Showing papers on "Disjoint sets published in 1986"

Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability

Abstract: Frontispiece Preface 1. Notation 2. Representing sets 3. Sperner systems 4. The Littlewood - Offord problem 5. Shadows 6. Random sets 7. Intersecting hypergraphs 8. The Turan problem 9. Saturated hypergraphs 10. Well-separated systems 11. Helly families 12. Hypergraphs with a given number of disjoint edges 13. Intersecting families 14. Factorizing complete hypergraphs 15. Weakly saturated hypergraphs 16. Isoperimetric problems 17. The trace of a set system 18. Partitioning sets of vectors 19. The four functions theorem 20. Infinite ramsey theory References Index.

On the single-operation worst-case time complexity of the disjoint set union problem

Abstract: We give an algorithm for the disjoint set union problem, within the class of algorithms defined by Tarjan, which has $O(\log n/\log \log n)$ single-operation time complexity in the worst case. Also we define a class of algorithms for the disjoint set union problem, which includes the class of algorithms defined by Tarjan. We prove that any algorithm from this class has at least $\Omega ({{\log n} / {\log \log n}})$ single-operation time complexity in the worst case.

Unification in Combinations of Collapse-Free Theories with Disjoint Sets of Function Symbols

Abstract: A unification algorithm for combinations of collapse-free equational theories with disjoint sets of function symbols is presented. The algorithm uses complete unification algorithms for the theories in the combination to compute complete sets of unifiers in the combined theory. It terminates if the algorithms for the theories in the combination terminate. The only restriction on the theories in the combination — apart from disjointness of function symbol sets — is that they must be collapse-free (i.e., they must not have axioms of the form t=x, where t is a non-variable term and x is a variable). This extends the class of equational theories for which the unification problem in combinations can be solved to collapse-free theories. The algorithm is based on a study of the properties of unifiers in combination of non-regular collapse-free theories.

Splitting strongly almost disjoint families

Abstract: We say that a family A C [A]K is strongly almost disjoint if something more than just \\A n B\\ < n, e.g. that \\A f! B\\ < a < k, is assumed for A, B G A. We formulate conditions under which every such strongly a.d. family is \"essentially disjoint\", i.e. for each A E A there is F(A) G [A]

A Best-First Search Algorithm for Optimal PLA Folding

Abstract: In this paper we propose a new algorithm for optimal PLA folding based on a graph theoretic formulation. An efficient best-first search (BFS) algorithm is presented which finds a near-optimal PLA folding. The proposed algorithm first constructs the longest paths on the associated disjoint graph generated from the PLA personality matrix, and then extracts the ordered folding sets from the constructed paths. The algorithm is shown to be effective for most test cases.

On the validity of the direct sum conjecture

Abstract: The direct sum conjecture states that the multiplicative complexity of disjoint sets of bilinear computations is the sum of their separate multiplicative complexities. This conjecture is known to hold for only a few specialized cases. In this paper, we establish its validity for large classes of computations. One such class can be defined as follows. Let $S_1 $ be a set of r$m \times n$ bilinear forms, and let $S_2 $ be a different set of s$p \times q$ bilinear forms. Then, if $2 \in \{ r,m,n,s,p,q\} $, we show that the direct sum conjecture holds over any field. The proof involves some nontrivial facts from linear algebra and relies on the theory of invariant polynomials. This result also settles the multiplicative complexity of pairs of bilinear forms over any field with large enough cardinality. It is also shown that the direct sum conjecture is true for the case when $r = mn - 2$.

Geometric applications of Davenport-Schinzel sequences

Abstract: We present efficient algorithms for the following geometric problems: (i) Preprocessing of a 2-D polyhedral terrain so as to support fast ray shooting queries from a fixed point. (ii) Determining whether two disjoint interlocking simple polygons can be separated from one another by a sequence of translations. (iii) Determining whether a given convex polygon can be translated and rotated so as to fit into another given polygonal region. (iv) Motion planning for a convex polygon in the plane amidst polygonal barriers. All our algorithms make use of Davenport Schinzel sequences and on some generalizations of them; these sequences are a powerful combinatorial tool applicable in contexts which involve the calculation of the pointwise maximum or minimum of a collection of functions.

Countable-dimensional universal sets

Abstract: The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension n > 1 (Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4). 1. Terminology and notation. All spaces considered in this paper are metrizable and separable. Our terminology follows Kuratowski [Ku] and Nagata [Na 2]. 1.1. Notation. We denote by I the interval [-1, 1], IX is the countable product of I, i.e. the Hilbert cube, px: I?? I is the projection onto the ith coordinate, P is the set of the irrationals from I and w is the set of natural numbers. Given a point t C=I, we let Q,= {(Xll X2,... ) C=I?: Xl1= t}.' 1.2. Partitions. A partition in a space X between a pair of disjoint sets A and B is a closed set L such that X\ L = U U V, where U and V are disjoint open sets with A c U and B c V. 1.3. Countable dimensional spaces and the transfinite inductive dimension ind. A space X is countable dimensional if it is a countable union X = U1Xi of zero dimensional sets Xi [Hu]. The transfinite dimension ind is the extension by transfinite induction of the classical Menger-Urysohn inductive dimension: ind X = -1 means X = 0, ind X < a if and only if each point x in X can be separated in X from any closed set not containing x by a partition L with ind L < a, a being an ordinal, we let ind X be the smallest ordinal a with ind X a if such an ordinal exists, and we put ind X = ox otherwise. If ind X 0 oc, then ind X is a countable ordinal, X having a countable base. The transfinite dimension ind was first discussed by Hurewicz [Hu, ?5], [H-W, p. 50] (although the idea goes back to Urysohn's memoir [Ur, p. 66]). A comprehensive survey of the topic is given by Engelking [En 2]. Hurewicz [Hu, En 2, 4.1, 4.15] proved that for a complete space X, ind X 0 xc if and only if X is countable dimensional and that each space X with ind X 0 xc has a countable dimensional compactification. Received by the editors June 21, 1985 and, in revised form, October 15, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F45. T'This paper was written while I was visiting Auburn University, Alabama. I would like to express my gratitude to the Department of Mathematics of Auburn University for its hospitality. ?01986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Geometric permutations and common transversals

Abstract: The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular, we consider the case where the family consists of pairwise disjoint translates of a single convex set.

A reduction of Keller's conjecture

Abstract: A family of translates of a closedn-dimensional cube is called a cube tiling if the union of the cubes is the wholen-space and their interiors are disjoint. According to a famous unsolved conjecture of O. H. Keller, two of the cubes in ann-dimensional cube tiling must share a complete (n − 1)-dimensional face. In this paper we shall prove that to solve Keller's conjecture it is sufficient to examine certain factorizations of direct sum of finitely many cyclic group of order four.

Tarski and Geometry

Abstract: Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

A reduction of keller's conjecture

Abstract: A family of translates of a closed n-dimensional cube is called a cube tiling if the union of the cubes is the whole n-space and their interiors are disjoint. According to a famous unsolved conjecture of O. H. Keller, two of the cubes in an n-dimensional cube tiling must share a complete (n -- 1)-dimensional face. In this paper we shall prove that to solve Keller's conjecture it is sufficient to examine certain factorizations of direct sum of finitely many cyclic group of order four. I.

Adjacency in binary matroids

Abstract: We say that two elements e , f of a binary matroid M are ‘adjacent’ if there is no minor of M isomorphic to ℳ( K 4 ) which uses both e and f and in which they correspond to opposite edges. We give a good characterization of when two elements are adjacent. In particular, we show that if M is 4-connected, elements e , f are adjacent if and only if M is either graphic or cographic and the elements correspond to adjacent edges of the graph. We deduce a theorem about disjoint paths in graphs.

A non-analytic proof of the Newman:80Zna´m result for disjoint covering systems

Abstract: A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes.

Studies of random number generators for parallel processing

Abstract: If Monte Carlo calculations are to be performed in a parallel processing environment, a method of generating appropriate sequences of pseudorandom numbers for each process must be available. Frederickson et al. proposed an elegant algorithm based on the concept of pseudorandom or Lehmer trees: the sequence of numbers from a linear congruential generator is divided into disjoint subsequences by the members of an auxilary sequence. One subsequence can be assigned to each process. Extensive tests show the algorithm to suffer from correlations between the parallel subsequences: this is a result of the small number of bits which particpate in the auxiliary sequence and illustrates the well-known discovery of Marsaglia. Two alternative algorithms are proposed, both of which appear to be free of interprocess correlations. One relaxes the conditions on the Lehmer tree by using an arbitrary auxiliary multiplier: it is not known to what extent the subsequences are disjoint. The other partitions the main sequence into disjoint subsequences by sending one member to each process in turn, minimizing interprocess communication by defining new sequence generating parameters. 10 refs., 4 figs.

The spectra of topological Markov shifts

Abstract: For every Perron number λ we construct an infinite collection of topological Markov shifts with entropy log λ whose spectra are disjoint except for the necessary conjugates of λ. This is used to show that Marcus' theorem about every Markov shift of entropy log n factoring onto the full n-shift does not extend to certain entropy values.

The packing measure of planar Brownian motion

Abstract: In a recent paper [4] a new fractal measure φ-p(E) with respect to the monotone function φ(s) was defined,1and it was shown that φ1(s) = s2[log log 1/s]−1 is the right function for measuring Bd(t), the Brownian motion in ℝd, d ≥ 3, in the sense that there are finite positive constants cd such that $${{\varphi }_{1}}-p\ {{B}^{d}}(A)={{c}_{d}}\left| A \right|a.s.$$ (1) for every Borel A ⊂ ℝ+ = [0,∞), where |A| denotes the Lebesgue measure of A. This new measure φ-p(E), which we called P-packing measure, involves maximising the φ content of a packing by disjoint balls centered in E, with radii at most δ, and taking the limit as δ ↓ 0. In the present note we consider the problem of determining a function φ2 which gives the analogue of (1) for the critical case of planar Brownian motion B2(t), which we will henceforth denote by Z(t). In fact, we show that no such φ2 exists, and give a test which determines whether φ-p Z[0,1] is 0 or + ∞. This provides a complete solution to Problem 1 of [5].

Monogenic inverse semigroups

Abstract: We give a survey of some of the realisations that have been given of monogenic inverse semigroups and discuss their relation to one another. We then analyse the representations by bijections, combined under composition, of monogenic inverse semigroups, and classify these into isomorphism types. This provides a particularly easy way of classifying monogenic inverse semigroups into isomorphism types. Of interest is that we find two quite distinct representations by bijections of free monogenic inverse semigroups and show that all such representations must contain one of these two representations. We call a bijection of the form a i ↦ a i+1 , i = 1,2,…, r − 1, a finite link of length r , and one of the form a i ↦ a i+1 , i = 1,2…, a forward link . The inverse of a forward link we call a backward link . Two bijections u: A → B and r : C → D are said to be strongly disjoint if A ∩ C , A ∩ D , B ∩ C and B ∩ D are each empty. The two distinct representations of a free monogenic inverse semigroup, that we have just referred to, are first, such that its generator is the union of a counbtable set os finite links that are pairwise storongly disjoint part of any representation of a free monogenic inverse semigroup, the remaining part not affecting the isomorphism type. Each representation of a monogenic inverse semigroup that is not free contains a strongly disjoint part, determining it to within isomorphism, that is generated by either the strongly disjoint union of a finite link and a permutation or the strongly disjoint union of a finite and a forward link.

On deterministic multi pass analysis

Abstract: Chains (or cascade composition) of push-down transducers are introduced as a model of multi-pass compilers. We focus on deterministic chains, since nondeterministic transducer chains of length two define the recursively enumerable sets. Deterministic chains recognize in linear time a superset of context-free deterministic languages. This family is $\mathcal{CH}$ closed under Boolean operations, disjoint shuffle,and reverse deterministic pushdown translation, but not under homomorphism. Equivalent definitions of the family in terms of composition of syntax-directed translation schemes and control languages are considered. The family is a strict hierarchy ordered by the length of the chain. The complexity of $\mathcal{CH}$ is obviously linear, but not all linear-time parsable languages are in $\mathcal{CH}$. On the other hand it strictly includes the Boolean closure of deterministic languages. Finally $\mathcal{CH}$ is not comparable with another classical Boolean algebra of formal languages, namely real-ti...