Disjoint Components Research Articles

An augmented Voronoi roadmap for 3D translational motion planning for a convex polyhedron moving amidst convex polyhedral obstacles

This paper concerns the development of a piecewise linear Voronoi roadmap for translating a convex polyhedron in a three-dimensional (3-D) polyhedral world. In general the Voronoi roadmap is incomplete for motion planning, i.e., it can have several disjoint components in one connected component of free space. An analysis of the roadmap shows that incompleteness is caused by the occurrence of the following simple geometric structure: a polygon in the Voronoi surface containing one or more polygons inside it. We formally bring out the details of this geometric structure and give an efficient augmentation of the roadmap that makes it complete. We show that the roadmap has size e = O( n 2 Q 2 l 2), where n is the total number of faces on the obstacles, Q is the total number of obstacles and l is the number of faces on the moving object. We also present an algorithm to construct the roadmap in O(( n + Ql) e + Q 2 log Q) time.

QUERY EVALUATION IN PARTITIONED DISJUNCTIVE DEDUCTIVE DATABASES

Query evaluation in disjunctive deductive databases is in general computationally hard. The class of databases for which the process is tractable is severely limited. The complexity of the process depends on the structure of the database as well as on the type of query being evaluated. In this paper we study the issue of simplified query processing in disjunctive deductive databases. We address the possibility of evaluating general queries by independently processing their atomic components and describe the class of databases for which this approach is possible. We also discuss the issue of dividing a disjunctive deductive database into a set of disjoint components then answering queries and computing database completions by combining the results obtained against the individual components. Some practical special cases are considered. The methods developed in this paper can be utilized to introduce parallelism into the query evaluation process.

A Lagrangian Relaxation Algorithm for Multidimensional Assignment Problems Arising from Multitarget Tracking

The central problem in multitarget tracking is the data association problem of partitioning the observations into tracks in some optimal way so that an accurate estimate of the true tracks can be recovered. This work considers what is perhaps the simplest multitarget tracking problem in a setting where the issues are easily delineated, i.e., straight lines in two-dimensional space-time with an error component introduced into the observations. A multidimensional assignment problem is formulated using gating techniques to introduce sparsity into the problem and filtering techniques to generate tracks which are then used to score each assignment of a collection of observations to a filtered track. Problem complexity is further reduced by decomposing the problem into disjoint components, which can then be solved independently. A recursive Lagrangian relaxation algorithm is developed to obtain high quality suboptimal solutions in real-time. The algorithms are, however, applicable to a large class of sparse mul...

The topology of interfaces

A discrete model of an interface separating two phases is considered. The interface is defined as the boundary of a self-avoiding n-omino with volume n embedded in the hypercubic lattice. The author proves that if there is no surface tension in the interface, then it has 'non-trivial' topology with probability one in the scaling limit. Here non-trivial topology means that the interface will consist of several disjoint components and will have non-zero genus. Moreover, the total area of the n-omino will be proportional to its volume.

Intent: an integrated environment for distributed heterogeneous databases

Intent: an integrated environment for distributed heterogeneous databases

A logical approach to asymptotic combinatorics I. First order properties

We shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics; this framework provides methods for determining the probability that a large, finite structure, ran- domly chosen from a given class, will have a given property. Our main concern is the asymptotic probability: the limiting value as the size of the structure increases. For example, a common problem in elementary probability texts is to show that the asymptotic probabity that a per- mutation will have no fixed point is l/e. We shall say nothing about the closely related problem of determining rates of convergence, although the methods presented here may extend to such problems. To develop a general approach we must fix a language for specifying properties of structures. Thus, our approach is logical; logic is the branch of mathematics that deals with problems of language. In this paper we con- sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties. In the sequel to this paper we consider properties expressible in the more general language of monadic second order logic. Clearly, we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant. Therefore, we choose to consider only classes closed under disjoint unions and components (see Sect. 1 for definitions). This condition is general enough to include the classes of graphs, permutations, unary functions, and many other important examples. The main theorems of the paper are about classes satisfying this condition: Theorems 5.7 and 5.8 state that the existence of asymptotic probabilities for a special set of sentences called component-bounded sentences (defined in Sect. 5) is equivalent to a con- dition on the growth of such a class; Theorem 5.9 gives necessary and suf- ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes. In Section 8 we present a long list of examples from the literature.

A computational model for interfaces

Decompositions of the plane into disjoint components separated by curves occur frequently. We describe a package of subroutines which provides facilities for defining, building, and modifying such decompositions and for efficiently solving various point and area location problems. Beyond the point that the specification of this package may be useful to others, we reach the broader conclusion that well-designed data structures and support routines allow the use of more conceptual or non-numerical portions of mathematics in the computational process, thereby extending greatly the potential scope of the use of computers in scientific problem solving. Ideas from conceptual mathematics, symbolic computation, and computer science can be utilized within the framework of scientific computing and have an important role to play in that area.

On Unimodality for Linear Extensions of Partial Orders

R. Rivest has recently proposed the following intriguing conjecture: Let $x^* $ denote an arbitrary fixed element in an n-element partially ordered set P, and for each k in $\{ 1,2, \cdots ,n \}$ let $N_k $ be the number of order-preserving maps from P onto $\{ 1,2, \cdots ,n \}$ that map $x^* $ into k. Then the sequence $N_1 , \cdots ,N_n $ is unimodal. This note proves the conjecture for the special case in which P can be covered by two linear orders. It also generalizes this result for P that have disjoint components, one of which can be covered by two linear orders.

An Extension of the Change-in-Ratio Method

The traditional change-in-ratio (CIR) method for estimating the size of animal populations assumes that the population is split into two disjoint and exhaustive components. A method for obtaining maximum likelihood estimates of population component sizes is presented for the more general situation in which the CIR method is utilized by splitting the population into three disjoint components. The large sample variance-covariance matrix of these estimators is derived.

On the adiabatic theorem of quantum mechanics

Let H(t) be a Hamiltonian whose spectrum has for all t a finite number of disjoint components sigma j(t). It is proved that when the change of H(t) is made infinitely slow the system, when started from a state corresponding to sigma k(0), passes through states corresponding to sigma k(t), for all t.

Estimates of general Mayer graphs. II. Long-range behavior of graphs with two root points occurring in the theory of ionized systems

We find the asymptotic behavior of general Mayer 2-graphs (Mayer graphs with two root points), which occur in the theory of ionized systems. This problem arises when one wants to compute corrections to the Debye length for large values of the plasma parameter. For a given 2-graph Γ(r) with Debye-Huckel linese−τ/r, we prove the inequalitiesCmr−λe−λτ ⩽Γ(r)⩽ Γ(r0)CMr3k−le−λτ, for anyr⩾r0, and whereCm andCM are positive and finite constants which depend only on Γ. These bounds are finite whenever Γ(r) is not infinite everywhere. The integersl, k, and λ denote, respectively, the number of lines of the graph Γ, its number of field points, and its local line connectivity (the maximum number of chains linking the root points, which have no line in common). From this result, we deduce that the simple irreducible 2-graphs dominant at large distances decay exponentially likee−τ and have an isthmus between the root points (an isthmus is a line whose deletion separates the graph into two disjoint components, each one containing a root point). We prove also that 2-graphs that have a number of linesl > 3k+λ are infinite. We exhibit simple, irreducible prototypes satisfying this condition, for anyk ⩾ 6. This implies that the Abe-Meeron theory of ionized gases as applied to a classical plasma is not free from divergences. Finally, we extend the preceding results to 2-graphs with lines FL=(e−τ/r)kL, withkL real positive. We prove that they still decay exponentially likee−λτ, where λ is now the maximal flow in a network associated to Γ by assigning the capacitykL to each lineL.