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Showing papers on "Disjoint sets published in 1991"


Journal ArticleDOI
TL;DR: An attempt is made to provide a unifying theoretical framework for this growing body of algorithms that have been proposed to solve the set union problem and its variants.
Abstract: This paper surveys algorithmic techniques and data structures that have been proposed tosolve thesetunion problem and its variants, Thediscovery of these data structures required anew set ofalgorithmic tools that have proved useful in other areas. Special attention is devoted to recent extensions of the original set union problem, and an attempt is made to provide a unifying theoretical framework for this growing body of algorithms.

200 citations


Proceedings ArticleDOI
01 Aug 1991

156 citations


Journal ArticleDOI
TL;DR: The method is different from the procedure suggested in [8], but uses the same combinatorial structures, namelyLD(n), which allows the proof for Lu's proof to be substantially shortened.

128 citations


Journal ArticleDOI
TL;DR: This paper offers an axiomatic characterization of the probabilistic relation X is independent of Y (written (X, Y)) where X and Y are two disjoint sets of variables and a polynomial membership algorithm is developed to decide whether any given independence statement (x, Y) logically follows from a set E of such statements.
Abstract: This paper offers an axiomatic characterization of the probabilistic relation “X is independent of Y (written (X, Y))”, where X and Y are two disjoint sets of variables. Four axioms for (X, Y) are presented and shown to be complete. Based on these axioms, a polynomial membership algorithm is developed to decide whether any given independence statement (X, Y) logically follows from a set Σ of such statements, i.e., whether (X, Y) holds in every probability distribution that satisfies Σ. The complexity of the algorithm is O(|Σ| · k2 + |Σ| · n), where |Σ| is the number of given statements, n is the number of variables in Σ ∪ {(X, Y)}, and k is the number of variables in (X, Y).

114 citations


Journal ArticleDOI
TL;DR: A deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices with asymptotically optimal running time is given, which has numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.
Abstract: We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An ?-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at most?n hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices (which is asymptotically optimal). Forr≤n1??, where ?>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn)O(1)rd?1). In the plane we achieve a time boundO(nr) forr≤n1??, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆Ed and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ? R; R is a simplex}), In timeO(n(logn)O(1)rd?1+rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ?-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.

113 citations


Proceedings ArticleDOI
01 Sep 1991
TL;DR: It is claimed that novel inferences can be drawn from CBD literatures, and the possibility is worth investigating that the combined arguments and the inferences to which they lead might not be made explicit anywhere within the published record of science.
Abstract: Difficult and intriguing information retrieval (IR) problems derive from what I call complementary but disjoint (CBD) structures within the literature of science. Complementary refers to the relationship between two separate scientific arguments which, when combined, yield important inferences and insights not apparent in the separate arguments. Corresponding to the two arguments are two complementary literatures. Each literature (ideally) is the “complete” set of articles that contain the argument in question. Disjoint literatures have no articles in common, do not cite or mention each other, and are not co-cited. If two complementary literatures are also disjoint, the possibility is worth investigating that the combined arguments and the inferences to which they lead might not be made explicit anywhere within the published record of science. The ever-increasing fragmentation of science into mutually-isolated specialties probably assures a limitless supply and combinatorial growth of implicit connections, some of which may be unknown solutions to important problems. These solutions are worth seeking. I have previously analyzed and reported three examples of CBD literature pairs, each of which led to a novel and plausible medical hypothesis that appeared to merit testing [1-5]. The universe of published medical literature, not a selected subset, was the target of a process that depended not only on extensive use of online information services but also on reading the text of many hundreds of medical articles. My claim that novel inferences can bc drawn from CBD literatures requires clarification. CBD literatures of course can never attain their ideal state of complclencss as specified in the definition, but they are presumed to approach as near to completion as attainable with a diligent search of the literature, On that basis, one can smkc a

106 citations


Book ChapterDOI
17 Jun 1991
TL;DR: It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine how many edges G has, and determineWhether G has a feedback vertex set of size ≤ k.
Abstract: It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size ≤ k. Also, every class \(\mathcal{G}\), that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of K3, has an \(\mathcal{O}\)(n) membership test algorithm.

103 citations


Proceedings ArticleDOI
02 Dec 1991
TL;DR: In this paper, an algorithm for computing disjoint paths in star graphs for two well known paradigms is presented. But the problem of finding a limited number of such paths in the star network is not addressed.
Abstract: Star networks have been recently proposed as an attractive choice for interconnection networks. They have sublogarithmic node degree and diameter and, like hypercubes, have a highly recursive structure. Several researchers have endeavored to prove that star networks are as versatile as hypercubes. The paper is an effort in the same direction. It presents optimal algorithms for computing disjoint paths in star graphs for two well known paradigms. It also studies the problem of disjoint connecting paths and presents an efficient algorithm for finding a limited number of such paths in the star network. >

101 citations


Journal ArticleDOI
TL;DR: It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system.
Abstract: In this paper structured systems are considered and the generic rank of the transfer matrix of such systems is introduced. It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system. This maximum number of disjoint paths can be calculated using techniques from combinatorics. As an application a structural version of the well-known almost disturbance decoupling problem is proposed.

85 citations


Journal ArticleDOI
TL;DR: It is shown that the abstract Voronoi diagram of n sites in the plane can be constructed in timeO(n logn) by a randomized algorithm based on Clarkson and Shor's randomized incremental construction technique.
Abstract: We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].

74 citations


Journal ArticleDOI
TL;DR: A weighting of set partitions which is analogous to the major index for permutations, and the corresponding weight generating function yields the q-Stirling numbers of the second kind of Carlitz and Gould.

01 Jan 1991
TL;DR: It is shown that for each fixed k there exists a linear-time algorithm for the problem of finding pairwise vertex-disjoint trees T1, Tp in G such that Ti covers Xi.
Abstract: We show that for each fixed k there exists a linear-time algorithm for the problem: given: an undirected plane graph G = (V,E) and subsets X1, . . . , Xp of V with |X1 ∪ · · · ∪ Xp| ≤ k; find: pairwise vertex-disjoint trees T1, . . . , Tp in G such that Ti covers Xi (i = 1, . . . , p).

Proceedings ArticleDOI
07 Apr 1991
TL;DR: An efficient SDP technique is utilized to run input files of paths/cuts preprocessed using (1) cardinality, (2) lexicographic, and (3) Hamming distance ordering methods and their combinations.
Abstract: Experimental results are presented showing the number of disjoint products and computer time involved in generating sum of disjoint product (SDP) terms. To help obtain the results, the authors have considered 19 benchmark networks containing paths (cuts) varying from 4 (4) to 780 (7376). Several SDP techniques are reviewed and are generalized into three propositions to find their inherent merits and demerits. An efficient SDP technique is, then, utilized to run input files of paths/cuts preprocessed using (1) cardinality, (2) lexicographic, and (3) Hamming distance ordering methods and their combinations. The experimental evaluation has been performed on an FPS 500 system. Results are analyzed, and it is shown that the preprocessing based on cardinality or its combinations with (2) and/or (3) performs better. >

Journal ArticleDOI
TL;DR: A graph-theoretic algorithm is proposed for decomposition of a linear system of equations into subsystems having a prescribed size of mutual interactions, suitable for conditioning large systems and achieving fast convergence rates in block-iterative computations via parallel multiprocessor schemes.
Abstract: A graph-theoretic algorithm is proposed for decomposition of a linear system of equations into subsystems having a prescribed size of mutual interactions The algorithm generates a whole range of nested decompositions with an apparent trade-off between levels of coupling and sizes of the subsystems Both disjoint and overlapping subsystems are considered Having a linear time complexity for a selected strength of coupling, the algorithm is suitable for conditioning large systems and achieving fast convergence rates in block-iterative computations via parallel multiprocessor schemes

Journal ArticleDOI
TL;DR: It is shown that the combined matching problem is in general undecidable but that it becomes decidable if all theories are regular and an efficient combination algorithm is developed.

Journal ArticleDOI
TL;DR: It is shown that an n -star can be decomposed into (n −2)! vertex disjoint cycles of lenght ( n −1) n . These cycles may be used in designing parallel algorithms on an interconnection network based on the star topology.

Journal ArticleDOI
TL;DR: Good characterizations in Edmonds' sense are given for the following problems for tournaments and a proof is given by C. Thomassen that problem (1) is NP-complete for digraphs in general.

Journal ArticleDOI
TL;DR: This work develops an efficient optimal covering algorithm for cacti and generalizes the results of Boesch, Gimpel, McHugh and Pinter, Wolfstahl where optimal covering algorithms for trees and graphs where no two cycles share a vertex were presented.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the classical non-dimensionalisation of many simple combusting systems is sufficiently nonlinear to conceal some of the differences in behaviour shown when different bifurcation parameters are varied quasi-statically.

Proceedings ArticleDOI
30 Jun 1991
TL;DR: It is proved that, for any class K chosen from a certain set of complexity classes, it holds that (1) if there is a sparse polynomial-time bounded-truth-table-hard set for K, then K=P, and (2) ifThere are a sparse strongly nondeterministic bounded- Truth- Table-Hard set forK,Then K contained in NP intersection co-NP.
Abstract: The authors study one-word-decreasing self-reducible sets, which are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query to a word lexicographically smaller than the input. It is first shown that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact it is proved that, for any class K chosen from a certain set of complexity classes, it holds that (1) if there is a sparse polynomial-time bounded-truth-table-hard set for K, then K=P, and (2) if there is a sparse strongly nondeterministic bounded-truth-table-hard set for K, then K contained in NP intersection co-NP. The main result also shows that the same facts hold for the class PSPACE. >

Journal ArticleDOI
TL;DR: In this paper, it was shown that every real function defined on a Baire space has a dense set of points of continuity, and that almost resolvable spaces are defined as the union of a first category set and a closed resolver.
Abstract: A space is said to be resolvable if it has two disjoint dense subsets. It is shown thatX is a Baire space with no resolvable open subsets iff every real function defined onX has a dense set of points of continuity. Thus almost resolvable spaces, as defined by Bolstein, are shown to be characterized as the union of a first category set and a closed resolvable set.

Journal ArticleDOI
TL;DR: In this article, the authors give a characterization of Possonian domains for which every bounded harmonic function is the harmonic extension of some function in L.............. ∞ of harmonic measure, and deduce several properties of such domains, including some results of Mountford and Port.
Abstract: We give a characterization of Possonian domains inR n , i.e., those domains for which every bounded harmonic function is the harmonic extension of some function inL ∞ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.

Journal ArticleDOI
TL;DR: In this article, a simple algorithm for introducing a set of interior points permits a simple partition of the volume mesh into two disjoint sets of tetrahedra such that one set determines the interior of the solid while the second determines the exterior.
Abstract: The reconstruction of a solid surface from a series of cross-sections and the generation of a volume mesh that conforms with a prescribed surface are issues that arise in numerous applications. We describe an approach to both these problems that is based on the generation of a tetrahedral mesh which automatically captures a triangulation of the surface points. A simple algorithm for introducing a set of interior points permits a simple partition of the volume mesh into two disjoint sets of tetrahedra such that one set determines the interior of the solid while the second determines the exterior.

Proceedings ArticleDOI
30 Apr 1991
TL;DR: This paper presents an analysis that leads to the complete characterization of the node disjoint (parallel) paths in a star graph and shows that by exploiting the cycle structure of permutations one can decentralize a routing algorithm for transfer of packets between a pair of nodes in these graphs.
Abstract: This paper presents an analysis that leads to the complete characterization of the node disjoint (parallel) paths in a star graph. It is shown that by exploiting the cycle structure of permutations (same as the conjugacy class) one can decentralize a routing algorithm for transfer of packets between a pair of nodes in these graphs. >


Journal ArticleDOI
TL;DR: In this paper, it was shown that if X and Y are compacta such that dim( X × Y ) m and 2 dim X + dim Y ⩽ 2 m - 2, then any two mappings X → R m and Y → Rm admit arbitrarily close approximations with disjoint images.

Book ChapterDOI
07 Aug 1991
TL;DR: In this article, the problem of partitioning the iterations of nested loops and data arrays accessed by the loops is addressed, and necessary and sufficient conditions for communication-free hyperplane partitioning are provided.
Abstract: This paper addresses the problem of partitioning the iterations of nested loops, and data arrays accessed by the loops. Hyperplane partitions of disjoint subsets of data arrays and loop iterations that result in the elimination of communication are sought. A characterization of necessary and sufficient conditions for communicationfree hyperplane partitioning is provided.

Journal ArticleDOI
TL;DR: In this article, the following theorem is proved: if X is a one-dimensional nonseparating plane continuum such that for any e > 0, X has two disjoint e-dense subcontinua, then X is either indecomposable or the union of two indecompositionable continua.

Journal ArticleDOI
TL;DR: A new algorithm is shown that generates the representation of completely or incompletely specified boolean functions in the form of arrays of disjoint ON- and DC- cubes that minimizes the number of usages of cube calculus operations.
Abstract: A new algorithm is shown that generates the representation of completely or incompletely specified boolean functions in the form of arrays of disjoint ON- and DC- ( if any) cubes. A peculiar feature of the algorithm, which speeds up its execution, is the fact that in comparison to known algorithms it minimizes the number of usages of cube calculus operations.

Proceedings ArticleDOI
03 Jan 1991
TL;DR: An 0(cZn3) steps algorithm A is described, and it is proved that it succeeds almost surely to find hidden Hamiltonian cycles in a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges.
Abstract: Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “high” the cycle. Is it possible to unravel the structures, that is, to efficiently find a Himiltonian cycle in G? We describe an O(n3 log n)-step algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots” of G by a collection of disjoint paths. (This is the hard part to analyze). Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems. © 1994 John Wiley & Sons, Inc.