An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on a recursive approach, uses novel routing techniques, and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p), which can be shown to be optimal on the hypercube by a corresponding lower bound. Tree contraction can be used to evaluate algebraic expressions consisting of operators +; ?; ; = and rational operands, as well as for the membership problem for certain subclasses of languages in DCFL. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
Consider the problem of finding a minimum length schedule for n unit execution time tasks on m pr... more Consider the problem of finding a minimum length schedule for n unit execution time tasks on m processors with tree-like precedence constraints. A sequential algorithm can solve this problem in linear time. The fastest known parallel algorithm needs O(log n) time using n 2 processors. For the case m = 2 we present two work optimal parallel algorithms that produce greedy optimal schedules for intrees and outtrees. Both run in O(log n) time using n= log n processors of an EREW PRAM.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p). The resulting speed-up of O(p= log p) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
ABSTRACT A polynomial ideal membership problem is an (n+1)-tuple P = p, p 1, p 2,..., p n where p... more ABSTRACT A polynomial ideal membership problem is an (n+1)-tuple P = p, p 1, p 2,..., p n where p and the p i are multivariate polynomials over some ring, and the problem is to determine whether p is in the ideal generated by the p i . For polynomials over the integers or rationals, it is known that this problem is exponential space hard. Here, we show that the problem for multivariate polynomials over the rationals is solvable in exponential space, establishing its exponential space completeness.
In this paper, an effective algorithm is presented which for any given arbitrary vector replaceme... more In this paper, an effective algorithm is presented which for any given arbitrary vector replacement system (VRS) [7] allows to decide whether it is persistent or not. This algorithm is self-contained, based on a recursive construction of semilinear representations for ...
Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almo... more Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almost all investigated graph classes are regular graphs such as meshes, complete trees, pyramids. In this paper, we present a general method for one-to-one embedding irregular graphs into their optimal hypercubes based on extended-edge-bisectors of graphs. An extended-edge-bisector is an edge-bisector with the additional property that a subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of the embedding depends on the quality of the extended-edge-bisector. Moreover, if the extended bisection can be efficiently computed on the hypercube, so can the embedding.
In this paper, a deterministic algorithm for dynamically embedding binary trees into next to opti... more In this paper, a deterministic algorithm for dynamically embedding binary trees into next to optimal hypercubes is presented. Due to a known lower bound, any such algorithm must use either randomization or migration, i.e., remapping of tree vertices, to obtain an embedding of trees into hypercubes with small dilation, load, and expansion simultaneously. The algorithm presented here uses migration of
In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optim... more In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a very irregular structure into hypercubes are investigated . The dilation of the presented embedding is bounded by , where denotes the treewidth of the graph and denotes the maximal degree of
Following ideas from (Hei83, DFGS91, MT97) and applying the tech- niques proposed in (May89, KM96... more Following ideas from (Hei83, DFGS91, MT97) and applying the tech- niques proposed in (May89, KM96, K¨ uh98), we present a deterministic al- gorithm for computing the dimension of a polynomial ideal requiring poly- nomial working space.
We investigate the stability of the modified difference scheme of Kim and Moin for numerical inte... more We investigate the stability of the modified difference scheme of Kim and Moin for numerical integration of two-dimensional incompressible Navier-Stokes equations by the Fourier method and by the method of discrete perturbations. The obtained analytic-form stability condition gives the maximum time steps allowed by stability, which are by factors from 2 to 58 higher than the steps obtained from previous empirical stability conditions. The stability criteria derived with the aid of CAS Mathematica are verified by numerical solution of two test problems one of which has a closed-form analytic solution.
We propose a new algorithm for the generation of orthogonal grids on regions bounded by arbitrary... more We propose a new algorithm for the generation of orthogonal grids on regions bounded by arbitrary number of polynomial inequalities. Instead of calculation of the grid nodes positions for a particular region, we perform all calculations for general polynomials given with indeterminate coefficients. The first advantage of this approach is that the calculations can be performed only once and then used to generate grids on arbitrary regions and of arbitrary mesh size with constant computational costs. The second advantage of our algorithm is the avoidance of singularities, which occur while using the existing algebraic grid generation methods and lead to the intersection of grid lines. All symbolic calculation can be performed with general purpose Computer Algebra Systems, and expressions obtained in this way can be translated in Java/C++ code.
Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known pro... more Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known problem. Considered as optimization task the problem may be reduced to finding a minimizer of the weighted combination of so-called length, area, and orthogonality functionals. Unfortunately, it has been shown that on the one hand, certain weights of the individual functionals do not admit the unique optimizer on certain geometric domains. On the other hand, some combinations of these functionals lead to the lack of ellipticity of corresponding Euler-Lagrange equations, and finding the optimal grid becomes computationally too expensive for practical applications. Choosing the right functional for the particular geometric domain of interest may improve the grid generation very much, but choosing the functional parameters is usually done in the trial and error way and depends very much on the geometric domain. This makes the automatic and robust grid generation impossible. Thus, in the present paper we consider the way to compute certain approximations of minimizer of grid functionals independently of the particular domain. Namely, we are looking for the approximation of the minimizer of the individual grid functionals in the local sense. This means the functional has to be satisfied on the possible largest parts of the domain. In particular, we shall show that the so called method of envelopes, otherwise called the method of rolling circle, that has been proposed in our previous paper, guarantees the optimality with respect to the area and orthogonality functionals in this local sense. In the global sense, the grids computed with the aid of envelopes, can be considered as approximations of the optimal solution. We will give the comparison of the method of envelopes with well established Winslow generator by presenting computational results on selected domains with different mesh size.
In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion ... more In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion of an object avoiding collisions with obstacles. The algorithm is based on the combination of configuration space and energy approaches. According to the configuration space approach, the position and orientation of the geometric object to be moved or placed is represented as an individual point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of this object. The configurations which, due to the presence of obstacles, are forbidden to the object, can be characterized as regions in this configuration space called configuration space obstacles. As will be demonstrated, configuration space obstacles can be computed symbolically using quantifier elimination over the reals and represented by polynomial inequalities. We propose to use the functional representation of semi-algebraic point sets defined by such inequalities, so-called R-functions, to describe nonlinear geometric objects in the configuration space. The potential field defined by R-functions can be used to "move" objects in such a way as to avoid collisions. Introducing the additional function, which forces the object towards the goal position, we reduce the problem of finding collision free path to a solution of the Newton's equations, which describes the motion of a body in the field produced by the superposition of "attractive" and "repulsive" forces. These equations can be solved iteratively in a computationally efficient manner. Furthermore, we investigate the differential properties of R-functions in order to construct a suitable superposition of attractive and repulsive potentials.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on a recursive approach, uses novel routing techniques, and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p), which can be shown to be optimal on the hypercube by a corresponding lower bound. Tree contraction can be used to evaluate algebraic expressions consisting of operators +; ?; ; = and rational operands, as well as for the membership problem for certain subclasses of languages in DCFL. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
Consider the problem of finding a minimum length schedule for n unit execution time tasks on m pr... more Consider the problem of finding a minimum length schedule for n unit execution time tasks on m processors with tree-like precedence constraints. A sequential algorithm can solve this problem in linear time. The fastest known parallel algorithm needs O(log n) time using n 2 processors. For the case m = 2 we present two work optimal parallel algorithms that produce greedy optimal schedules for intrees and outtrees. Both run in O(log n) time using n= log n processors of an EREW PRAM.
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubi... more An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shu e exchange or the butter y network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O( l n p m log p). The resulting speed-up of O(p= log p) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound. The same algorithmic ingredients can also be used to solve the term matching problem, one of the fundamental problems in logic programming.
ABSTRACT A polynomial ideal membership problem is an (n+1)-tuple P = p, p 1, p 2,..., p n where p... more ABSTRACT A polynomial ideal membership problem is an (n+1)-tuple P = p, p 1, p 2,..., p n where p and the p i are multivariate polynomials over some ring, and the problem is to determine whether p is in the ideal generated by the p i . For polynomials over the integers or rationals, it is known that this problem is exponential space hard. Here, we show that the problem for multivariate polynomials over the rationals is solvable in exponential space, establishing its exponential space completeness.
In this paper, an effective algorithm is presented which for any given arbitrary vector replaceme... more In this paper, an effective algorithm is presented which for any given arbitrary vector replacement system (VRS) [7] allows to decide whether it is persistent or not. This algorithm is self-contained, based on a recursive construction of semilinear representations for ...
Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almo... more Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almost all investigated graph classes are regular graphs such as meshes, complete trees, pyramids. In this paper, we present a general method for one-to-one embedding irregular graphs into their optimal hypercubes based on extended-edge-bisectors of graphs. An extended-edge-bisector is an edge-bisector with the additional property that a subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of the embedding depends on the quality of the extended-edge-bisector. Moreover, if the extended bisection can be efficiently computed on the hypercube, so can the embedding.
In this paper, a deterministic algorithm for dynamically embedding binary trees into next to opti... more In this paper, a deterministic algorithm for dynamically embedding binary trees into next to optimal hypercubes is presented. Due to a known lower bound, any such algorithm must use either randomization or migration, i.e., remapping of tree vertices, to obtain an embedding of trees into hypercubes with small dilation, load, and expansion simultaneously. The algorithm presented here uses migration of
In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optim... more In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a very irregular structure into hypercubes are investigated . The dilation of the presented embedding is bounded by , where denotes the treewidth of the graph and denotes the maximal degree of
Following ideas from (Hei83, DFGS91, MT97) and applying the tech- niques proposed in (May89, KM96... more Following ideas from (Hei83, DFGS91, MT97) and applying the tech- niques proposed in (May89, KM96, K¨ uh98), we present a deterministic al- gorithm for computing the dimension of a polynomial ideal requiring poly- nomial working space.
We investigate the stability of the modified difference scheme of Kim and Moin for numerical inte... more We investigate the stability of the modified difference scheme of Kim and Moin for numerical integration of two-dimensional incompressible Navier-Stokes equations by the Fourier method and by the method of discrete perturbations. The obtained analytic-form stability condition gives the maximum time steps allowed by stability, which are by factors from 2 to 58 higher than the steps obtained from previous empirical stability conditions. The stability criteria derived with the aid of CAS Mathematica are verified by numerical solution of two test problems one of which has a closed-form analytic solution.
We propose a new algorithm for the generation of orthogonal grids on regions bounded by arbitrary... more We propose a new algorithm for the generation of orthogonal grids on regions bounded by arbitrary number of polynomial inequalities. Instead of calculation of the grid nodes positions for a particular region, we perform all calculations for general polynomials given with indeterminate coefficients. The first advantage of this approach is that the calculations can be performed only once and then used to generate grids on arbitrary regions and of arbitrary mesh size with constant computational costs. The second advantage of our algorithm is the avoidance of singularities, which occur while using the existing algebraic grid generation methods and lead to the intersection of grid lines. All symbolic calculation can be performed with general purpose Computer Algebra Systems, and expressions obtained in this way can be translated in Java/C++ code.
Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known pro... more Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known problem. Considered as optimization task the problem may be reduced to finding a minimizer of the weighted combination of so-called length, area, and orthogonality functionals. Unfortunately, it has been shown that on the one hand, certain weights of the individual functionals do not admit the unique optimizer on certain geometric domains. On the other hand, some combinations of these functionals lead to the lack of ellipticity of corresponding Euler-Lagrange equations, and finding the optimal grid becomes computationally too expensive for practical applications. Choosing the right functional for the particular geometric domain of interest may improve the grid generation very much, but choosing the functional parameters is usually done in the trial and error way and depends very much on the geometric domain. This makes the automatic and robust grid generation impossible. Thus, in the present paper we consider the way to compute certain approximations of minimizer of grid functionals independently of the particular domain. Namely, we are looking for the approximation of the minimizer of the individual grid functionals in the local sense. This means the functional has to be satisfied on the possible largest parts of the domain. In particular, we shall show that the so called method of envelopes, otherwise called the method of rolling circle, that has been proposed in our previous paper, guarantees the optimality with respect to the area and orthogonality functionals in this local sense. In the global sense, the grids computed with the aid of envelopes, can be considered as approximations of the optimal solution. We will give the comparison of the method of envelopes with well established Winslow generator by presenting computational results on selected domains with different mesh size.
In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion ... more In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion of an object avoiding collisions with obstacles. The algorithm is based on the combination of configuration space and energy approaches. According to the configuration space approach, the position and orientation of the geometric object to be moved or placed is represented as an individual point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of this object. The configurations which, due to the presence of obstacles, are forbidden to the object, can be characterized as regions in this configuration space called configuration space obstacles. As will be demonstrated, configuration space obstacles can be computed symbolically using quantifier elimination over the reals and represented by polynomial inequalities. We propose to use the functional representation of semi-algebraic point sets defined by such inequalities, so-called R-functions, to describe nonlinear geometric objects in the configuration space. The potential field defined by R-functions can be used to "move" objects in such a way as to avoid collisions. Introducing the additional function, which forces the object towards the goal position, we reduce the problem of finding collision free path to a solution of the Newton's equations, which describes the motion of a body in the field produced by the superposition of "attractive" and "repulsive" forces. These equations can be solved iteratively in a computationally efficient manner. Furthermore, we investigate the differential properties of R-functions in order to construct a suitable superposition of attractive and repulsive potentials.
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Papers by Ernst W Mayr