Tries-Based Parallel Solutions for Generating Perfect Crosswords Grids

A general crossword grid generation is considered an NP-complete problem and theoretically it could be a good candidate to be used by cryptography algorithms. In this article, we propose a new algorithm for generating perfect crosswords grids (with no black boxes) that relies on using tries data structures, which are very important for reducing the time for finding the solutions, and offers good opportunity for parallelisation, too. The algorithm uses a special tries representation and it is very efficient, but through parallelisation the performance is improved to a level that allows the solution to be obtained extremely fast. The experiments were conducted using a dictionary of almost 700,000 words, and the solutions were obtained using the parallelised version with an execution time in the order of minutes. We demonstrate here that finding a perfect crossword grid could be solved faster than has been estimated before, if we use tries as supporting data structures together with parallelisation. Still, if the size of the dictionary is increased by a lot (e.g., considering a set of dictionaries for different languages—not only for one), or through a generalisation to a 3D space or multidimensional spaces, then the problem still could be investigated for a possible usage in cryptography.

About grid generation in constructions bounded by the surfaces of revolution

For constructions bounded by the surfaces of revolution, structured grid generation technique is presented. Its technology has been elaborated within the variational approach for constructing optimal grids satisfying optimality criteria: closeness of grids to uniform ones, closeness of grids to orthogonal ones and adaptation to a given function. Grid generation has been designed for numerical solution of the differential equations modeling the vortex processes of multi-component hydrodynamics. In the technology, the three-dimensional construction in which it is required to construct a grid is represented in the form of the curvilinear hexahedron defining its configuration. The specific feature of the required configurations is that some faces of a curvilinear hexahedron lie in one plane and along edges of adjoining faces grid cells degenerate into prisms. Grid generation in the considered constructions has started to be developed by the elaboration of algorithms for the volume of revolution which has become the basic construction. The basic construction is obtained by the rotation through 180? around the axis of a generatrix consisting of straight line segments, arcs of circles and ellipses. Then the deformed volumes of revolutions are considered along with the generalizations of the volume of revolution which represent constructions obtained by the surfaces of revolution with parallel axis of rotation. The aim of the further development of the technology is to consider more and more complicated constructions and elaborate the technology for them. In the presentation, the current state of the development of the technology is given. Examples of generated grids are supplied.

Acute PEBI Grid Generation for Reservoir Geometries

An unstructured grid generation method is presented that automates control-volume boundary alignment to geological objects and control point alignment to complex wells. The grid generation method is coupled with an iterative acute mesh reconstruction technique, to construct essentially acute triangulations, while satisfying quite general geometric constraints. For well aligned grids control points are constrained to the well trajectory and protection circles are used, whereas for boundary aligned grids halo construction is performed. Unstructured Delaunay triangulations (DT) have the desirable locally orthogonal perpendicular bisectional (PEBI) property, required by the industry standard two-point flux approximation for consistency for isotropic fields. The PEBI property can only be exploited if such grids are comprised of acute simplexes, with circumcentres located inside their respective elements. The method presented enables acute DT layered mesh generation while honoring internal boundaries and wells in a two dimensional space. A dual (Voronoi) grid derived from a feature honored simplicial mesh is then projected in the vertical direction generating 2.5D PEBI grids. Effectiveness of the method to construct acute PEBI grids honoring geological objects and complex wells is demonstrated by meshing representative reservoir geometries. Numerical results are presented that verify consistency of the two-point flux on the resulting boundary-aligned acute PEBI grids. Development of an unstructured grid generation method which 1) Automates interior boundary alignment, 2) Honors features with respect to control point and/or control volume, and 3) Generates acute PEBI grids for reservoir geometries is presented. A unique workflow is presented to generate boundary aligned acute PEBI grids for complex geometries. Development of boundary aligned grids that honor both geological objects and multilateral complex wells, together with mesh reconstruction to ensure circumcenter containment is presented. Further, 3D PEBI grid generation method which can limit refinement to well perforations and geological objects is also described.

PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.