Showing papers in "International Journal of Computer Mathematics in 1990"
TL;DR: In this paper, the problem of finding a maximum clique of an undirected graph is formulated and solved as a linearly constrained indefinite quadratic global optimization problem, and theoretical upper and lower bounds on the size k of the maximum cliques are derived from the global optimization formulation.
Abstract: The problem of finding a maximum clique of an undirected graph is formulated and solved as a linearly constrained indefinite quadratic global optimization problem. Theoretical upper and lower bounds on the size k of the maximum clique are derived from the global optimization formulation, and a relationship between the set of distinct global maxima of the optimization problem and the set of distinct maximum cliques of the graph is discussed. In addition, some preliminary computational results are also presented.
95 citations
TL;DR: A new characterization of interval graphs is exploited for the purpose of obtaining a linear-time algorithm for computing both the center and the diameter of an interval graph.
Abstract: The computational problem of finding the center of a graph is motivated by a number of facility-location problems. We exploit a new characterization of interval graphs for the purpose of obtaining a linear-time algorithm for computing both the center and the diameter of an interval graph.
68 citations
TL;DR: This paper shows that in many cases the problem of drawing a hypergraph can be reduced to the problem to drawing normal graphs, i.e. when the hyperedges connecting the vertices are drawn as curves.
Abstract: There is an increasing amount of applications in computer science and other fields in which hypergraphs are used. This paper shows that in many cases the problem of drawing a hypergraph can be reduced to the problem of drawing normal graphs. This holds true especially when considering hypergraphs drawing in the edge standard, i.e. when the hyperedges connecting the vertices are drawn as curves.
61 citations
TL;DR: The behavior of a class of block-iterative projection algorithms for solving convex feasibility problems is studied and a limit characterization theorem and a convergence criterion are proven.
Abstract: The behavior of a class of block-iterative projection algorithms for solving convex feasibility problems is studied. A limit characterization theorem and a convergence criterion are proven. Ways of accelerating the computational procedures are pointed out.
51 citations
TL;DR: This paper presents an algorithm to compute the (min, +) product of two n × n matrices, which follows the approach described by Fredman, but is faster than Fredman's own algorithm: its time complexity is either O(n 3/√log2 n) or even O( n 2.5√ log2 n), if one adheres to the uniform-cost RAM model faithfully.
Abstract: The (min, + ) product C of two n × n matrices A and B is defined as C ij = min1≦k≦n A ik + B kj . This paper presents an algorithm to compute the (min, +) product of two n × n matrices. The algorithm follows the approach described by Fredman, but is faster than Fredman's own algorithm: its time complexity is either O(n 3/√log2 n) or even O(n 2.5√log2 n), if one adheres to the uniform-cost RAM model faithfully. This result implies the existence of O(n 3/√log2 n) algorithms for the problems that (min, +) matrix multiplication is equivalent to, such as the all-pairs shortest paths problem. As the presented algorithm uses operations on sets, the formal analysis of its time complexity raises a few interesting questions about the applicability of the standard RAM complexity model.
51 citations
TL;DR: Direct constructions for the usual language theoretic operations in terms of alternating finite automata are presented and minimization and direct transformations between alternating, non-deterministic, and deterministic finite Automata are discussed.
Abstract: Alternation is a natural generalization of nondeterminism. The model of alternating finite automata was first introduced and studied by Chandra et al. in [2]. Although alternating finite automata are no more powerful than deterministic finite automata with respect to language recognition, special features of alternating finite automata may provide new approaches and techniques for solving theoretical and practical problems concerning regular languages. In this paper we present direct constructions for the usual language theoretic operations in terms of alternating finite automata. Moreover, we discuss minimization and direct transformations between alternating, non-deterministic, and deterministic finite automata.
50 citations
TL;DR: In this paper, the similarity properties of nonlinear ordinary free boundary value problems are considered and a method for finding the location of the free boundary s through the first numerical integration and the numerical solution by means of a second integration is presented.
Abstract: We consider the similarity properties of nonlinear ordinary free boundary value problems, i.e., u″ = f(x,u,u′) x∊(0,s) s>0 u(0) = α; u(s) = u′(s) = 0; α≠0. By making use of group properties we show that for the two classes of problems it is possible to define a method that allows us to find the location of the free boundary s through the first numerical integration and the numerical solution by means of a second integration. Moreover, by requiring invariance of some parameter, we give an important extension of the method to solve a problem that does not belong to the two classes in point. Finally we remark that the method is self-validating.
42 citations
TL;DR: It is shown that the baryccntcr heuristic clearly outperforms the median heuristic, although only the latter has a proved bound for the maximum error done when two vertices are ordered.
Abstract: This paper studies different heuristics for drawing 2-Ievel hierarchical graphs. Especially, we compare the barycenter and the median heuristics. We show that the baryccntcr heuristic clearly outperforms the median heuristic, although only the latter has a proved bound for the maximum error done when two vertices are ordered. Moreover, we improve a known heuristic, called the greedy switching, by introducing the barycenter heuristic as a preprocessing phase for it.
38 citations
TL;DR: In this article, a two-step P-stable method for numerical integration of periodic initial value problems is derived, which can be made explicit by use of an analogous lower order method.
Abstract: A two-step P-stable method for the numerical integration of periodic initial value problems is derived. By the analysis of the error term a good estimate for the present period can be made. It is shown that the obtained implicit method can be made explicit by use of an analogous lower order method. In both cases a stability analysis is given. The superiority of these methods over other available methods is illustrated by two examples.
34 citations
TL;DR: In this article, a class of numerical methods for the approximate solution of ODEs with retarded argument is presented, which are essentially based on the spline functions and the study of existence and uniqueness are considered for such methods.
Abstract: This paper presents a class of numerical method for the approximate solution of ordinary differential equations with retarded argument. These methods are essentially based on the spline functions. The study of existence and uniqueness are considered for such methods. Numerical examples and comparisons with other methods are given.
31 citations
TL;DR: In this paper, an expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method, which yields more accurate results than those computed by P. M. Beckett [2] and A. R. Payne [6] as indicated from solving the Falkner-Skan equation.
Abstract: An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method [1], which yields more accurate results than those computed by P. M. Beckett [2] and A. R. Wadia and F. R. Payne [6] as indicated from solving the Falkner-Skan equation, which uses a boundary value technique. This method is accomplished by starting with Chebyshev approximation for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative.
TL;DR: The solution of Helmhohz's equation in regions of arbitrary cross-section is obtained by the finite element method.
Abstract: In this paper the solution of Helmhohz's equation in regions of arbitrary cross-section is obtained by the finite element method.
TL;DR: Finite difference methods of 0(k2+kh2+h4) for solving the system of 1-D nonlinear parabolic partial differential equations using three spatial grid points subject to Dirichlet boundary conditions are presented.
Abstract: In this paper, we present finite difference methods of 0(k2+kh2+h4 ) for solving the system of 1-D nonlinear parabolic partial differential equations using three spatial grid points subject to Dirichlet boundary conditions. The method for scalar equation has been tested on Burgers' equation. The numerical results show that the proposed method produces accurate and oscillation free solutions for large Reynolds numbers.
TL;DR: This work extends the application of group analysis approach to determining the numerical solution of free boundary value problems by introducing the concept of normal variables and giving an iterative method applicable to anyfree boundary value problem.
Abstract: We extend the application of group analysis approach to determining the numerical solution of free boundary value problems. If the differential problem is invariant under a translation group of transformations we will formulate a non-iterative method of solution. This is done by introducing the concept of normal variables. Application of the method to two problems in the class characterized produces correct numerical results. Moreover, introducing a parameter into the differential problem and requiring invariance under an extended stretching group we give an iterative method applicable to any free boundary value problem. As further result of the knowledge of the group properties we point out that these methods are self-validating. Finally we suggest application of numerical transformation methods to boundary value problems.
TL;DR: In this paper, the effectiveness of single term Walsh series method is demonstrated through different types of systems and it is observed that for singular systems the method is more suitable than any other systems.
Abstract: The effectiveness of single term Walsh series method is demonstrated through different types of systems. It is observed that for singular systems the method is more suitable than any other systems.
TL;DR: It is shown that these grammars with productions having associated only words consisting of one or two symbols characterize type 0 languages.
Abstract: Each production of a generalized forbidding grammar has an associated finite set of words. Such a production can be applied only if none of its associated words is a substring of a given rewritten sentential form. It is shown that these grammars with productions having associated only words consisting of one or two symbols characterize type 0 languages
TL;DR: It is shown that a modified Numerov method can be used to reduce the error of the order k 6 h 4 of the classical NumerOV method of the kth eigenvalue with uniform step length h, to an error of order k 3 h 4.
Abstract: The computation of eigenvalues of regular Sturm-Liouville problems is considered. It is shown that a modified Numerov method can be used to reduce the error of the order k 6 h 4 of the classical Numerov method of the kth eigenvalue with uniform step length h, to an error of order k 3 h 4 . By an appropriate minimization of the local error term of the method one can obtain even more accurate results. A comparison of the simple correction techniques of Andrew and Paine to Numerov's method is given. Numerical examples demonstrate the usefulness of the present approach even for low values of k.
TL;DR: This method reduces the dimensionality of the system in such a way that it can lead to an iterative approximate formula for the computation of n−1 components of the solution, while the remaining component is evaluated separately using the final approximations of the other components.
Abstract: A method for the numerical solution of systems of nonlinear algebraic and/or transcendental equations in is presented. This method reduces the dimensionality of the system in such a way that it can lead to an iterative approximate formula for the computation of n−1 components of the solution, while the remaining component of the solution is evaluated separately using the final approximations of the other components. This (n−1)-dimensional iterative formula generates a sequence of points in which converges quadratically to n−1 components of the solution. Moreover, it does not require a good initial guess for one component of the solution and it does not directly perform function evaluations, thus it can be applied to problems with imprecise function values. A proof of convergence is given and numerical applications are presented.
TL;DR: The investigated topics are: closure properties, the efficiency of generating a (linear) language by such a system compared with usual grammars, hierarchies, and so on.
Abstract: We continue the study of parallel communicating grammar systems introduced in P[acaron]un and Sântean [7] as a grammatical model of parallel computing. The investigated topics are: closure properties, the efficiency of generating a (linear) language by such a system compared with usual grammars, hierarchies.
TL;DR: This paper presents a new method for the analysis of nonlinear singular systems of the form K [xdot](t) = A x(t) + f(x(t)) via single-term Walsh series (STWS) approach.
Abstract: This paper presents a new method for the analysis of nonlinear singular systems of the form K [xdot](t) = A x(t) + f(x(t)) via single-term Walsh series (STWS) approach. Block pulse and discrete solutions are obtained for any length of time. The method is very simple and direct and can easily be implemented on a digital computer
TL;DR: It is shown that the two sweeps of the AGE algorithm can be combined into one, thus forming a stable explicit algorithm with reduced computational effort and suitable for implementation on a parallel computer.
Abstract: In this paper, parabolic differential equations with periodic boundary conditions in one space dimension (on a cylinder) are solved numerically by using the Alternating Group Explicit (AGE) iterative method [1]. It is shown that the two sweeps of the AGE algorithm can be combined into one, thus forming a stable explicit algorithm with reduced computational effort and suitable for implementation on a parallel computer.
TL;DR: An interesting new integral expression involving two Associated Laguerre Polynomials is derived, and seven special cases of this expression are discussed.
Abstract: An interesting new integral expression involving two Associated Laguerre Polynomials is derived, and seven special cases of this expression are discussed.
TL;DR: A parallel algorithm for the set partitioning problem which has applications in the minimization of finite state automata is presented and runs on a CREW PRAM in O(log2 n)z time and requires 0(n) processors.
Abstract: A parallel algorithm for the set partitioning problem which has applications in the minimization of finite state automata is presented. The algorithm runs on a CREW PRAM in O(log2 n)z time and requires 0(n) processors.
TL;DR: The performance of some explicit block methods as a grid smoother in the multigrid method is investigated and compared with the point Gauss-Seidel (GS) and successive over relaxation (SOR) methods.
Abstract: In Yousif and Evans [1], different explicit block methods for solving elliptic pde's were studied. In this paper, the performance of some of these methods as a grid smoother in the multigrid method is investigated and compared with the point Gauss-Seidel (GS) and successive over relaxation (SOR) methods.
TL;DR: In this paper, the authors consider the case of the linear complementarity problem where all or some of the variables are required to take integer values and discuss several applications to economic equilibrium problems and polymatrix games.
Abstract: In this paper we consider the case of the linear complementarity problem where all or some of the variables are required to take integer values. We discuss several applications to economic equilibrium problems and polymatrix games. When the integer variables are bounded, then the problem can be solved using an equivalent linear integer formulation. For the general problem (unbounded case) the problem can be solved using enumeration of the feasible points of a set of mixed zero-one linear inequalities.
TL;DR: The main contributions are the supply of further detail to remove ambiguities, a determination of the minimum number of extra bits required during the calculation, a verification of the more detailed system, and its extension to an integer division procedure.
Abstract: This paper refers to the algorithm and its hardware implementation described by Brickell [1] for modular multiplication in N+10 clock pulses where N is the number of bits in the binary integers involved. Brickell [1] uses a delayed carry representation which consists of two registers of N bits each—one for the uncarried carries. Of course, up to N clocks ticks may eventually be required to assimilate the carries at the end of the computation. Several sources of possible error are reported here—one in the hardware, one in the specification which the intended hardware satisfies, and one in the definition of the control variables T 1 and T 2. Our main contributions are the supply of further detail to remove such ambiguities, a determination of the minimum number of extra bits required during the calculation, a verification of the more detailed system, and its extension to an integer division procedure. The existence of a proof enables it to be used reliably for its intended purpose in applications such as cr...
TL;DR: By the use of repeated partitioning of the matrix into (2 × 2) subsystems it is shown that the linear system can be recursively decoupled into an explicit form suitable for solving on parallel or vector computers.
Abstract: In many numerical methods it is necessary to solve repeatedly tridiagonal linear systems of a certain form, i.e. diagonally dominant. By the use of repeated partitioning of the matrix into (2 × 2) subsystems it is shown that the linear system can be recursively decoupled into an explicit form suitable for solving on parallel or vector computers.
TL;DR: A matrix separation of the variables method for solving initial-boundary value problems for coupled systems of second order partial differential equations and finite computable approximate solutions and upper error bounds for them in terms of data are given.
Abstract: In this paper a matrix separation of the variables method for solving initial-boundary value problems for coupled systems of second order partial differential equations is presented. Finite computable approximate solutions and upper error bounds for them in terms of data are given.
TL;DR: This proof establishes pre- and post-conditions for all steps of the linear-time algorithm in Joe and Simpson for computing the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon or in its blocked exterior.
Abstract: We prove the correctness of the linear-time algorithm in Joe and Simpson [7] for computing the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon or in its blocked exterior. This proof establishes pre- and post-conditions for all steps of the algorithm. The algorithm and the proof are identical for interior and blocked exterior viewpoints. Other algorithms for computing the visibility polygon (usually for interior viewpoints only) are either nonlinear-time and harder to implement or can fail for polygons that wind sufficiently.
TL;DR: A set of alternative smoothing operators to the usual relaxation methods for multigrid algorithms is presented and analyzed in terms of frequency domain behavior and results prove effective in solving Frobenius matrix norm minimization problem.
Abstract: A set of alternative smoothing operators to the usual relaxation methods for multigrid algorithms is presented and analyzed in terms of frequency domain behavior. The operations presented are inherently parallel and fit well onto hypercube multiprocessors: they can be readily calculated and applied in a parallel manner. We start by interpreting multigrid smoothers as approximate inverses. In particular, a least squares approximate inverse obtained by solving a Frobenius matrix norm minimization problem proves effective. This approximate inverse also has a least squares interpretation in the frequency domain for the special case of circulant operators, or in the case of local mode Fourier analysis for the discrete operator in the central part of the domain over which the discretization is performed. Experimental results are presented for one and two dimensional problems. Convergence rates as determined by direct iteration are compared with local mode Fourier analysis results.