Showing papers on "Disjoint sets published in 1988"

Systems of differential equations which are competitive or cooperative: III. Competing species

Abstract: For pt.II, see J. Math. Anal., vol.16, p.423 (1985). Persistent trajectories of the n-dimensional system xi=xiNi(x1, . . ., xn), xi>or=0, are studied under the assumptions that the system is competitive and dissipative with irreducible community matrices ( delta Ni/ delta xj). The main result is that there is a canonically defined countable (generically finite) family of disjoint invariant open (n-1) cells which attract all non-convergent persistent trajectories. These cells are Lipschitz submanifolds and are transverse to positive rays. In dimension 3 this implies that an omega limit set of a persistent orbit is either an equilibrium, a cycle bounding an invariant disc, or a one-dimensional continuum having a non-trivial first Cech cohomology group and containing an equilibrium. Thus the existence of a persistent trajectory in the three-dimensional case implies the existence of a positive equilibrium. In any dimension it is shown that if the community matrices are strictly negative then there is a closed invariant (n-1) cell which attracts every persistent trajectory. This shows that a seemingly special construction by Smale (1976) of certain competitive systems is in fact close to the general case.

New methods for computing visibility graphs

Abstract: Let S be a set of n non-intersecting line segments in the plane The visibility graph GS of S is the graph that has the endpoints of the segments in S as nodes and in which two nodes are adjacent whenever they can “see” each other (ie, the open line segment joining them is disjoint from all segments or is contained in a segment) Two new methods are presented to construct GS Both methods are very simple to implement The first method is based on a new solution to the following problem: given a set of points, for each point sort the other points around it by angle It runs in time O(n2) The second method uses the fact that visibility graphs often are sparse and runs in time O(m log n) where m is the number of edges in GS Both methods use only Ogr;(n) storage

P-printable sets

Abstract: P-printable sets arise naturally in the.studies of generalized Kolmogorov complexity and data compression, as well as in other areas. We present new characterizations of the P-printable sets and present necessary and sufficient conditions for the existence of sparse sets in P that are not P-printable. As a corollary to one of our results, we show that the class of sets of small generalized Kolmogorov complexity is exactly the class of sets which are P-isomorphic to a tally language.

A fast algorithm to minimize multi-output mixed-polarity generalized Reed-Muller forms

Abstract: A very fast computer program that accepts a Boolean function as an array of multi-output disjoint cubes and returns a mixed-polarity Generalized Reed-Muller Form is presented. Such circuits often have gates and interconnections than classical sum-of-product realizations and are easily testable. The program was tested on many examples from literature as well as on many large arithmetic functions with up to 8 inputs, 8 output and 255 minterms. On all the examples from the literature the solutions were either the same or better than those generated by other methods. The algorithm is based on a new cube operation, called xlinking, that generalizes known operations of merger, exclusion and other logic operations specified by previous authors.

Disjoint Products and Efficient Computation of Reliability

Abstract: In this paper, we analyze the problem of computing the probability of the union of a set of events, where each event is given as the product of a set of Boolean variables. Each Boolean variable represents the operation or failure of a particular component. The problem has direct applications to the reliability analysis of complex systems as well as more general applications. After showing that the problem is NP-hard in general, we define simple, computationally efficient procedures for generating upper and lower bounds on the required probability. We show that these procedures produce the exact answer, i.e., the upper bound equals the lower bounds, for two classes of systems, matroids and threshold systems. These results draw on the relationship between this problem and the notion of shellability studied in the context of simplicial polytopes. Shellability is shown to have a very interesting and useful interpretation in this problem setting.

On sets truth-table reducible to sparse sets

Abstract: We study sets that are truth-table reducible to sparse sets in polynomial time. The principal results are as follows: (1) For every integer $k > 0$, there is a set L and a sparse set S such that $L...

Parallel algorithms for some functions of two convex polygons

Abstract: Let P andQ be two convex,n-vertex polygons. We consider the problem of computing, in parallel, some functions ofP andQ whenP andQ are disjoint. The model of parallel computation we consider is the CREW-PRAM, i.e., it is the synchronous shared-memory model where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location (even if they are trying to write the same thing). We show that a CREW-PRAM havingn 1/k processors can compute the following functions in O(k1+ɛ) time: (i) the common tangents betweenP andQ, and (ii) the distance betweenP andQ (and hence a straight line separating them). The positive constant ɛ can be made arbitrarily close to zero. Even with a linear number of processors, it was not previously known how to achieve constant time performance for computing these functions. The algorithm for problem (ii) is easily modified to detect the case of zero distance as well.

An approach to homotopy classification of links

Abstract: A reformulation and refinement of the i-invariants of Milnor are used to give a homotopy classification of 4 component linlis and suggest a possible general homotopy classification. The main idea is to use the (reduced) group of a link and its "geometric" automorphisms to define the precise indeterminacy of these invariants. Introduction. In 1954, Milnor [M] introduced the concept of link homotopy, a notion considerably weaker than the usual notion of link type. In fact it is also weaker than the relations of PL and topological isotopy and I-equivalence introduced later (see [R, Go, G1]), but equivalent to the apparently weaker relation which asks for disjoint singular cylinders in I x S3 connecting the links (see [Li]). Recently link homotopy has come up in a reformulation of the 4-dimensional topological surgery conjecture (see [FL]). Unlike usual link type the classification problem for link homotopy seems quite tractable in [M] Milnor gives a complete homotopy classification of links with 3 or fewer components, and shows that a link is homotopically trivial if and only if a certain collection of numerical invariants is defined and vanishes alternatively this can be interpreted in terms of the reduced link group *9(L) defined in [M] (see below). The numerical homotopy invariants used in [M] are part of a wider collectionthe ,u-invariants which appear in [M1] and are there shown to be invariants of topological isotopy. In [S] they are shown to be invariants of topological Iequivalence, but are not, except for the subset used in [M], homotopy invariants. Although the ,u-invariants do not serve to classify links under I-equivalence, it is still an open question whether a link with trivial ,u-invariants is cobordant to a boundary link. A boundary link is one whose components bound disjoint surfaces in S3-they have trivial ,u-invariants and include all split links (links whose components lie in disjoint balls in S3) and parallel links (links formed by several "parallel" copies of a single knot). As already mentioned, the homotopy ,u-invariants are trivial if and only if the link is homotopy trivial but it is not true that any links with the same ,u-invariants are homotopic [L] their indeterminacy is too large. In [L] a classification of links up to surgery equivalences is given using certain of the homotopy ,u-invariants with a refinement of their indeterminacy. In fact, it seems implicit in [M] that one should be able to use the ,u-invariants to give a general homotopy classification. It is shown in [M, P] that the ,u-invariants Received by the editors December 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. Partially supported by the National Science Foundation. (r)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach

Abstract: We deal with the coupling of hyperbolic and parabolic systems in a domain Ω divided into two disjoint subdomains Ω+ and Ω− . Our main concern is to find out the proper interface conditions to be fulfilled at the surface separating the two domains. Next, we will use them in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, that is, the hyperbolic system is rendered parabolic by adding a small artificial “viscosity”. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These we take as interface conditions for the hyperbolic-parabolic problem. Actually, we discuss two alternative sets of interface conditions according to whether the regularization procedure is variational or nonvariational. We show how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, we discuss a method of resolution which alternates the resolution of the hyperbolic problem within Ω− and of the parabolic one within Ω+ . The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows.

A new hierarchy of hypercube interconnection schemes for parallel computers

Abstract: This paper introduces a new hierarchy of cube-based interconnection schemes, called the base-b cube (which properly contains the well-known binary cube), for the design of parallel computers. This hierarchy admits a recursive definition and allows many more reconfigurations than are possible with the binary cube. Our analysis addresses the inherent cost-delay trade-off for this hierarchy along with a number of related topological properties such as sparsity, diameter, existence of node disjoint paths, and odd and even cycles. Embeddings of standard interconnection schemes including linear and two-dimensional arrays, rings, and complete binary trees in a base-b cube are illustrated.

On restricted two-factors

Abstract: A two-factor of G consists of disjoint cycles that cover $V( G )$. The authors consider the existence problem for two-factors in which the cycles are restricted to having lengths from a prescribed (possibly infinite) set of integers. Theorems are presented which derive the existence of such restricted two-factors in G from their existence in $G - u$ and $G - v $. The possibility of such theorems is then related to the complexity of the corresponding existence problem. In particular, the only four cases in which polynomial algorithms can be expected (in the sense that all other cases are shown to be NP-hard) are identified.

An algebra for data flow diagram process decomposition

Abstract: Data flow diagram process decomposition, as applied in the analysis phase of software engineering, is a top-down method that takes a process, and its input and output data flows, and logically implements the process as a network of smaller processes. The decomposition is generally performed in an ad hoc manner by an analyst applying heuristics, expertise, and knowledge to the problem. An algebra that formalizes process decomposition is presented using the De Marco representation scheme. In this algebra, the analyst relates the disjoint input and output sets of a single process by specifying the elements of an input/output connectivity matrix. A directed acyclic graph is constructed from the matrix and is the decomposition of the process. The graph basis, grammar matrix, and graph interpretations, and the operators of the algebra are discussed. A decomposition procedure for applying the algebra, prototype, and production tools and outlook are also discussed. >

Partitioning polyhedral objects into nonintersecting parts

Abstract: An algorithm is described for partitioning intersecting polyhedrons into disjoint pieces and, more generally, removing intersections from sets of planar polygons embedded in three space. Polygons, or faces, need not be convex and may contain multiple holes. Intersections are removed by considering pairs of faces and slicing the faces apart along their regions of intersection. To reduce the number of face pairs examined, bounding boxes around groups of faces are checked for overlap. The intersection algorithm also computes set-theoretic operations on polyhedrons. Information gathered during face cutting is used to determine which portions of the original boundaries may be present in the result of an intersection, a union, or a difference of solids. The method includes provisions to detect and in some cases overcome, the effects of numerical inaccuracy on the topological decisions that the algorithm must make. The regions in which ambiguous results are possible are flagged so that the user can take appropriate action. >

On a Problem upon Configurations Contained in Graphs with Given Chromatic Number

Abstract: We consider the following problem: Do simple graphs with chromatic number γ contain the following configuration: γ vertices pairwise linked by a system of paths pairwise edge disjoint? We prove this is the case for γ = 5 and γ = 6.

Representing orders on the plane by translating convex figures

Abstract: Given a finite collection of disjoint, convex figures on the plane, is it possible to assign to each a single direction of motion so that this collection of figures may be separated, through an arbitrary large distance, by translating each figure one at a time, along its assigned direction? We present a computational model for this separability problem based on the theory of ordered sets.

Computational Complexity of Coherent Systems and the Reliability Polynomial

Abstract: : There are three general methods for system reliability evaluation, namely: 1) Inclusion-Exclusion, 2) Sum of Disjoint Products, and 3) Pivoting. Of these, only pivoting can be applied directly to a logic tree or network graph representation without first finding minimal path (or cut) sets. Domination theory provides the basis for selecting optimal pivoting strategies. Simple proofs of domination theory results for coherent systems are given, based on the reliability polynomial. These results are related to the problem of finding efficient strategies for computing coherent system reliability. The original results for undirected networks are due to Satyanarayana and Chang (1983). Many of the original set theoretic results are due to Huseby (1984). However, he does not use the reliability polynomial to prove his results. Additional keywords: Operation's research. (Author)

Iterative disjoint cluster and discriminant function processing of formation log responses and other data

Abstract: A set of representative subpopulations of a dataset such as a set of depth related formation log responses is selected by (a) classifying the dataset into n=a disjoint clusters; (b) determining discriminant function for the thus produced n=a disjoint clusters; (c) generating a measure respresentative of agreement/disagreement betweeen the classification generated by cluster analysis in step (a) and the classification generated by discriminant analysis in step (b); (d) repeating steps a-c for other numbers n=a, b, c, . . . ; and (e) selecting the set n=k where k is a, b, c, . . . , where the measure shows best agreement.

Amortized Analysis of Algorithms for Set Union with Backtracking. Revision

Abstract: : Mannila and Ukkonen have studied a variant of the classical disjoint set union (equivalence) problem in which an extra operation, called deunion, can undo the most recently performed union operation not yet undone They proposed a way to modify standard set union algorithms to handle deunion operations This document analyzes several algorithms based on their approach The most efficient such algorithms have an amortized running time of O(log n/log log n) per operation, where n is the total number of elements in all the sets These algorithms use O(n log n) space, but the space usage can be reduced to O(n) by a simple change It is proven that any separable pointer-based algorithm for the problem required omega(log n/log log n) time per operation, thus showing that our upper bound an amortized time is tight (KR)

Integer sets with distinct subset-sums

Abstract: In Section 1 we introduce the problem of finding minimal-height sets of n natural numbers with distinct subset-sums (SSD), and in Section 2 review the well-known Conway-Guy sequence u, conjectured to yield a minimal SSD set for every n. We go on (Section 3) to prove that u certainly cannot be improved upon by any "greedy" sequence, to verify numerically (Section 4) that it does yield SSD sets for n < 80, and (Section 5) by direct search to show that these are minimal for n S 8. There is a brief interlude (Section 6) on the problem of decoding the subset from its sum. In Section 7 generalizations of u are constructed which are asymptotically smaller: Defining the Limit Ratio of a sequence w to be a = 1imn wn/2l, the Atkinson-Negro-Santoro sequence v (known to give SSD sets) has a = 0.6334, ConwayGuy (conjectured to) has a = 0.4703, and our best generalization has a = 0.4419. We also (Section 8) discuss when such sequences have the same a, and (Section 9) how a may efficiently be computed to high accuracy. 1. The Distinct Subset-Sum Problem. A well-known problem in combinatorial number theory [2, pp. 64-65] involves the construction for given n of a set p = { pi}, i = 1, .. ., n, of natural numbers possessing the property we shall call Subset-Sum Distinctness, or SSD for short: That is, (l.l) ~ ~ F pi P= E, Pi * S = T c {l, ... ,n} i(S i(T or, distinct subsets of p have distinct sums. Evidently, choosing pi = 21 satisfies (1.1); the interest lies in how much the maximum element Pn can be reduced below 2n-1. In order to explore the SSD property, it is convenient to consider a more general concept: We say that x has a representation by p, with length k and signature 1, when for some S and T as before (1.2) x= E PE Po iES ieT where S n T = 0, SI + IT = k, ISI ITI = 1; that is, x is the difference of two disjoint subset-sums, of which the first has 1 terms more than the second, and the two together have k terms. Alternatively, n (1.3) x eip i=l Received June 23, 1986; revised October 20, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11B13, 05-04, 05A99, 11J13, 94A60, 65B05.

On Independent Circuits in Finite Graphs and a Conjecture of Erdös and Pósa.

Abstract: We state and prove some results about the existence of many mutually disjoint circuits in a graph. The results were obtained during thesis work supervised by G. A. Dirac, and most of them have been announced by him several years ago. In some cases, however, proofs have never before appeared.

Unification in a Combination of Arbitrary Disjoint Equational Theories

Abstract: The unification problem for terms in a disjoint combination E1 +... +En of arbitrary theories is reduced to a combination of pure unification problems in Ej, where free constants may occur in terms, and to constant elimination problems like: find all substitutions σ such that (the free constant) ci no longer occurs in the term σt (modulo Ej), where t is a term in the theory Ej.

On perfect binary arrays

Abstract: The perfect binary arrays introduced by Calabro and Wolf correspond to a class of difference sets, viz. H sets. Existence and construction of H sets on various grids are considered. In particular, H sets on 12*12 grids are presented and a number of two-dimensional grids are indicated where H sets do not exist.< >

Green Functions and Conditioned Gauge Theorem for a 2-Dimensional Domain

Abstract: In this paper we investigate the properties of the classical Green function GD(•,•), i.e., the kernel function for the operator (− ∆/2)−1, in a domain D ⊂ R2, where D is a Jordan domain, namely, a bounded domain in R2 with the boundary ∂D which consists of finitely many disjoint Jordan curves. It is easy to see that any bounded Lipschitz domain in R2 is a Jordan domain.

On the 2-Linkage Problem for Semicomplete Digraphs

Abstract: We consider the problem of finding two disjoint directed paths with prescribed ends in a semicomplete digraph. The problem is NP - complete for general digraphs as proved in [4]. We obtain best possible sufficient conditions in terms of connectivity for a semicomplete digraph to be 2-linked (i.e., it contains two disjoint paths with prescribed ends for any four given endvertices). We also consider the algorithmically equivalent problem of finding a cycle through two given disjoint edges in a semicomplete digraph. For this problem it is shown that if D is a 5–connected semicomplete digraph, then D has a cycle through any two disjoint edges, and this result is best possible in terms of the connectivity. In contrast to this we prove that if T is a 3–connected tournament, then T has a cycle through any two disjoint edges. This is best possible, too. Finally we give best possible sufficient conditions in terms of local connectivities for a tournament to contain a cycle through af given pair of disjoint edges.

A Correlation Inequality for the Symmetric Exclusion Process

Abstract: We prove a correlation inequality concerning the occupation of disjoint sets in the symmetric exclusion process. As an application we derive a pointwise ergodic theorem for this same process.

Using trace theory to model discrete events

Abstract: In this paper discrete processes are defined by means of trace structures. Every symbol in a trace denotes (the occurrence of) some discrete event. The trace alphabet is split into two disjoint sets, one denoting the communication events, the other denoting the exogenous events. Control of a discrete process means constructing a second discrete process having as alphabet the communication events only, so that the connection of the two discrete processes results in a desired exogenous trace set. Connection of discrete processes means blending of the corresponding trace structures.