Showing papers on "Disjoint sets published in 1993"
TL;DR: A deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd−1) time is presented, based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes.
Abstract: Givenn hyperplanes inEd, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverEd and such that the interior of each simplex intersects at mostn/r hyperplanes. We present a deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd?1) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.
314 citations
01 Jan 1993
TL;DR: In this paper, the authors discuss applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory.
Abstract: This chapter discusses applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory. These theorems stand at the origin of what is known today as the combinatorial geometry of convex sets. Helly's theorem states that: letting K be a family of convex sets in ℝ d , and supposing K is finite or each member of K is compact; if every d + 1 or fewer members of K have a common point, then there is a point common to all members of K . Radon's theorem states that: letting X be a set of d + 2 or more points in ℝ d ; then X contains two disjoint subsets whose convex hulls have a common point. Caratheodory's theorem states that: letting X be a set in ℝ d and p a point in the convex hull of X . Then there is a subset Y of X consisting of d + 1 or fewer points such that p lies in the convex hull of Y .
298 citations
TL;DR: An eigenvalue upper boundϕ(G) on the maximum cut mc (G) of a weighted graph has several interesting properties that resemble the behaviour ofmc (G), and ϕ is subadditive with respect to amalgam, and additive withrespect to disjoint sum and 1-sum.
Abstract: We introduce and study an eigenvalue upper boundź(G) on the maximum cut mc (G) of a weighted graph. The functionź(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented.
We show thatź is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove thatź(G) is never worse that 1.131 mc(G) for a planar, or more generally, a weakly bipartite graph with nonnegative edge weights. We give a dual characterization ofź(G), and show thatź(G) is computable in polynomial time with an arbitrary precision.
204 citations
01 Jul 1993
TL;DR: The visibility complex of a collection of pairwisedisjoint convex objects in the plane is introduced and can be used to compute the view from a point or a convex object with respect toinline-equation.
Abstract: We introduce the visibility complex of a collection O of n pairwise disjoint convex objects in the plane. This 2 dimensional cell complex may be considered as a generalization of the tangent visibility graph of 0. Its space complexity k is proportional to the size of the tangent visibility graph. We give an O(nlog n+k) algorithm for its construction. Furthermore we show how the visibility complex can be used to compute the view from a point or a convex object with respect to O in O(m log n) time, where m is the size of the view. The view from a point is a generalization of the visibility polygon of that point with respect to O.
191 citations
TL;DR: It is shown that sets that are complete for any level of the polynomial hierarchy are not nonadaptively random-self-reducible, unless the hierarchy collapses, and it is show that sets complete for the classes PP andMOD are random- Self-Reducible.
Abstract: This paper generalizes the previous formal definitions of random-self-reducibility. It is shown that, even under a very general definition, sets that are complete for any level of the polynomial hierarchy are not nonadaptively random-self-reducible, unless the hierarchy collapses. In particular, NP-complete sets are not nonadaptively random-self-reducible, unless the hierarchy collapses at the third level. By contrast, we show that sets complete for the classes PP and ${\text{MOD}}_m {\text{P}}$ are random-self-reducible.
187 citations
TL;DR: It is shown that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability, which implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerians graphs.
Abstract: In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family ℱ of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingP≠NP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.
142 citations
TL;DR: Techniques for implicitly building and searching convolutions are developed and applied to derive efficient algorithms for fundamental problems that arise in robotics and computer vision.
Abstract: Given two intersecting polyhedraP, Q and a directiond, find the smallest translation ofQ alongd that renders the interiors ofP andQ disjoint. The same problem can also be posed without specifying the direction, in which case the minimum translation over all directions is sought. These are fundamental problems that arise in robotics and computer vision. We develop techniques for implicitly building and searching convolutions and apply them to derive efficient algorithms for these problems.
141 citations
01 Jan 1993
TL;DR: Theorem 1.1 (Helly's Theorem) of transversal theory has its origins in Helly's theorem as mentioned in this paper, which states that if every d + 1 members of a convex set have a common point, then there is a point common to all the members of the set.
Abstract: Geometric transversal theory has its origins in Helly’s theorem: Theorem 1.1 (Helly’s Theorem) [49]. Suppose A is a family of at least d + 1 convex sets in IR d , and A is finite or each member of A is compact. Then if every d + 1 members of A have a common point, there is a point common to all the members of A.
138 citations
01 Jan 1993
TL;DR: The notions of packing and covering are well defined in hyperbolic geometry as mentioned in this paper, and packing in or covering of the whole space is called packing or a covering, respectively, and an arrangement that is a packing and a covering at the same time is called a tiling.
Abstract: Publisher Summary This chapter describes packing and covering with convex sets and discusses arrangements of sets in a space E, which should have a structure admitting the notions of congruence, measure, and convexity. Given a domain in E, a packing in the domain is an arrangement the members of which are all contained in the domain and have mutually disjoint interiors, and a covering of the domain is an arrangement whose union contains the domain. A packing in or a covering of the whole space E is called a packing or a covering, respectively. An arrangement that is a packing and a covering at the same time is called a tiling. All the known proofs of the theorem of Minkowski–Hlawka and its refinements are nonconstructive. The concepts of packing and covering are well defined in hyperbolic geometry. While high-density packings and low-density coverings can be considered efficient, the definition of density allows some undesired local deviations to occur that go contrary to the intuitive concept of efficiency.
130 citations
TL;DR: In this article, a countable regular crowded space no point of which is a limit point of two disjoint sets is constructed, and for each k ϵ [2, ω], there is a k-irresolvable space.
123 citations
TL;DR: The NP-complete problem of determining whether two disjoint point sets in then-dimensional real spaceRn can be separated by two planes is cast as a bilinear program, that is minimizing the scalar product of two linear functions on a polyhedral set.
Abstract: The NP-complete problem of determining whether two disjoint point sets in then-dimensional real spaceRn can be separated by two planes is cast as a bilinear program, that is minimizing the scalar product of two linear functions on a polyhedral set. The bilinear program, which has a vertex solution, is processed by an iterative linear programming algorithm that terminates in a finite number of steps a point satisfying a necessary optimality condition or at a global minimum. Encouraging computational experience on a number of test problems is reported.
27 May 1993
TL;DR: The notion of interval sets is introduced as a new kind of sets represented by a pair of sets, namely, the lower and upper bounds, and operations on interval sets are defined, based on the corresponding set-theoretic operations on their members.
Abstract: The notion of interval sets is introduced as a new kind of sets, represented by a pair of sets, namely, the lower and upper bounds. The interval-set algebra may be regarded as a counterpart of the interval-number algebra. It provides a useful tool to represent qualitative information. Operations on interval sets are also defined, based on the corresponding set-theoretic operations on their members. In addition, basic properties of interval-set algebra are examined, and the relationships between interval sets, rough sets and fuzzy sets are analyzed. >
TL;DR: It is proved that for circular arc graphs the size of a maximum oo-irredundant set equals 2β ∗ or 2β∗ +1 (depending on parity) where β ∗ is the strong matching number.
TL;DR: It is shown that this problem can be reduced to the problem of finding a minimal shortest path from each node to the destination in a modified network, and a distributed algorithm on the original network that simulates a shortest-paths algorithm running on the modified network is presented.
Abstract: Distributed algorithms for finding two disjoint paths of minimum total length from each node to a destination are presented. The algorithms have both node-disjoint and link-disjoint versions and provide each node with information sufficient to forward data on the disjoint paths. It is shown that this problem can be reduced to the problem of finding a minimal shortest path from each node to the destination in a modified network, and a distributed algorithm on the original network that simulates a shortest-paths algorithm running on the modified network is presented. The algorithm has a smaller space complexity than any previous distributed algorithm for the same problem, and a method for forwarding packets is presented that does not require any additional space complexity. A synchronous implementation of the algorithm is also presented and studied. >
TL;DR: Algorithms with efficient, output-size-sensitive query times are presented for the generalized versions of a number of intersection searching problems, including: interval intersection searching, orthogonal segment intersection searched, Orthogonal range searching, point enclosure searching, rectangle intersection searching and segment intersection searching.
Abstract: A new class of geometric intersection searching problems is introduced, which generalizes previously-considered intersection searching problems and is rich in applications. In a standard intersection searching problem, a set S of n geometric objects is to be preprocessed so that the objects that are intersected by a query object q can be reported efficiently. In a generalized problem, the objects in S come aggregated in disjoint groups and what is of interest are the groups, not the objects, that are intersected by q. Although this problem can be solved easily by using an algorithm for the standard problem, the query time can be Ω(n) even though the output size is just O(1). In this paper, algorithms with efficient, output-size-sensitive query times are presented for the generalized versions of a number of intersection searching problems, including: interval intersection searching, orthogonal segment intersection searching, orthogonal range searching, point enclosure searching, rectangle intersection searching, and segment intersection searching. In addition, the algorithms are also space-efficient.
TL;DR: Researchers have proposed cardinality-, lexicographic-, and Hamming-distance-order methods to preprocess the path terms in sum of disjoint products (SDP) techniques for network reliability analysis, showing that preprocessing based on cardinality or its combinations with lexicography- and/or Hamming distance-ordering performs better.
Abstract: Researchers have proposed cardinality-, lexicographic-, and Hamming-distance-order methods to preprocess the path terms in sum of disjoint products (SDP) techniques for network reliability analysis. For cutsets, an ordering based on the node partition associated with each cut is suggested. Experimental results showing the number of disjoint products and computer time involved in generating SDP terms are presented. Nineteen benchmark networks containing paths varying from 4 to 780, and cuts from 4 to 7376, are considered. Several SDP techniques are generalized into three propositions to find their inherent merits and drawbacks. An efficient SDP technique is then used to run input files of paths/cuts preprocesses using cardinality-, lexicographic-, and Hamming-distance-ordering, and their combinations. The results are analyzed, showing that preprocessing based on cardinality or its combinations with lexicographic-, and/or Hamming-distance-ordering performs better. >
TL;DR: A characterization of necessary and sufficient conditions for communication-free hyperplane partitioning of disjoint subsets of data arrays and loop iterations that result in the elimination of communication is provided.
01 Mar 1993
TL;DR: In this article, the maximal zero entropy factor of a topological flow is defined using entropy pairs and explicitly given for some simple Cartesian products, and it is proved that only the trivial flow is disjoint from all flows whose maximal zero-entropy factor is trivial.
Abstract: The maximal zero entropy factor of a topological flow is defined using entropy pairs and explicitly given for some simple cartesian products. As a consequence, it is proved that only the trivial flow is disjoint from all flows whose maximal zero entropy factor is trivial
TL;DR: A uniform approach to problems involving lines in 3-space based on mapping lines inR3 into points and hyperplanes in five-dimensional projective space (Plücker space) is presented.
Abstract: We present a uniform approach to problems involving lines in 3-space. This approach is based on mapping lines inR3 into points and hyperplanes in five-dimensional projective space (Plucker space). We obtain new results on the following problems:
1.
Preprocessn triangles so as to answer efficiently the query: “Given a ray, which is the first triangle hit?” (Ray- shooting problem). We discuss the ray-shooting problem for both disjoint and nondisjoint triangles.
2.
Construct the intersection of two nonconvex polyhedra in an output sensitive way with asubquadratic overhead term.
3.
Construct the arrangement ofn intersecting triangles in 3-space in an output-sensitive way, with asubquadratic overhead term.
4.
Efficiently detect the first face hit by any ray in a set of axis-oriented polyhedra.
5.
Preprocessn lines (segments) so as to answer efficiently the query “Given two lines, is it possible to move one into the other without crossing any of the initial lines (segments)?” (Isotopy problem). If the movement is possible produce an explicit representation of it.
TL;DR: An algorithm is presented for solving equations in a combination of arbitrary theories over disjoint sets of function symbols that consists of a set of transformation rules that simplify a unification problem until a solved form is obtained.
TL;DR: This article surveys most of the results and techniques associated with properties of r.e. sets in ordinary recursion theory.
TL;DR: This article provides a much shorter proof of a common generalization of the edge- and node-connectivity versions showing that the subgraph Gk has a certain mixed connectivity property.
Abstract: From a theorem of W. Mader [“Uber minimal n-fach zusammenhangende unendliche Graphen und ein Extremal problem,” Arch. Mat., Vol. 23 (1972), pp. 553–560] it follows that a k-connected (k-edge-connected) graph G = (V,E) always contains a k-connected (k-edge-connected) subgraph G′ = (V,E′) with O(k|V|) edges. T. Nishizeki and S. Poljak “K-Connectivity and Decomposition of Graphs into Forests,” Discrete Applied Mathematics, submitted) showed how G′ can be constructed as the union of k forests. H. Nagamochi and T. Ibaraki [A Linear Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph, Algorithmica, Vol. 7 (1992), pp. 583–596] constructed such a subgraph Gk in linear time and showed for any pair x,y of nodes that Gk contains k openly disjoint (edge-disjoint) paths connecting x and y if G contains k openly disjoint (edge-disjoint) paths connecting x and y (even if G is not k-connected (k-edge-connected)). In this article we provide a much shorter proof of a common generalization of the edge- and node-connectivity versions showing that the subgraph Gk has a certain mixed connectivity property. © 1993 John Wiley & Sons, Inc.
TL;DR: In this paper, the authors proved that a Hanani triple system exists for all v≡1 (mod 6) elements except for v ∈ {7, 13].
Abstract: Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.
TL;DR: Support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in R 3 have been proved in this article, using analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support.
Abstract: In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in R 3 . Let f be a distribution of compact support on R 3 . Assume Y is a real analytic admissible line complex and Y 0 is an open connected subset of Y with one line in Y 0 disjoint from supp f. Under weak geometric assumptions, if the Radon transform of f is zero for all lines in Y 0 , then supp f intersects no line in Y 0 . These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support
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TL;DR: Any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopy to a mirror image of the other.
Abstract: We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown:
(i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free.
(ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of $K_5$ or $K_{3,3}$.
(iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by ``3-switches'', an analog of 2-switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other.
(iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from $K_6$ by means of $Y\Delta$- and $\Delta Y$-exchanges.
TL;DR: This paper deals with variety of problems in pcf theory and infinitary combinatorics, looking at normal filters and prc, measures of the size of [lambda]^{ with |B_i|
Abstract: This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{ with |B_i|
TL;DR: Most of the data structures presented here use close to linear space and have query time close to O (√ n + K ) or O ( n 2/3 + K ), where K is the size of the output.
01 Jan 1993
TL;DR: In this paper, the following conjecture of Richard Rado was studied: a family of intervals of a linearly ordered set is the union of countably many disjoint subfamilies iff every subfamily of size ℵ1 has this property.
Abstract: We study the following conjecture of Richard Rado: a family of intervals of a linearly ordered set is the union of countably many disjoint subfamilies iff every subfamily of size ℵ1 has this property. We connect it with a well-known two-cardinal transfer principle of model theory known as Chang’s Conjecture and show that it solves the Singular Cardinals Problem.
30 Sep 1993
TL;DR: A new approach to the problem of finding edge-disjoint paths in a planar, undirected graph with “classical” case where an instance must additionally fulfill the so-called evenness-condition is introduced.
Abstract: In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph s.t. each path connects two specified vertices on the outer face boundary. We will focus on the “classical” case where an instance must additionally fulfill the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires \(\mathcal{O}\left( {n^{{5 \mathord{\left/{\vphantom {5 3}} \right.\kern-
ulldelimiterspace} 3}} \left( {\log \log n} \right)^{{1 \mathord{\left/{\vphantom {1 3}} \right.\kern-
ulldelimiterspace} 3}} } \right)\) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which yields an \(\mathcal{O}\left( n \right)\) algorithm.