Showing papers on "Disjoint sets published in 1992"
TL;DR: A single linear programming formulation is proposed which generates a plane that of minimizes an average sum of misclassified points belonging to two disjoint points sets in n-dimensional real space, without the imposition of extraneous normalization constraints that inevitably fail to handle certain cases.
Abstract: A single linear programming formulation is proposed which generates a plane that of minimizes an average sum of misclassified points belonging to two disjoint points sets in n-dimensional real space. When the convex hulls of the two sets are also disjoint, the plane completely separates the two sets. When the convex hulls intersect, our linear program, unlike all previously proposed linear programs, is guaranteed to generate some error-minimizing plane, without the imposition of extraneous normalization constraints that inevitably fail to handle certain cases. The effectiveness of the proposed linear program has been demonstrated by successfully testing it on a number of databases. In addition, it has been used in conjunction with the multisurface method of piecewise-linear separation to train a feed-forward neural network with a single hidden layer.
771 citations
TL;DR: The complexity of these problems when G is restricted to be a partial k -tree is discussed, and a polynomial time algorithm is given for the n disjoint connecting paths problem restricted topartial k -trees (with n part of input).
187 citations
TL;DR: This estimate provides a missing link of the proof that the number hd(n) of halving hyperplanes for a set of n points in Rd satisfies the inequality hd (n) ⩽ O(nd−e) for some e > 0.
180 citations
TL;DR: In this article, the notion of clump in a matroid sum is introduced, and efficient algorithms for clumps are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.
Abstract: This paper presents improved algorithms for matroid-partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.
157 citations
TL;DR: A combinatorial bijection is given between pairs of permutations in S n the product of which is a given n -cycle and two-coloured plane edge-rooted trees on n edges, when the numbers of cycles in the disjoint cycle representations of the permutations sum to n + 1.
Abstract: A combinatorial bijection is given between pairs of permutations in S n the product of which is a given n -cycle and two-coloured plane edge-rooted trees on n edges, when the numbers of cycles in the disjoint cycle representations of the permutations sum to n + 1 Thus the corresponding connection coefficient for the symmetric group is determined by enumerating these trees with respect to appropriate characteristics This is extended to the case of m -tuples of permutations in S n the product of which is a given n -cycle, in which the combinatorial objects replacing trees are cacti of m -gons
134 citations
15 Jun 1992
TL;DR: This paper describes a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions—a slight generalization of unificationblems with constants—is decidable for these theories.
Abstract: Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions—a slight generalization of unification problems with constants—is decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol.
106 citations
TL;DR: The limitation of the conventional signal-to-noise ratio as a performance measure in matched-filter-based optical pattern recognition for input-scene noise that is disjoint (or nonoverlapping) with the target is investigated.
Abstract: The limitation of the conventional signal-to-noise ratio as a performance measure in matched-filter-based optical pattern recognition for input-scene noise that is disjoint (or nonoverlapping) with the target is investigated.
88 citations
TL;DR: All four versions of the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized are shown to be strongly NP-complete even for k = 2.
Abstract: Consider a network G = (V,E) with distinguished vertices s and t, and with k different costs on every edge. We consider the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized, where the j th edge-cost is associated with the j th path. The problem has several variants: The paths may be vertex-disjoint or arcdisjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP-complete even for k = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.
87 citations
TL;DR: This is an approximate version of the well-known conjecture of Erdős, Faber, and Lovasz stating that the chromatic index of a nearly-disjoint hypergraph on n vertices is at most n + o ( n ).
85 citations
TL;DR: The new algorithm is empirically compared with a best possible algorithm, with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen, and requires only a few percent more function evaluations than the best possible one.
Abstract: We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P′: find a globallye-optimal value off and a corresponding point; Problem Q″: find a set of disjoint subintervals of [a, b] containing only points with a globallye-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P′. In phase I, this algorithm obtains rapidly a solution which is often globallye-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves thee-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globallye-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For smalle, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q″. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.
79 citations
TL;DR: The algorithm and its implementation provide the fastest and most comprehensive program (having many options) known to the authors for the calculation of the Rademacher-Walsh transform.
Abstract: A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from nondisjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. The programs for both algorithms use advantages of C language to speed up the execution. The comparison of different versions of the algorithm has been carried out. The algorithm and its implementation provide the fastest and most comprehensive program (having many options) known to the authors for the calculation of the Rademacher-Walsh transform. It successfully overcomes all drawbacks in the calculation of the transform from the design automation system based on spectral method-the SPECSYS system from Drexel University, which uses fast Walsh transform. >
TL;DR: Three efficient algorithms, one that constructs an optimal set and two that construct approximations, are presented, one of which not only constructs a larger set of short disjoint paths than an iterated version of the standard Dijkstra algorithm, but also offers a major reduction in computation time for large networks.
Abstract: The motives for seeking a set of short disjoint paths in a communication network are explained. A sequentially constructed optimal set is defined. Three efficient algorithms, one that constructs an optimal set and two that construct approximations, are presented. One of the latter algorithms not only constructs a larger set of short disjoint paths than an iterated version of the standard Dijkstra algorithm, but also offers a major reduction in computation time for large networks. >
TL;DR: In this article, the authors consider the following global optimization problems for a univariate Lipschitz function defined on an interval: Problem P: find a globally optimal value off and a corresponding point; Problem Q: localize all globally optimal points.
Abstract: We consider the following global optimization problems for a univariate Lipschitz functionf defined on an interval [a, b]: Problem P: find a globally optimal value off and a corresponding point; Problem Pź: find a globallyź-optimal value off and a corresponding point; Problem Q: localize all globally optimal points; Problem Qź: find a set of disjoint subintervals of small length whose union contains all globally optimal points; Problem Qź: find a set of disjoint subintervals containing only points with a globallyź-optimal value and whose union contains all globally optimal points.
We present necessary conditions onf for finite convergence in Problem P and Problem Q, recall the concepts necessary for a worst-case and an empirical study of algorithms (i.e., those ofpassive and ofbest possible algorithms), summarize and discuss algorithms of Evtushenko, Piyavskii-Shubert, Timonov, Schoen, Galperin, Shen and Zhu, presenting them in a simplified and uniform way, in a high-level computer language. We address in particular the problems of using an approximation for the Lipschitz constant, reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globallyź-optimal value. New algorithms for Problems Pź and Qź and an extensive computational comparison of algorithms are presented in a companion paper.
02 Sep 1992
TL;DR: It is proved that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high which shows that interaction of systems in such disj joints may be very subtle, and provides the basis for two derived modularity results.
Abstract: Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union \(\mathcal{R}_1 \oplus \mathcal{R}_2\) of two (finite) terminating term rewriting systems \(\mathcal{R}_1 ,\mathcal{R}_2\) is non-terminating, then one of the systems, say \(\mathcal{R}_1\), enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. \(\mathcal{R}_1\)⊕G({x,y}) → x,G({x, y}) → y is non-terminating, and the other system \(\mathcal{R}_2\) is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient syntactical criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high which shows that interaction of systems in such disjoint unions may be very subtle. Finally, extensions and generalizations of our main results in various directions are discussed and sketched.
TL;DR: It is shown that if the number of vectors in each of the sets is constrained, then a weaker conclusion holds, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls.
Abstract: LetA1,?,An be distinctk-dimensional vectors. We consider the problem of partitioning these vectors intom sets so as to maximize an objective which is a quasi-convex function of the sum of vectors in each set. We show that there exists an optimal partition whose sets have (pairwise) disjoint conic hulls. We also show that if the number of vectors in each of the sets is constrained, then a weaker conclusion holds, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls. The results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope.
Journal Article•
TL;DR: On the basis of the class of linear frames, it is demonstrated that the expressive power of the language is considerably stronger than that of classical modal logic.
Abstract: We enrich propositional modal logic with operators ◇ (n ∈ N) which are interpreted on Kripke structures as “there are more than n accessible worlds for which ...”, thus obtaining a basic graded modal logic GrK. We show how some familiar concepts (such as subframes, p-morphisms, disjoint unions and filtrations) and techniques from modal model theory can be used to obtain results about expressiveness (like graded modal equivalence, correspondence and definability) for this language. On the basis of the class of linear frames we demonstrate that the expressive power of the language is considerably stronger than that of classical modal logic. We give a class of formulas for which a first-order equivalent can be systematically obtained, but also show that the set of formulas for which such an equivalence exists is in some sense a proper subset of the set of so called Sahlqvist formulas, a syntactically defined set of modal formulas for which a corresponding formula is guaranteed to exist. Finally we show how, combining the technique of ‘filtration’ with a notion of ‘copying worlds’ — in view of the “more than n” interpretation, one cannot simply collapse worlds — for some graded modal logics (GrK, GrT, . . . ), the finite model property (and also decidability) is obtained.
TL;DR: This paper presents an approximation algorithm with worst-case ratio 1.324 for the optimization version of this problem and shows this problem to be NP-complete.
TL;DR: The following asymptotic result is proved: for every fixed graphH withh vertices, any graphG withn vertices and with minimum degree contains (1−o(1) n/h vertex disjoint copies ofH.
Abstract: The following asymptotic result is proved. For every fixed graphH withh vertices, any graphG withn vertices and with minimum degree $$d \geqslant \frac{{\chi (H) - 1}}{{\chi (H)}}n$$ contains (1?o(1))n/h vertex disjoint copies ofH.
TL;DR: The decompose the image plane into disjoint sets, restrict the domain of definition of the functionals to these sets, and use the hypotheses to deform and to move the boundaries of the sets within theimage plane.
Abstract: A well-known method for the reconstruction of motion fields from noisy image data is to determine flow fields by the minimization of a quadratic functional. The first approach of this class has been proposed by Horn and Schunck (1981). A drawback of such approaches is that an explicit representation of the discontinuities of the motion field is lacking and that, in general, the resulting flow fields approximate the motion fields only badly at the corresponding locations in the image plane. In this article, we discuss the possibility to improve the results by hypothesizing the qualitative structure of the motion field in terms of certain parameters. We decompose the image plane into disjoint sets, restrict the domain of definition of the functionals to these sets, and use the hypotheses to deform and to move the boundaries of the sets within the image plane. We discuss the range of applicability of this new technique and illustrate the algorithm by numerical examples. This article is a revised and extended version of Schnorr (1990).
TL;DR: This paper considers the problem of listing all linear extensions of a partial order so that successive extensions differ by the transposition of a single pair of elements.
TL;DR: An algorithm is provided that given n rectangles builds a linear sized Binary Space Partition with no empty regions by means of at most 4n-1 cuts, running in O(n log n) worst-case space and time.
TL;DR: In this article, it was shown that the hemimaximal sets form an orbit by extending Soare's proof for maximal sets and detail the properties of splittings of maximal sets which led to this result.
TL;DR: It is proved that there is an oracle A such that every two disjoint sets in NP A are P-separable, and Σ 2 P = U DTIME(2 P )| p is a polynomial.
01 Jul 1992
TL;DR: It is proved sufficient conditions for the existence of edge-disjoint paths connecting any set of q ≤ n/(logn)κ disjoint pairs of vertices on any n vertex bounded degree expander.
Abstract: Given an expander graph G = (V,E) and a set of q disjoint pairs of vertices in V , we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of q paths so found is edge-disjoint. (For general graphs the related decision problem is NPcomplete.) We prove sufficient conditions for the existence of edge-disjoint paths connecting any set of q ≤ n/(logn)κ disjoint pairs of vertices on any n vertex bounded degree expander, where κ depends only on the expansion properties of the input graph, and not on n. Furthermore, we present a randomized o(n3) time algorithm, and a random NC algorithm for constructing these paths. (Previous existence proofs and construction algorithms allowed only up to nǫ pairs, for some ǫ ≪ 1/3, and strong expanders [19].) In passing, we develop an algorithm for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.
TL;DR: In this article, it was shown that the spectrum of a periodic Schrodinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property.
Abstract: It was observed in [Su5] that the spectrum of a periodic Schrodinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.
TL;DR: The enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane is complete, and a family of d+1 translates in ℝd admitting (d+1)!/2 geometric permutation is constructed.
Abstract: We construct a family ofn disjoint convex set in ?d having (n/(d?1))d?1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ?d, and construct a family ofd+1 translates in ?d admitting (d+1)!/2 geometric permutations.
TL;DR: With this construction and the relationship between DDS and DTS, many new upper bounds for DTS and some better orthogonal codes are obtained.
Abstract: Disjoint difference sets (DDS), difference triangle sets (DTS), and related codes are discussed and a recursive construction for DDS is given With this construction and the relationship between DDS and DTS, many new upper bounds for DTS and some better orthogonal codes are obtained >
15 Jul 1992
TL;DR: A unification algorithm provides a symbolic constraint solver in the combination of algebraic structures whose finite domains of values are non disjoint and correspond to constants.
Abstract: We extend the results on combination of disjoint equational theories to combination of equational theories where the only function symbols shared are constants. This is possible because there exist finitely many proper shared terms (the constants) which can be assumed irreducible in any equational proof of the combined theory. We establish a connection between the equational combination framework and a more algebraic one. A unification algorithm provides a symbolic constraint solver in the combination of algebraic structures whose finite domains of values are non disjoint and correspond to constants. Primal algebras are particular finite algebras of practical relevance for manipulating hardware descriptions.
01 Nov 1992
TL;DR: An approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language which uses consistency algorithms which can actively process a wider variety of numeric constraints than most other CLP systems, including constraints containing some common nonlinear functions.
Abstract: There have been many proposals for adding sound implementations of numeric processing to Prolog. This paper describes an approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language. Echidna uses consistency algorithms which can actively process a wider variety of numeric constraints than most other CLP systems, including constraints containing some common nonlinear functions. A unique feature of Echidna is that it implements domains for real-valued variables with hierarchical data structures and exploits this structure using a hierarchical arc consistency algorithm specialized for numeric constraints. This gives Echidna two advantages over other systems. First, the union of disjoint intervals can be represented directly. Other approaches require trying each disjoint interval in turn during backtrack search. Second, the hierarchical structure facilitates varying the precision of constraint processing. Consequently, it is possible to implement more effective constraint processing control algorithms which avoid unnecessary detailed domain analysis. These advantages distinguish Echidna from other CLP systems for numeric constraint processing.