Coarse-Grained Parallel Geometric Search

We present a parallel algorithm for solving thenext element search problemon a set of line segments, using a BSP-like model referred to as thecoarse grained multicomputer(CGM). The algorithm requiresO(1) communication rounds (h-relations withh=O(n/p)),O((n/p) log n) local computation, andO((n/p) log p) memory per processor, assumingn/p⩾p. Our result implies solutions to the point location, trapezoidal decomposition, and polygon triangulation problems. A simplified version for axis-parallel segments requires onlyO(n/p) memory per processor, and we discuss an implementation of this version. As in a previous paper by Develliers and Fabri (Int. J. Comput. Geom. Appl.6(1996), 487–506), our algorithm is based on a distributed implementation of segment trees which are of sizeO(n log n). This paper improves onop. cit.in several ways: (1) It studies the more general next element search problem which also solves, e.g., planar point location. (2) The algorithms require onlyO((n/p) log n) local computation instead ofO(log p*(n/p) log n). (3) The algorithms require onlyO((n/p) log p) local memory instead ofO((n/p) log n).     

A Simple Parallel Algorithm for the Single-Source Shortest Path Problem on Planar Digraphs

We present a simple parallel algorithm for the single-source shortest path problem in planar digraphs with nonnegative real edge weights. The algorithm runs on the EREW PRAM model of parallel computation in O((n^2ε+n^1−ε) log n) time, performing O(n^1+ε log n) work for any 0<ε<1/2. The strength of the algorithm is its simplicity, making it easy to implement and presumable quite efficient in practice. The algorithm improves upon the work of all previous parallel algorithms. Our algorithm is based on a region decomposition of the input graph and uses a well-known parallel implementation of Dijkstra's algorithm. The logarithmic factor in both the work and the time can be eliminated by plugging in a less practical, sequential planar shortest path algorithm together with an improved parallel implementation of Dijkstra's algorithm.     

Tetrel bonds between PySiX3 and some nitrogenated bases: Hybridization,substitution, and cooperativity

Ab initio calculations have been performed to study the influence of hybridization, substitution, and cooperativity on the tetrel bond in the complexes of PySiX₃ (Py = pyridine and X = halogen). The tetrel bond becomes stronger in the order of p-PySiF₃⋯NCH(sp) < p-PySiF₃⋯NHCH₂(sp²) < p-PySiF₃⋯NH₃(sp³). The electron-donating group in the electron donor strengthens the tetrel bond, and this effect is mainly achieved through electrostatic and polarization interactions. Tetrel bonding displays cooperative effects with triel bonding and chalcogen bonding, characterized by shorter binding distances and greater electron densities. The cooperative effects between triel/chalcogen bond and tetrel bond have been analyzed by molecular electrostatic potentials and charge transfer. Energy decomposition indicates that many-body effects are mainly caused by polarization energy. The geometries of Si⋯N interaction and its applications in crystal materials have been characterized and evidenced by a CSD research.     

Scalable 2D Convex Hull and Triangulation Algorithms for Coarse Grained Multicomputers

In this paper we describe scalable parallel algorithms for building the convex hull and a triangulation ofncoplanar points. These algorithms are designed for thecoarse grained multicomputermodel:pprocessors withO(n/p)⪢O(1) local memory each, connected to some arbitrary interconnection network. They scale over a large range of values ofnandp, assuming only thatn⩾p^1+ε(ε>0) and require timeO((T_sequential/p)+T_s(n, p)), whereT_s(n, p) refers to the time of a global sort ofndata on approcessor machine. Furthermore, they involve only a constant number of global communication rounds. Since computing either 2D convex hull or triangulation requires timeT_sequential=Θ(n log n) these algorithms either run in optimal time,Θ((n log n)/p), or in sort time,T_s(n, p), for the interconnection network in question. These results become optimal whenT_sequential/pdominatesT_s(n, p) or for interconnection networks like the mesh for which optimal sorting algorithms exist.     

Some Applications of Magma in Designs and Codes: Oval Designs,Hermitian Unitals and Generalized Reed–Muller Codes

We describe three applications of Magma to problems in the area of designs and the associated codes:  •  Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs;  •  the Hermitian unital as a 2-(q³ +  1, q +  1, 1) design, and the code overF_p where p divides q +  1;  •  a basis of minimum-weight vectors for the code over F_pof the design of points and hyperplanes of the affine geometry AG_d(F_p), where p is a prime.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Tighter Lower Bounds for Nearest Neighbor Search and Related Problems in the Cell Probe Model

We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n,d) cells, each containing poly(d, log n) bits. We get a lower bound of Ω(d/log n) on the search time, a significant improvement over the recent bound of Ω(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.     

On reliability of the folded hypercubes

In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Solutions to the functional equation I(x, y) = I(x, I(x, y)) for three types of fuzzy implications derived from uninorms

This paper investigates an iterative Boolean-like law with fuzzy implications derived from uninorms. More precisely, we characterize the solutions to the functional equation I(x, y) = I(x, I(x, y)) that involve RU-, (U, N)- and QLU-implications generated by the most usual classes of uninorms.     

PetRBF — A parallel O(N) algorithm for radial basis function interpolation with Gaussians

We have developed a parallel algorithm for radial basis function (rbf) interpolation that exhibits O(N) complexity, requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method (rasm) as a preconditioner and a fast matrix-vector algorithm. Previous fast rbf methods — achieving at most O(NlogN) complexity — were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance with respect to the domain, but with sufficient overlap. This is a common choice in particle methods for fluid simulation, our main target application. The fast decay of the Gaussian basis function allows rapid convergence of the iterative solver even when the subdomains in the rasm are very small. At the same time we show that the accuracy of the interpolation can achieve machine precision. The present method was implemented in parallel using the petsc library (developer version). Numerical experiments demonstrate its capability in problems of rbf interpolation with more than 50 million data points, timing at 106 s (19 iterations for an error tolerance of 10^? 15) on 1024 processors of a Blue Gene/L (700 MHz PowerPC processors). The parallel code is freely available in the open-source model.     

Integrating Factors for Second-order ODEs

A systematic algorithm for building integrating factors of the form μ(x, y), μ(x, y^′) or μ(y, y^′) for second-order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ (x, y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.     

Mason: morphological simplification

The traditional rounding and filleting morphological filters are biased. Hence, as r grows, the rounding R_r (S) of S shrinks and the filleting F_r (S) grows. A shape S is r-regular when R_r (S) = F_r (S) = S. The combinations F_r (R_r (S)) and R_r (F_r (S)) produce nearly r-regular shapes, but retain a bias: F_r (R_r (S)) is usually smaller than S and R_r (F_r (S)) is larger. To overcome this bias, we propose a new filter, called Mason. The r-mortar M_r (S) of S is F_r (S)–R_r (S), and the stability of a point P with respect to S is the smallest value of r for which P belongs to M_r (S). Stability provides important information about the shape’s imbedding that cannot be obtained through traditional topological or differential analysis tools. F_r (R_r (S)) and R_r (F_r (S)) only affect space in M_r (S). For each maximally connected component of M_r (S), Mason performs either F_r (R_r (S)) or R_r (F_r (S)), choosing the combination that alters the smallest portion of that component. Hence, Mason acts symmetrically on the shape and on its complement. Its output is guaranteed to have a smaller symmetric difference with the original shape than that of either combination F_r (R_r (S)) or R_r (F_r (S)). Many previously proposed shape simplification algorithms were focused on reducing the combinatorial storage or processing costs of a shape at the expense of the smoothness and regularity or altered the shape in regular portions that did not exhibit any high frequency complexity. Mason is the first shape simplification operator that is independent of the particular representation and offers the advantage of preserving portions of the boundary of S that are regular at the desired scale.     

Differences in computer exposure between university administrators and CAD draftsmen

This study utilized an external logger system for onsite measurements of computer activities of two professional groups—twelve university administrators and twelve computer-aided design (CAD) draftsmen. Computer use of each participant was recorded for 10 consecutive days—an average of 7.9 ± 1.8 workdays and 7.8 ± 1.5 workdays for administrators and draftsmen, respectively. Quantitative parameters computed using recorded data were daily dynamic duration (DD) and static duration, daily keystrokes, mouse clicks, wheel scrolling counts, mouse movement and dragged distance, average typing and clicking rates, and average time holding down keys and mouse buttons. Significant group differences existed in the number of daily keystrokes (p < 0.0005) and mouse clicks (p < 0.0005), mouse distance moved (p < 0.0005), typing rate (p < 0.0001), daily mouse DD (p < 0.0001), and keyboard DD (p < 0.005). Both groups had significantly longer mouse DD than keyboard DD (p < 0.0001). Statistical analysis indicates that the duration of computer use for different computer tasks cannot be represented by a single formula with same set of quantitative parameters as those associated with mouse and keyboard activities. Results of this study demonstrate that computer exposure during different tasks cannot be estimated solely by computer use duration. Quantification of onsite computer activities is necessary when determining computer-associated risk of musculoskeletal disorders. Other significant findings are discussed.     

A non-linear fuzzy regression for estimating reliability in a degradation process

Since some assumptions such as the function ϕ(·) needs to be completely specified and the relationship between μ and ϕ(s) must have linear behavior in the model μ = a + bϕ(S) used in the accelerated life testing analysis, generally do not hold; the estimation of stress level contains uncertainty. In this paper, we propose to use a non-linear fuzzy regression model for performing the extrapolation process and adapting the fuzzy probability theory to the classical reliability including uncertainty and process experience for obtaining fuzzy reliability of a component. Results show, that the proposed model has the ability to estimate reliability when the mentioned assumptions are violated and uncertainty is implicit; so that the classical models are unreliable.     

Irregular Primes and Cyclotomic Invariants to 12 Million

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overF_pof degree  <  (p −  1) / 2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log²p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schönhage–Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp −  1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer–Vandiver conjecture and for computing the cyclotomic invariants of a prime.     

Broadcast in the rendezvous model

In many large, distributed or mobile networks, broadcast algorithms are used to update information stored at the nodes. In this paper, we propose a new model of communication based on rendezvous and analyze a multi-hop distributed algorithm to broadcast a message in a synchronous setting. In the rendezvous model, two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously. Thus nodes u and v obtain a rendezvous at a meeting point. If m is the number of meeting points, the network can be modeled by a graph of n vertices and m edges. At each round, every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities. Rendezvous enable an exchange of information between the two entities. We get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast: we show that, for any graph, the expected number of rounds is between ln n and O (n²). For these two bounds, we prove that there exist some graphs for which the expected number of rounds is either O (ln (n)) or Ω (n²). For specific topologies, additional bounds are given.     

Deadline-based scheduling in support of real-time data delivery

The use of deadline-based scheduling in support of real-time delivery of application data units (ADUs) in a packet-switched network is investigated. Of interest is priority scheduling where a packet with a smaller ratio of T/H (time until delivery deadline over number of hops remaining) is given a higher priority. We refer to this scheduling algorithm as the T/H algorithm. T/H has time complexity of O(log N) for a backlog of N packets and was shown to achieve good performance in terms of the percentage of ADUs that are delivered on-time. We develop a new and efficient algorithm, called T/H − p, that has O(1) time complexity. The performance difference of T/H, T/H − p and FCFS are evaluated by simulation. Implementations of T/H and T/H − p in high-speed routers are also discussed. We show through simulation that T/H − p is superior to FCFS but not as good as T/H. In view of the constant time complexity, T/H − p is a good candidate for high-speed routers when both performance and implementation cost are taken into consideration.     

Gröbner Bases Applied to Finitely Generated Field Extensions

Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x) / k(g), resp. how to determine the (separable) degree if k(x) / k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x) / k(g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group.     

A new class of massively parallel direction splitting for the incompressible Navier–Stokes equations

We introduce in this paper a new direction splitting algorithm for solving the incompressible Navier–Stokes equations. The main originality of the method consists of using the operator (I ? ?_xx)(I ? ?_yy)(I ? ?_zz) for approximating the pressure correction instead of the Poisson operator as done in all the contemporary projection methods. The complexity of the proposed algorithm is significantly lower than that of projection methods, and it is shown the have the same stability properties as the Poisson-based pressure-correction techniques, either in standard or rotational form. The first-order (in time) version of the method is proved to have the same convergence properties as the classical first-order projection techniques. Numerical tests reveal that the second-order version of the method has the same convergence rate as its second-order projection counterpart as well. The method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors. The method has been validated on the three-dimensional lid-driven cavity flow using grids composed of up to 2 × 10⁹ points.