankrupt or solvent regions, where similar companies are clustered. The time evolution of the companies and other important conclusions can be drawn from the resulting maps.Characters and symbols used and their meaning nx x dimension of the neuron grid, in number of neurons - ny y dimension of the neuron grid, in number of neurons - n dimension of the input vector, number of input variables - (i, j) indices of a neuron on the map - k index of the input variables - w _ijk synaptic weight that connects thek input with the (i, j) neuron on the map - W _ij weight vector of the (i, j) neuron - x _k input vector - X input vector - (t) learning rate - _o starting learning rate - _f final learning rate - R(t) neighbourhood radius - R₀ starting neighbourhood radius - R _f final neighbourhood radius - t iteration counter - t _rf number of iterations until reachingR _f - t _f number of iterations until reaching _f - h(·) lateral interaction function - standard deviation - for every - d (x, y) distance between the vectors x and y     

An algorithm for recognizing palm polygons

A polygonP is said to be apalm polygon if there exists a pointxP such that the Euclidean shortest path fromx to any pointyP makes only left turns or only right turns. The set of all such pointsx is called thepalm kernel. In this paper we propose an O(E) time algorithm for recognizing a palm polygonP, whereE is the size of the visibility graph ofP. The algorithm recognizes the given polygonP as a palm polygon by computing the palm kernel ofP. If the palm kernel is not empty,P is a palm polygon.The extended abstract of this paper was reported at the Second Canadian Conference in Computational Geometry, pp. 246–251, 1990     

Fast algorithms for minimum matrix norm with application in computer graphics

In this paper we consider the following problem. Given (r ₁,r ₂, ...,r _n) R ⁿ, for anyI= (I ₁,I ₂,...,I _n) Z ⁿ, letE ₁=(e _ij), wheree _ij=(r _i–r_j)–(I _i–I_j), findI Z ⁿ such that |E _I| is minimized, where |·| is a matrix norm. This problem arises from optimal curve rasterization in computer graphics, where minimum distortion of curve dynamic context is sought. Until now, there has been no polynomial-time solution to this computer graphics problem. We present a very simpleO(n lgn)-time algorithm to solve this problem under various matrix norms.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0046373.