Showing papers on "Disjoint sets published in 1984"
TL;DR: This paper considers the problem of finding, for each possible sink vertex v, a pair of edge-disjoint paths from s to v of minimum total edge cost, and gives an implementation of Suurballe's algorithm that runs in O(m log(1+ m/n)n) time and O( m) space.
Abstract: Let G be a directed graph containing n vertices, one of which is a distinguished source s, and m edges, each with a non-negative cost. We consider the problem of finding, for each possible sink vertex v, a pair of edge-disjoint paths from s to v of minimum total edge cost. Suurballe has given an O(n2 logn)-time algorithm for this problem. We give an implementation of Suurballe's algorithm that runs in O(m log(1+ m/n)n) time and O(m) space. Our algorithm builds an implicit representation of the n pairs of paths; given this representation, the time necessary to explicitly construct the pair of paths for any given sink is O(1) per edge on the paths.
758 citations
TL;DR: It is shown that two one-pass methods proposed by van Leeuwen and van der Weide are asymptotically optimal, whereas several other methods, including one proposed by Rein and advocated by Dijkstra, are slower than the best methods.
Abstract: This paper analyzes the asymptotic worst-case running time of a number of variants of the well-known method of path compression for maintaining a collection of disjoint sets under union. We show that two one-pass methods proposed by van Leeuwen and van der Weide are asymptotically optimal, whereas several other methods, including one proposed by Rein and advocated by Dijkstra, are slower than the best methods.
475 citations
TL;DR: An algorithm is given that provides information about the topological structure of a cad of the plane that is not contained in the original version of that algorithm in (Collins 1975).
Abstract: In Part I of the present paper we defined cylindrical algebraic decompositions (cad’s), and described an algorithm for cad construction. In Part II we give an algorithm that provides information about the topological structure of a cad of the plane. Informally, two disjoint cells in E r , r ≥ 1, are adjacent if they touch each other; formally, they are adjacent if their union is connected. In a picture of a cad, eg Fig. 2 of Part I, it is obvious to the eye which pairs of cells are adjacent. However, the cad algorithm of Part I does not actually produce this information, nor did the original version of that algorithm in (Collins 1975).
150 citations
01 Jan 1984
TL;DR: In this paper, the authors discuss normality and collectionwise normality in Zermelo-Fraenkel set theory and show that one can get from normality to ZFC if one or both of two circumstances hold: the canonical cover has a nice refinement or the elements of the collection are nice.
Abstract: Publisher Summary This chapter discusses normality and collectionwise normality. A topological space is normal if there exist open sets UA and UB that separate disjoint closed sets A and B. Normality implies that any finite number of disjoint closed sets can be simultaneously separated. A convergent sequence with its limit point shows that “finite” cannot be extended to “countable.” A space is defined to be collectionwise normal if any discrete collection of closed sets can be separated. The chapter further presents theorems of Zermelo–Fraenkel set theory (ZFC). One can get from normality to collectionwise normality in ZFC if one or both of two circumstances hold: the canonical cover has a “nice” refinement or the elements of the collection are “nice.”
87 citations
TL;DR: In this paper, the authors give an algorithm to decide, starting with W, whether C has a simple representative, that is a representative without self-intersections, and apply it to infinite geodesies on M, corresponding to bi-infinite words in F. The algorithm is purely mechanical and combinatorial.
Abstract: Let M be a compact orientable surface with non-empty boundary and with X{M) < 0, and let T = nlM. Let C be the free homotopy class of a closed loop on M and let W = W{C) be a word in a fixed set of generators T which represents C. In this paper we give an algorithm to decide, starting with W, whether C has a simple representative, that is a representative without self-intersections. Such a word will be said to be simple. As an application, we begin a study of simple words in F. Our results also apply to infinite geodesies on M, corresponding to biinfinite words in F, where now we ask which finite blocks appear in such a word when the corresponding infinite homotopy class has a simple representative. For finite words there are, of course, other such algorithms, see for example [8, 9, 2, 3, 4]. Our algorithm most resembles that in [2] in that it is purely mechanical and combinatorial. It is simpler than that in [2] but what is more important is that it reveals the underlying mechanism which determines whether self-intersections occur; the combinatorics of that mechanism seem quite interesting and non-trivial. We represent M as U/T where U ^ D is the universal covering space of M, and where D is the unit disc with the Poincare metric and F is a discrete group of hyperbolic isometries. Poincare showed in [7] that C contains a simple representative if and only if the unique smooth geodesic representative C of C is simple, and that C is simple if and only if for each lift y of C to D the curves in the infinite family {fy}fer a r e pairwise disjoint. Now, to see if geodesies yl5 y2 e {fy} are disjoint in D it is enough to know whether the ideal endpoints of yx on 3D separate those of y2. Crucial to our work is a scheme for parametrizing points on 3D by infinite words in F, first developed by Nielsen in [6]. The idea of this paper is to show how information on the order of the points 3yl53y2 on 3D is encoded in Nielsen's 'boundary expansion' (Theorem A) and then to examine consequences. When dM ± 0 the group F is a free group so that each conjugacy class has a unique shortest representative which is obtained by cyclic reduction of any word in the class. However, if dM = 0 the shortest word in the conjugacy class is in general not unique. If dM = 0 and W e T has a shortest representative which does not contain any pieces which are half of the defining relator in F, then the problem of deciding whether W is simple is identical with that on the surface with a disc removed, that is one simply regards F as if it were a free group. On the other hand, the exceptional cases when W contains half a relator involve some subtle points which are not without interest, but are somewhat tangential to the main idea in the paper. For that reason, we shall omit the case in which dM = 0 . Here is an outline of this paper. The tools we need are set up in §§2 and 3 where we prove Theorem A. The algorithm (Theorem B) is given in §4. In §5 we give
73 citations
TL;DR: A convex structure is binary if every finite family of pairwise intersecting convex sets has a non-empty intersection as mentioned in this paper and the hull of a finite set is a polytope.
Abstract: A convex structure is binary if every finite family of pairwise intersecting convex sets has a non-empty intersection. Distributive lattices with the convexity of all order-convex sublattices are a prominent type of example, because they correspond exactly to the intervals of a binary convex structure which has a certain separation property. In one direction, this result relies on a study of so-called base-point orders induced by a convex structure. These orderings are used to construct an 'intrinsic' topology. For binary convexities, certain basic questions are answered with the aid of some results on completely distributive lattices. Several applications are given. Dimension problems are studied in a subsequent paper. 0. Introduction A convex structure consists of a set X, together with a collection ^ of subsets of X, henceforth called convex sets, such that (1.1) the empty set and the set X are convex; (1.2) the intersection of convex sets is convex; (1.3) the union of an updirected collection of convex sets is convex. The collection € itself is called a convexity on X. Axiom (1.2) allows the construction of an associated (convex) hull operator (usually denoted by h) in the obvious way. The hull of a finite set is called a polytope, and the hull of a two-point set is also called an interval. A half-space is a convex set with a convex complement. The following separation axioms—comparable with the axioms Tl 5 . . . ,T4, in topology—are used frequently: Sx: singletons are convex (which we will assume throughout); S2: two distinct points are in complementary half-spaces; S3: a convex set is an intersection of half-spaces; S4: two disjoint convex sets extend to complementary half-spaces. In this paper we will concentrate largely on a particular, though fundamental, class of convexities with the following binarity property: each finite collection of pairwise intersecting convex sets has a non-empty intersection. The basic types of examples are described in §1. Many of these examples arise in a topological context; for other examples, a natural topology can be constructed. Binary convex structures on a topological space have been studied extensively in the past years [15, 17, 19, 22, 29, 34, 38]. The topology and the convexity are always assumed to be compatible in the sense that polytopes are closed. A triple, consisting of a set X, a convexity €, and a compatible topology ST, is called a topological convex Proc. London Math. Soc. (3), 48 (1984), 1-33. 5388.3.48 A
64 citations
01 Jan 1984
TL;DR: In this paper, the authors focus on the preservation of separation and covering properties under the product operation and show that the Lindelof property is not preserved by products, unlike all of the weaker separation properties.
Abstract: Publisher Summary
This chapter focuses on the preservation of separation and covering properties under the product operation. Unlike all of the weaker separation properties, normality, paracompactness, and the Lindelof property are not preserved by products. A space is said to be κ-paracompact if its every open covering of cardinality has a locally finite open refinement, and a space is κ-collectionwise normal if for every discrete family {Fα}α < x of its closed subsets there is a family{Uα}α < x of mutually disjoint open sets such that Fα ⊂ Uα. A space is perfect if all of its open subsets are Fα sets and a space is perfectly normal if it is perfect and normal.
62 citations
01 Jan 1984
TL;DR: Strongly code-disjoint (SCD) checkers are defined and shown to include totally self-checking (TSC) code- Disjoint checkers, the largest class of checkers with which a combinational system may achieve the TSC goal.
Abstract: Strongly code disjoint (SCD) checkers are defined and shown to include totally self-checking code disjoint checkers. This type of checker is necessary companion of the strongly fault-secure networks defined by J.E. Smith and G. Metze (1978). SCD checkers are the largest class of checkers with which a combinatorial system may achieve the totally self-checking goal. Some examples are given to illustrate the design of SCD checkers.
60 citations
01 Jan 1984
TL;DR: In this article, the vertex set of a chemical graph is partitioned into disjoint subsets on the basis of the equivalence of various orders of closed neighborhoods and the information indices (ICν, SICN, CIν, and Rν) are calculated.
Abstract: Quantitative measures of molecular complexity are calculated through the application of information-theoretic formalism on chemical graphs. The vertex set of a chemical graph is partitioned into disjoint subsets on the basis of the equivalence of various orders of closed neighborhoods and the information indices (ICν, SICν, CIν, and Rν) are calculated. The applications of these indices in structure-activity correlations are discussed.
53 citations
01 Apr 1984
TL;DR: A number of greedy algorithms are examined and are shown to be probably inherently sequential, which means that it is unlikely that these sequential algorithms can be sped up significantly using parallelism.
Abstract: A number of greedy algorithms are examined and are shown to be probably inherently sequential. Greedy algorithms are presented for finding a maximal path, for finding a maximal set of disjoint paths in a layered dag, and for finding the largest induced subgraph of a graph that has all vertices of degree at least k. It is shown that for all of these algorithms, the problem of determining if a given node is in the solution set of the algorithm is P-complete. This means that it is unlikely that these sequential algorithms can be sped up significantly using parallelism.
47 citations
TL;DR: In this article, the concept of local automorphism is introduced, where a bijective function f is a local automomorphism of A, if Dom f and Rng f (range) are subsets of the base, and if f is an isomorphism from the restriction A/Dom f onto A/Rng f.
Abstract: Given a chain, or total ordering ⩽, an interval can be defined in two ways : absolute and relative. An absolute interval is a subset I of the base, such that if x, y ∈ I and x ⩽ z ⩽ y, then z ∈ I. A relative interval with bound {a, b} where a t. A key for generalization to an arbitrary relation A, is the notion of local automorphism. A bijective function f is a local automorphism of A, if Dom f and Rng f (range) are subsets of the base, and if f is an isomorphism of the restriction A/Dom f onto A/Rng f. Then a subset I of the base |A| is an absolute A-interval, if any local automorphism f of the restriction A/I is extensible by the identity map on |A|-I; i.e. the union of f and of the latter identity map is still a local automorphism of A. Any intersection of A-intervals is an A-interval. A characterization is given for the exterval (complement of an interval). A subset D of the base is a relative A-interval with bound F (subset of the base), if D is disjoint from F and is maximal, with respect to inclusion, among those sets D such that any local automorphism of A/D is extenseible by the identity map on F. Every absolute A-interval is a relative A-interval. Both notions are identical for a chain A, but already differ for a partial ordering A. Two related notions are introduced : those of finite-val and subval. The finite-val is a boolean notion : union, intersection, complement of finite-vals are finite-vals. For a circular ordering C defined from a chain A by C (x, y, z) = + iff x ⩽ y ⩽ z (mod A) or any condition obtained from this by a circular permutation; then I is a C-subval iff I is an A-absolute interval or exterval; intuitively the subval is the circular segment, or cake-portion. Every interval or exterval is a subval; every subval is a finite-val, but not conversely. An A-filter (A-ultrafilter) is defined as usually, by replacing sets by A-absolute intervals. A compactable relation is defined in two equivalent ways : (1) for every A-interval I, the complement of I is a finite union of A-intervals; (2) to each A-ultrafilter F, associate an element h (F) ∈ F; then there exist finitely many F such that the union of corresponding h (F) covers the base. Finally A-ultrafilters and usual ultrafilters are used to extend to arbitrary relations: (1) the closure of rationals by real numbers; (2) the Stone closure of a boolean lattice.
TL;DR: In this paper, the probabilistic properties are described in terms of non-commutative probability theory based on von Neumann algebras and examples of this type are generated, by means of second quantisation, from a unitary dilation of a given self-adjoint contraction semigroup, called the time orthogonal unitary Dilation, whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary Hilbert spaces.
Abstract: An analysis of Feynman-Kac formulae reveals that, typically, the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line. We consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non-commutative probability theory based on von Neumann algebras. Examples of this type are generated, by means of second quantisation, from a unitary dilation of a given self-adjoint contraction semigroup, called the time orthogonal unitary dilation, whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary Hilbert spaces.
TL;DR: The polynomial heuristic algorithm is proved to give optimal solution for length less than five and experiments show very good results for the algorithm.
Abstract: We consider the following problem: Given an integer k and a network G with two distinct vertices s and t, find a maximum number of vertex disjoint paths from s to t of length bounded by k. In a recent work [9] it was shown that for length greater than four this problem is NP-hard. In this paper we present a polynomial heuristic algorithm for the problem for general length. The algorithm is proved to give optimal solution for length less than five. Experiments show very good results for the algorithm.
TL;DR: In this paper, the authors describe an operation that transforms a g-graph with n vertices and e edges into one with n + 1 vertices, and e + 2 edges.
Abstract: Publisher Summary Graphs (V, E) are presented in this chapter, where V is a finite point set in ℝ2 in general position and E is a collection of nondegenerate closed straight line segments with endpoints in V. Such graphs are called “geometric graphs” (g-graphs). Denote by D a configuration comprising two disjoint segments and by CD a D configuration whose segments are sides of a convex quadrilateral. A g-graph is D-free (CD-free) if it has no subgraph of type D (CD). A g-graph G = (V, E) is maximal D-free (maximal CD-free) if it is D-free (CD-free) and has the maximum possible number of edges T(n, D) (T(n, CD)). The chapter describes an operation that transforms a g-graph with n vertices and e edges into one with n + 1 vertices and e + 2 edges.
TL;DR: It is proved that if |ℓ|>3.5 then ℓ contains four distinct members A, B, C, D such that A∪B=C∪D and A∩B =C∩D=0.
Abstract: Let l be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |l|>3.5\(\left( {\begin{array}{*{20}c} n \\ {r - 1} \\ \end{array} } \right)\) then l contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.
TL;DR: In this paper, the existence of disjoint invariant regions on the same constant energy surface has been investigated in nonlinear hamiltonian systems with more than two degrees of freedom.
TL;DR: The partitioning problem is studied, consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized, which is equivalent to minimizing the completion time of n jobs on m parallel identical processors.
Abstract: We study the partitioning problem, consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized. This is equivalent to minimizing the completion time of n jobs on m parallel identical processors. We establish upper bounds on the deviation from optimality of two heuristics: the well-known LPT heuristic, and the on-line RLP heuristic. These bounds serve to establish a probabilistic analysis of these heuristics; for both of them, the absolute deviation from optimality remains finite, when the size of the list of numbers becomes infinite. This is a stronger result than previous convergence theorems, and it is valid whenever the processing times are iid random variables with finite mean and arbitrary distributions.
TL;DR: A significant association was found between morphologically alike and geographically near species of Medicago, which is interpreted as evidence confirming the prediction.
Abstract: The pairwise relationships among a set or objects can be divided into two subsets, namely, those indicating the objects which are relative neighbors, and those which do not. If it is reasonable to expect that the same subsets should be obtained from a different source for the relationships, then a contingency table can be formed, and a hypothesis of marginal independence can be examined. In an example in which the spoken letters of the alphabet were compared with what is heard, it was found that there was a high degree of association, but none between either of these two sets and their lowercase visual appearance. In a second example, a hypothesis was examined which postulates that species which recently diverged should tend to be geographically close. Before this could be evaluated, it was necessary to define a distance between distinct taxa occupying not necessarily disjoint geographical regions. With the hypothesis that the set of species morphologically similar to each includes some or all of those wh...
TL;DR: An alternative proof to a theorem of Cohn and Lempel relating the number of orbits of a product of a full cycle by disjoint transpositions to the rank over GF (2) of the associated chord-intersection-matrix is yielded.
TL;DR: In this article, a continuation of Harrington and Shelah [3] is presented, where the contents of this paper are self-contained, and the authors present a discussion of their work.
Abstract: This is a continuation of Harrington and Shelah [3]; however, the contents of this paper are self-contained.
TL;DR: This work shows that any creative set has the U.S.P. of r.e. sets B and C such that B has degree β and that there are complete r.
Abstract: An re set A of degree α is said to have the universal splitting property (USP) if for each re degree β ≤ α, there is a splitting of A into disjoint re sets B and C such that B has degree β We show that any creative set has the USP and there are complete re sets A which fail to have the USP We show that there are re degrees a such that no re set A of degree a has the USP We also explore the possible degrees of re bases of re vector spaces
TL;DR: In this article, the authors consider finitely additive random measures taking independent values on disjoint Borel sets in $R^k, and ask when such measures possess versions which are right continuous with left limits, in an appropriate sense.
Abstract: We consider finitely additive random measures taking independent values on disjoint Borel sets in $R^k$, and ask when such measures, restricted to some subclass $\mathscr{A}$ of closed Borel sets, possess versions which are "right continuous with left limits", in an appropriate sense. The answer involves a delicate relationship between the "Levy measure" of the random measure and the size of $\mathscr{A}$, as measured via an entropy condition. Examples involving stable measures, Dudley's class $I(k, \alpha, M)$ of sets in $R^k$ with $\alpha$-times differentiable boundaries, and convex sets are considered as special cases, and an example given to show what can go wrong when the entropy of $\mathscr{A}$ is too large.
TL;DR: This time-and space-optimal algorithm to compute the edges of the disjoint polygons, that is, the contour, uses the scan-line paradigm as in two previous approaches to this problem for rectangles, but requires a simpler data structure.
01 Jan 1984
TL;DR: In this paper, the authors present a systematic introduction to certain recent developments in the study of measures in topological spaces and prove that CH holds if and only if in every nonempty compact space, the intersection of fewer than c open dense sets is notempty.
Abstract: Publisher Summary
This chapter presents a systematic introduction to certain recent developments in the study of measures in topological spaces. A finite Borel measure in X is regular whenever it is either inner regular or outer regular. A space X satisfies the countable chain condition (ccc) if each disjoint family of open subsets of X is countable. A space X satisfies the ccc whenever either of the following conditions holds: (1) X is separable or if (2) X is the support of a σ-finite Borel measure in X. This chapter further presents a proof of the proposition that CH holds if and only if in every nonempty compact space, the intersection of fewer than c open dense sets is nonempty.
TL;DR: In this article, the conditions for inclusion are provided under which a part of the motion of the larger system is reproduced by the smaller system, that is, the large system “includes” the smaller one.
TL;DR: In this paper, the authors analyze the representation of any relation R by a union of matching relations, and present a general discussion of the relation representation and various results concerning the representation.
01 Jan 1984
TL;DR: The problem of representing a given gaussian zero mean random vector y by linear statistical models is considered and it is shown that to every random vector x, making y1 and y2 conditionally independent given x there corresponds an (essentially unique) model of y of the form (0).
Abstract: The problem of representing a given gaussian zero mean random vector y by linear statistical models is considered. This is a concrete formulation of a simple stochastic realization problem. Let y=[y′1],y′2]′ be any partition of y into two disjoint subvectors y1, y2. It is shown that to every random vector x, making y1 and y2 conditionally independent given x there corresponds an (essentially unique) model of y of the form
$$\begin{gathered}y_1 = H_1 x + n_1 \hfill \\y_2 = H_2 x + n_2 \hfill \\\end{gathered} $$
(0)
where H1 and H2 are deterministic matrices, n1 and n2 are mutually independent noise terms and each ni(i=1,2) is independent of x. The family of all realizations of y of the form (0) is analyzed both probabilistically and from the point of view of explicit computation of the parameters. Possible applications especially to the theory of Factor Analysis are discussed.
TL;DR: An analytical method to control by multiparameter output feedback the characteristic values of a closed loop multi-input multi-output system inside (or outside) general aubregions of the complex plane is presented in this paper.
Abstract: An analytical method to control by multiparameter output feedback the characteristic values of a closed loop multi-input multi-output system inside (or outside) general aubregions of the complex plane, is presented. The method pertains to analogue as well as discrete systems. The design method is robust in the sense that the characteristic values are designed to cluster inside the desired subregion, even though any number of the specifying parameters are only known to within a given interval. The interval may possibly be disjoint and possibly infinite (completely unknown). Examples are provided which illustrate the broadness and effectiveness of the method.