List Of Vertices Research Articles

Fast versions of the Gilbert-Johnson-Keerthi distance algorithm: additional results and comparisons

Considers fast algorithms for computing the Euclidean distance between objects that are modeled by convex polytopes in three-dimensional space. The algorithms, designated by RGJK, are modifications of the Gilbert-Johnson-Keerthi algorithm that follow the scheme originated by Cameron (1997). Each polytope is represented by its vertices and a list of adjacent vertices for each vertex. When the algorithms are appropriately applied to a pair of objects that have small incremental motions, they share the advantage of the closest-feature algorithm introduced by Lin and Canny (1991): computational time is very small and does not depend significantly on the total number of object vertices. However, when the objects contain complex vertices or faces, the time can increase drastically. Reasons for this problem are analyzed and algorithmic fixes for them are given. Other contributions to algorithmic performance include a procedure for reducing computational time in the presence of collisions. Comprehensive numerical experiments illuminate the dependence of computational time on algorithmic details, object complexity, and the size of incremental motions. The experiments include direct comparisons of RGJK with the closest-feature algorithms of Lin and Canny and of Mirtich (1998).

Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs

Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall-Halmos-Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut-vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut-vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000

On the complexity of a restricted list-coloring problem

We investigate a restricted list-coloring problem. Given a graph G = ( V, E), a non empty set of colors L, and a nonempty subset L( v) of L for each vertex v, find an L-coloring of G with the size of each class of colors being equal to a given integer. This restricted list-coloring problem was proposed by de Werra. We prove that this problem is N P-Complete even if the graph is a path with at most two colors on each vertex list. We then give a polynomial algorithm which solves this problem for the case where the total number of colors occurring in all lists is fixed.

Cartographic line simplification and polygon CSG formulæ in [formula omitted] time

A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. By using a data structure that maintains convex hulls of polygonal lines under splits, both were known to have O( n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log ∗ n) algorithm for maintaining convex hulls under splits at extreme points. It opens the question of whether there are practical, linear-time solutions to these problems.

Application of structural numbers to generating the characteristics of mechanical systems

This paper presents the application of structural numbers to the symbolic generation of mechanical systems characteristics. Symbolic approach is based on the method of numeration, method of graphs, algebra of structural numbers and SALS (computer package designed to perform symbolic computations). Theorems of algebra of structural numbers have been transformed into Turbo-Prolog predicates. As the base for symbolic approach the method of graphs and structural numbers of the 1st category has been applied as algorithm of this method is directly connected with the topological structure of the system. Moreover it eliminates the necessity of formulating and solving the equations of motion. When analyzing the mechanical system one looks for characteristic functions describing the response of this system: dynamic stiffness and flexibility, dynamic mobility, force and velocity transformation function, characteristic equation, coefficients of vibrations. Those coefficients can be obtained in symbolic form using the cascade symbolic model. Starting from the theory of numeration the hierarchic graph of mechanical system and the structural number for this graph can be created. Structural numbers can be transformed into form of lists. Graph structure can also be written as the set of lists: vertex list, edge list, incidence list. Applying general rules of algebra of structural numbers, to hierarchical structural numbers symbolical characteristics of mechanical systems can be determined without the necessity of formulating equations of motion. The obtained method can be used in future for synthesis of mechanical systems having required parameters and used as application in Computer Aided Design systems.

A Priori Optimization of the Probabilistic Traveling Salesman Problem

The probabilistic traveling salesman problem (PTSP) is defined on a graph G = (V, E), where V is the vertex set and E is the edge set. Each vertex vi has a probability pi of being present. With each edge (vi, vj) is associated a distance or cost cij. In a first stage, an a priori Hamiltonian tour on G is designed. The list of present vertices is then revealed. In a second stage, the a priori tour is followed by skipping the absent vertices. The PTSP consists of determining a first-stage solution that minimizes the expected cost of the second-stage tour. The problem is formulated as an integer linear stochastic program, and solved by means of a branch-and-cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function. Problems involving up to 50 vertices are solved to optimality.

S-MATRIX APPROACH TO MASSLESS HIGHER SPINS THEORY II: THE CASE OF INTERNAL SYMMETRY

The system of higher massive spins s = 0, 1, 2,… (every spin lies in o(n)-matrix algebra) is studied in the framework of the light-cone formalism. It is shown that the conditions of closure of the Poincaré-Lorentz (PL) algebra for a four-legged diagram allow one to express all the cubic interaction constants in terms of some unique universal constant. We also shown that the systems in which even (odd) spins are realized as n × n (anti) symmetric matrices admit the consistent formulation. A complete list of four-legged vertices of the theory is given.

On the Complexity of Computing the Volume of a Polyhedron

We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.

Surface-Shadow Generation For The Preparation Of Synthetic Imagery

Shadows can add significant edge and surface detail to imagery and thus substantially increase the performance of automated correlation guidance systems. A shadow-generation algorithm was implemented to increase the accuracy of synthetic imagery used to simulate visible, near-infrared, and far-infrared sensors. Initially, a data base was established in which all surfaces were represented by a list of vertices and material codes and arranged according to a scheme of a priori masking priority. Each surface was then clipped against updated clipping polygons representing the silhouette of all previous surfaces that had higher masking priorities as viewed from the position of the light source. The resulting hidden surface was inserted into the data base and flagged as a shadow for gray-scale prediction by the appropriate sensor model. Because each surface is compared to a union of polygons rather than to individual surfaces, this algorithm is computationally efficient for use with large data bases.

A computer program for the cartesian translation of crystallographic data

A computer program has been developed to enable a Cartesian description of any crystal. This program requires standard crystallographic data (system, number of forms, Miller indices, axial ratios and interaxial angles) to calculate the general equations of the faces present, a cyclic list of vertices on each face, and the volume and total surface area of the observed crystal. For verification or illustration purpose, a graphic treatment can be selected. This graphic treatment rotates, translates and scales the crystal projection according to specifications. For many purposes, crystals of low symmetry or complex crystallographic properties are more easily treated once expressed in a Cartesian manner.