Showing papers on "Disjoint sets published in 1994"
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15 Nov 1994
TL;DR: The interplay of algebra and order convex lattice-ordered subgroups polars and disjoint elements complete distibutivity totally ordered structures order preserving permutations of a chain classes of latticeordered groups as discussed by the authors.
Abstract: The interplay of algebra and order convex lattice-ordered subgroups polars and disjoint elements complete distibutivity totally ordered structures order preserving permutations of a chain classes of lattice-ordered groups normal-valued lattice-ordered groups representable and abelian lattice-ordered groups Archimedian lattice-ordered groups lattice-ordered varieties.
248 citations
TL;DR: A systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP shows that there exists a disjoint pair of NP-complete sets such that every separator is NP-hard.
180 citations
TL;DR: In this article, a single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn, where the conical hulls of the κ sets are not (κ;−1)-point disjoins in Rn + 1.
Abstract: A single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn . When the conical hulls of the κ sets are (κ−1)-point disjoint in Rn +1, a κ-piece piecewise-linear surface generated by the linear program completely separates the κ sets. This improves on a previous linear programming approach which required that each set be linearly separable from the remaining κ−1 sets. When the conical hulls of the κ sets are not (κ;−1)-point disjoint, the proposed linear program generates an error-minimizing piecewise-linear separator for the κ Sets. For this case it is shown that the null solution is never a unique solver of the linear program and occurs only under the rather rare condition when the mean of each point set equals the mean of the means of the other κ−l sets. This makes the proposed linear computational programming formulation useful for approximately discriminating between κ sets that are not piecewise-linear separable. Computationa...
174 citations
23 Jan 1994
TL;DR: In this article, a polynomial-time algorithm was proposed to compute a piecewise linear polyhedral surface of size O(Ko logKo), where Ko is the complexity of an optimal surface satisfying the constraints of the problem.
Abstract: Motivated by applications in computer graphics, visualization, and scientic compu- tation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x;y) and an input parameter "> 0, compute a piecewise-linear function (x;y) of minimum complexity (that is, an xy-monotone polyhedral surface, with a mini- mum number of vertices, edges, or faces) such thatj(xp;yp) i zp j" for all (xp;yp;zp)2 S.W e give hardness evidence for this problem, by showing that a closely related problem is NP-hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise- linear surface of size O(Ko logKo), where Ko is the complexity of an optimal surface satisfying the constraints of the problem. The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: givenn \red" andm \blue" points in the plane, nd a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles.
162 citations
TL;DR: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems and shows that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy.
Abstract: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets $B^k $ called blocks, and M nonnegative-valued convex block-separable inequalities called coupling or resource constraints. The algorithms are based on an exponential potential function reduction technique. It is shown that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy $\varepsilon$ in $O( K\ln M ( \varepsilon^{ - 2} + \ln K ) )$ iterations, each of which solves K block problems to a comparable accuracy, either sequentially or in parallel. The same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration. An extension to objective and constraint functions of arbitrary sign is also presented. The above results yield fast approximatio...
147 citations
TL;DR: It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle.
Abstract: It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle. Representations using contacts of T- of Y-shaped objects follow. Moreover, there is a one-to-one mapping between all the triangular contact representations of a maximal plane graph and all its partitions into three Schnyder trees.
145 citations
TL;DR: In this paper, the authors study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge group SU(2) and derive disjoint classes of solutions with a regular origin or a horizon.
Abstract: We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a regular origin or a horizon. The 3-spaces (t=const.) of the first, generic class are compact and singular. The second class consists of an infinite family of globally regular, resp. black hole solutions. The third type is an oscillating solution, which although regular is not asymptotically flat.
142 citations
TL;DR: In this article, Razborov et al. studied disjoint NP-pairs representable in a theory T of Bounded Arithmetic in the sense that T proves U intersection V = \emptyset, and showed that such a separation is obvious for the theory S(S_2) + S Sigma^b_2 -PIND considered in [1].
Abstract: In this paper we study the pairs (U,V) of disjoint NP-sets representable in a theory T of Bounded Arithmetic in the sense that T proves U intersection V = \emptyset. For a large variety of theories T we exhibit a natural disjoint NP-pair which is complete for the class of disjoint NP-pairs representable in T. This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [1]. Namely, in order to prove the independence result from a theory T, it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separation is obvious for the theory S(S_2) + S Sigma^b_2 - PIND considered in [1], and this gives an alternative proof of the main result from that paper. [1] A. Razborov. Unprovability of lower bounds on circuit size in certain fragments of Bounded Arithmetic. To appear in Izvestiya of the RAN , 1994.
116 citations
TL;DR: In this article, a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors was discovered, in which each attractor attracts a set of positive $3$-dimensional Lebesgue measure whose points of Lebesge density are dense in the whole manifold.
Abstract: We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive $3$-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations.
110 citations
TL;DR: In this article, the authors give a linear-time algorithm for finding ham-sandwich cuts in R 3 when the three sets are suitably separated, with complexity O(n 3/2 ) for R 2 and O(nd?1?a(d)) for R 3 for R 4.
Abstract: Given disjoint setsP1,P2, ...,Pd inRd withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects thePi. We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond?1. This, in turn, is related to the number ofk-sets inRd?1. With the current estimates, we get complexity close toO(n3/2) ford=3, roughlyO(n8/3) ford=4, andO(nd?1?a(d)) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR3 when the three sets are suitably separated.
110 citations
TL;DR: A variety of problems on the interaction between two sets of line segments in two and three dimensions are considered, including counting the number of intersecting pairs between m blue segments andn red segments in the plane.
Abstract: We consider a variety of problems on the interaction between two sets of line segments in two and three dimensions. These problems range from counting the number of intersecting pairs between m blue segments andn red segments in the plane (assuming that two line segments are disjoint if they have the same color) to finding the smallest vertical distance between two nonintersecting polyhedral terrains in three-dimensional space. We solve these problems efficiently by using a variant of the segment tree. For the three-dimensional problems we also apply a variety of recent combinatorial and algorithmic techniques involving arrangements of lines in three-dimensional space, as developed in a companion paper.
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TL;DR: It is shown, that for each constant k≥1, the following problems can be solved in time: given a graph G, determine whether G has k vertex disjoint cycles, determine Whether 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 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 , that is closed under minor taking, taking, and that does not contain the graph consisting of k disjoint copies of K3, has an membership test algorithm.
TL;DR: In this article, it was shown that the problem of finding pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.
Abstract: It is shown that, for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.
12 Jun 1994
TL;DR: The paper considers two major types of deviations from the traditional graph-theoretic network, and provides an algorithm for the shortest pair of physically-disjoint paths between a given pair of nodes in the network.
Abstract: Telecommunication fiber networks can be more complicated than the traditional graph-theoretic networks of nodes and links. This is due to practical and economic considerations. The paper considers two major types of deviations from the traditional graph-theoretic network, and provide an algorithm for the shortest pair of physically-disjoint paths between a given pair of nodes in the network. Such disjoint paths can be used for improving the reliability of the network, e.g., one path may be used as a back up while the other is actually used for transmission of data. Alternatively, the entire traffic between the given pair of nodes in the network may be divided equally over the two disjoint paths so that if a node or link on one of the paths fails, not all of the traffic is lost. Optimizing the length of disjoint paths helps in reducing the amount of fiber usage and network costs. >
TL;DR: The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\ kappa$ depends only on the expansion properties of the input graph, and not on $n$.
Abstract: Given an expander graph $G=(V,E)$ and a set of $q$ disjoint pairs of vertices in $V$, the authors are interested in finding for each pair $(a_i, b_i)$ a path connecting $a_i$ to $b_i$ such that the set of $q$ paths so found is edge disjoint. (For general graphs the related decision problem is NP complete.)
The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\kappa$ depends only on the expansion properties of the input graph, and not on $n$. Furthermore, a randomized $o(n^3)$ time algorithm, and a random $\cal NC$ algorithm for constructing these paths is presented. (Previous existence proofs and construction algorithms allowed only up to $n^\epsilon$ pairs, for some $\epsilon\ll \frac{1}{3}$, and strong expanders [D. Peleg and E. Upfal, Combinatorica, 9 (1989), pp.~289--313.].)
In passing, an algorithm is developed for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.
TL;DR: It is shown that the following statement is true fork=4 if the authors restrict ourselves to planar graphs and similar statements are considered for weaklys — independent spanning trees and for directed graphs.
Abstract: IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT 1 andT 2 inG are calleds -- independent if for each vertexx inG the paths fromx tos inT 1 andT 2 are openly disjoint. It is known that the following statement is true fork≤3: IfG isk-connected, then there arek pairwises -- independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys -- independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.
01 Jan 1994
TL;DR: This paper proves several new modularity results for unconditional and conditional term rewriting systems and refutes a conjecture of Middeldorp (1990, 1993).
Abstract: In this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructor-sharing combinations. Here we focus on a more general kind of combination: so-called composable systems. As far as conditional term rewriting systems are concerned, all known modularity result but one apply only to disjoint systems. Here we investigate conditional systems which may share constructors. Furthermore, we refute a conjecture of Middeldorp (1990, 1993).
TL;DR: An efficient method is given for triangulating a collection of p disjoint Jordan polygonal chains in time O (n + p (log p)1+e), for any fixed e > 0, where n is the total number of vertices.
Abstract: Recent advances on polygon triangulation have yielded efficient algorithms for a large number of problems dealing with a single simple polygon. If the input consists of several disjoint polygons, however, it is often desirable to merge them in preprocessing so as to produce a single polygon that retains the geometric characteristics of its individual components. We give an efficient method for doing so, which combines a generalized form of Jordan sorting with the efficient use of point location and interval trees. As a corollary, we are able to triangulate a collection of p disjoint Jordan polygonal chains in time O (n + p (log p)1+e), for any fixed e > 0, where n is the total number of vertices. A variant of the algorithm gives a running time of O ((n + p log p) log log p). The performance of these solutions approaches the lower bound of Ω (n + p log p).
TL;DR: The main results are generalized to some restricted form of non-disjoint combinations of term rewriting systems, namely for ‘combined systems with shared constructors’ and 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.
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 ℛ1 ⊕ ℛ2 of two (finitely branching) terminating term rewriting systems ℛ1, ℛ2 is non-terminating, then one of the systems, say ℛ1, enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. ℛ1 ⊕ {itG(x, y)→ x, G(x, y) → y} is non-terminating, and the other system ℛ2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient 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. In particular, we show how to generalize the main results to some restricted form of non-disjoint combinations of term rewriting systems, namely for ‘combined systems with shared constructors’.
26 Jun 1994
TL;DR: In this article, it was shown that a convexity structure satisfies the Pasch axiom of plane geometry if every pair of disjoint polytopes with at most n vertices can be separated by complementary half spaces.
Abstract: A convexity structure satisfies the separation propertyS4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces,n-ary convexities, and graphs. In particular, it is proven that
a)
ann-ary convexity isS4 iff every pair of disjoint polytopes with at mostn vertices can be separated by complementary half spaces, and
b)
an interval convexity isS4 iff it satisfies the analogue of the Pasch axiom of plane geometry.
26 Sep 1994
TL;DR: A data structure that can store a set of disjoint fat objects in 2- and 3-space such that point location and bounded-size range searching with arbitrarily shaped regions can be performed efficiently.
Abstract: We present a data structure that can store a set of disjoint fat objects in 2- and 3-space such that point location and bounded-size range searching with arbitrarily shaped regions can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or non-convex polygonal/polyhedral objects. For dimension d=2,3, the multi-purpose data structure supports point location and range searching queries in time O(logd−1n) and requires O(n logd−1n) storage, after O(n logd−1n log log n) preprocessing. The data structure and query algorithm are rather simple. The results are likely to be extendible in many directions.
TL;DR: In this paper, the strong link between matroids and matching was used to extend the ideas that resulted in the design of random $NC$ $(RNC)$ algorithms for matching to obtain $RNC$ algorithm for the matroid union, intersection, and matching problems.
Abstract: The strong link between matroids and matching is used to extend the ideas that resulted in the design of random $NC$ $(RNC)$ algorithms for matching to obtain $RNC$ algorithms for the matroid union, intersection, and matching problems, and for linearly representable matroids. As a consequence, $RNC$ algorithms for the well-known problems of finding an arboresence and a maximum cardinality set of edge-disjoint spanning trees in a graph are obtained. The key tools used are linear algebra and randomization.
TL;DR: This algorithm uses a plane sweep to presort the segments; then it operates on a list of slabs that efficiently stores a single level of a segment tree that performs well even with inadequate physical memory.
25 Aug 1994
TL;DR: The first widespread use of variations of generic oracles to achieve the necessary relativized worlds is made.
Abstract: We settle all relativized questions of the relationships between the following five propositions:
P = NP.
P = UP.
P = NP $\cap$ coNP.
All disjoint pairs of NP sets are P-separable.
All disjoint pairs of coNP sets are P-separable.
We make the first widespread use of variations of generic oracles to achieve the necessary relativized worlds.
06 Jun 1994
TL;DR: A genetic algorithm (GA) for partitioning a hypergraph into two disjoint graphs of least ratio-cut is presented, which is a fast local optimizer, and a preprocessing step.
Abstract: A genetic algorithm (GA) for partitioning a hypergraph into two disjoint graphs of least ratio-cut is presented. Two notable features of this algorithm are: (i) a fast local optimizer, and (ii) a preprocessing step. Some supporting combinatorial arguments for the preprocessing heuristic are also provided. Experimental results on industrial benchmarks circuits are favorable when compared with recently published algorithms [25], [26], [19].
TL;DR: A graph-theoretic decomposition algorithm is used to partition the Jacobian into weakly coupled, possibly overlapping blocks, and it is shown that it suffices to invert only the diagonal blocks to carry out the Newton iterates.
Abstract: The purpose of this paper is to present a block-parallel Newton method for solving large nonlinear systems. A graph-theoretic decomposition algorithm is first used to partition the Jacobian into weakly coupled, possibly overlapping blocks. It is then shown that it suffices to invert only the diagonal blocks to carry out the Newton iterates. A rigorous justification of this practice is provided by using a convergence result of Kantorovich in the expanded space of the iterates, where overlapping blocks appear as disjoint. The individual blocks, or a group of blocks, can be inverted by a dedicated processor, making the new block-diagonal Newton method ideally suited for parallel processing. Applications to the power flow problem are presented and parallelization issues are discussed.
TL;DR: This work shows that the (two-output) strongly code disjoint checkers do not allow such reduction of their output code space and on this basis it is shown that the strongly fault secure property is not necessary.
Abstract: The final checker of a self-checking system is an embedded double-rail checker (the partial checkers have in general two outputs). The self-testing or the strongly code disjoint property of this embedded checker can be lost if it is not exercised by an appropriate set of inputs. If some partial checkers are strongly code disjoint, then some undetectable faults can modify the input/output mapping of these checkers. This can compromise the exercising of the final double-rail checker especially if this modification leads to the reduction of the partial checker's output code space. In that case it will be required that the partial strongly code disjoint checkers must also be strongly fault secure. In this work we show that the (two-output) strongly code disjoint checkers do not allow such reduction of their output code space and on this basis we show that the strongly fault secure property is not necessary. We also give some techniques that ensure exercising the final checker. >
TL;DR: Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders.
Abstract: A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.