Showing papers on "Disjoint sets published in 1990"

Programming in Martin-Lo¨f's type theory: an introduction

Abstract: Part 1 Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product of two sets disjoint union of two sets disjoint union of a family of sets the set of small sets (the first universe) well-orderings general trees. Part 2 Subsets: subsets in the basic set theory the subset theory. Part 3 Monomorphic sets: types defining sets in terms of types. Part 4 Examples: some small examples program derivation specification of abstract data types.

Facets of the clique partitioning polytope

Abstract: A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW 1,⋯,W k such that eachW i induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = ∪ 1{uv|u, v ∈ W i ,u ≠ v}. Given weightsw e ∈ℝ for alle ∈ E, the clique partitioning problem is to find a clique partitioningA ofG such that ∑ e∈A w e is as small as possible. This problem—known to be -hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphK n , i.e., is the convex hull of the incidence vectors of the clique partitionings ofK n . We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofK n induce facets of . The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grotschel and Wakabayashi (1989).

Boolean Properties of Sets

Abstract: Summary. The text includes a number of theorems about Boolean operations on sets: union, intersection, difference, symmetric difference; and relations on sets: meets (having non-empty intersection), misses (being disjoint) and subset (inclusion).

An improved minimizing algorithm for sum of disjoint products (reliability theory)

Abstract: The Abraham-Locks-revised (ALR) sum-of-disjoint products (SDP) algorithm is an efficient method for obtaining a system reliability formula. The author describes a minor modification of the ALR algorithm called the Abraham-Locks-Wilson (ALW) method. The new feature is an alternative method of ordering paths and terms. ALW obtains a shorter disjoint system formula on a test example than any previous SDP method and allows small computational savings in processing large paths of complex networks. As there are different ways to obtain a reliability formula it is useful to use an approach which yields the smallest formula relative to computational effort expended. The extra effort in ordering the terms should be reasonably small and usually leads to improved efficiency in the later stages of the algorithm. ALW allows the analyst to operate in a more efficient way on many problems, particularly if the overlap ordering is used in the early stages of processing but is probably ignored for terms that contain a majority of the Boolean variables. >

A class of bounded approximation algorithms for graph partitioning

Abstract: This paper considers the problem of partitioning the nodes of a weighted graph into k disjoint subsets of bounded size, such that the sum of the weights of the edges whose end vertices belong to the same subset is maximized. A class of approximation algorithms based on matching is presented. These algorithms are shown to yield practical worst-case bounds when k is large. Extensive empirical experimentation indicates that the methods produce consistently good solutions to an important VLSI design problem in a fraction of the time required by competing methods.

Query size estimation by adaptive sampling (extended abstract)

Abstract: We present an adaptive, random sampling algorithm for estimating the size of general queries. The algorithm can be used for any query Q over a database D such that 1) for some n, the answer to Q can be partitioned into n disjoint subsets Q1, Q2, …, Qn, and 2) for 1 ≤ i ≤ n, the size of Qi is bounded by some function b(D, Q), and 3) there is some algorithm by which we can compute the size of Qi, where i is chosen randomly. We consider the performance of the algorithm on three special cases of the algorithm: join queries, transitive closure queries, and general recursive Datalog queries.

On the periodicity of symbolic observations of piecewise smooth discrete-time systems

Abstract: A study is made of the behavior of discrete-time systems composed of a set of smooth transition maps coupled by a quantized feedback function. The feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region. The main result is that under a constraint on the norm of the derivative of the transition maps, a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic. Under these assumptions, it is shown that eventually the feedback function is determined by a finite state automaton. A similar result is proved in the case of finite state dynamic feedback. >

Permutations for convex sets

Abstract: LetA be a family ofn pairwise disjoint compact convex sets inRd. Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0JaaGOm% aiabfo6atnaaDaaaleaacaWGPbGaeyypa0JaaGimaaqaaiaadsgacq% GHsislcaaIXaaaaOWaaeWaaeaadaqhaaWcbaGaiaiG0caaamyAaaqa% aiaad2gacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaaaa!4A12! $$\Phi _d (m) = 2\Sigma _{i = 0}^{d - 1} \left( {_i^{m - 1} } \right)$$ . We show that the directed lines inRd, d ? 3, can be partitioned into % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOWaaeWaaeaadaqadaqaamaaDaaaleaacaaI% YaaabaGaamOBaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa!3CFF! $$\Phi _d \left( {\left( {_2^n } \right)} \right)$$ sets such that any two directed lines in the same set which intersect anyA?⊆A generate the same ordering onA?. The directed lines inR2 can be partitioned into 12n such sets. This bounds the number of geometric permutations onA by 1/2?d ford?3 and by 6n ford=2.

On the μ-colorings of a matroid

Abstract: Let Mbe a matroid on S. Let μ = (μ1,⋯, μK) be a partition of ∣S∣ (:∣ ∣ denotes cardinality). We state a necessary and sufficient condition for the existence of pairwise disjoint independent sets of M I 1, [cddot]I 1, satisfying the following conditions. We use this result to restate a theorem of CGamas [2], giving necessary and sufficient conditions for nonvanishing of decomposable symmetrized tensors.

Quasiextremal distance domains and conformal mappings onto circle domains

Abstract: This paper contributes to the theory of quasiextremal distance domains. We present some new properties for these domains and point out results concerning the extension of quasiconformal homeomorphisms.For example, we establish a continuity property for mod(E, F; D) and use this to demonstrate that mod(E, F; D) = cap(E, F; D) whenever D is a QED domain and E, F are disjoint compacta in D. Our final result is that if each boundary component of a plane domain is either a point or a Jordan curve and if the domain satisfies a boundary quasiextremal distance property, then there exists a quasiconformal self-homeomorphism of the entire plane which maps the given domain conformally onto a circle domain.

On link maps and their homotopy classification

Abstract: is called a link map if the two spheres S p and S o have disjoint images, f l (S p) nf2(S ~) = 0. Two such link maps are called link homotopic if they are homotopic through link maps. This type of equivalence relation was introduced by Milnor [M] in the classical case m = 3, p = q . . . . . 1 and has recently found much renewed interest (see e.g. [MR; FR; K i l 3 ; Ko2-5]). In this paper we study and determine in a number of cases the set LM~. q of link homotopy classes of link maps f as above.

Exact robot navigation in geometrically complicated but topologically simple spaces

Abstract: Navigation functions on forests of stars, geometrically complicated C-spaces (configuration spaces) that are topologically indistinguishable from a simple disc punctured by disjoint smaller discs representing model obstacles, are constructed. For reasons of mathematical tractability, each C-space obstacle is approximated by a Boolean combination of linear and quadratic polynomial inequalities (with sharp corners allowed), and a calculus of implicit representations is used to effectively represent such obstacles. Evidence is provided of the effectiveness of this technology of implicit representations in the form of several simulation studies. >

A maximum stable matching for the roommates problem

Abstract: The stable roommates problem is that of matchingn people inton/2 disjoint pairs so that no two persons, who are not paired together, both prefer each other to their respective mates under the matching. Such a matching is called “a complete stable matching”. It is known that a complete stable matching may not exist. Irving proposed anO(n 2) algorithm that would find one complete stable matching if there is one, or would report that none exists. Since there may not exist any complete stable matching, it is natural to consider the problem of finding a maximum stable matching, i.e., a maximum number of disjoint pairs of persons such that these pairs are stable among themselves. In this paper, we present anO(n 2) algorithm, which is a modified version of Irving's algorithm, that finds a maximum stable matching.

Convex polytopes without triangular faces

Abstract: LetP be a convexd-polytope without triangular 2-faces. Forj=0,…,d−1 denote byf j(P) the number ofj-dimensional faces ofP. We prove the lower boundf j(P)≥f j(C d) whereC d is thed-cube, which has been conjectured by Y. Kupitz in 1980. We also show that for anyj equality is only attained for cubes. This result is a consequence of the far-reaching observation that such polytopes have pairs of disjoint facets. As a further application we show that there exists only one combinatorial type of such polytopes with exactly 2d+1 facets.

Variations on the theme of repeated distances

Abstract: We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ℝd, d≥4. We also give asymptotically tight upper bounds on the total number of occurrences of the “favourite” distances fromn points in ℝd, d≥4. Related results are proved for distances determined byn disjoint compact convex sets in ℝ2.

Convexly independent sets

Abstract: A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family ℱ of mutually disjoint compact convex sets such that any subfamily of at mostk members of ℱ is convexly independent, but no subfamily of sizen is.

A lock technique for disjoint and non-disjoint complex objects

Abstract: Using database systems in the field of non-standard applications like engineering, robotics, etc. leads to many new requirements. Some of the major ones are support of (disjoint and non-disjoint) complex objects and of long transactions. These requirements disclose severe drawbacks of traditional concurrency control techniques: Transactions are either serialized unnecessarily or the concurrency control overhead grows drastically. Furthermore, traditional lock protocols cannot be applied in a straightforward way to non-disjoint complex objects.

The domatic number problem interval graphs

Abstract: A set of vertices D is a dominating set of a graph $G = ( V,E )$ if every vertex in $V - D$ is adjacent to a vertex in D. The domatic number $d ( G )$ of a graph $G = ( V,E )$ is the maximum number k such that V can be partitioned into k disjoint dominating sets $D_1 , \cdots ,D_k $. The main purpose of this paper is to give linear algorithms for the domatic number problem in interval graphs. This paper also proves that $d ( G ) = \delta ( G ) + 1$ for any interval graph G, where $\delta ( G )$ is the minimum degree of a vertex in G.

A generalization of Hadwiger's transversal theorem to intersecting sets

Abstract: Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal of the family. This generalizes Hadwiger's Transversal Theorem to families of compact convex sets which are not necessarily pairwise disjoint. If every six sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering. If the family is pairwise disjoint and every four sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering.

Implicitly Searching Convolutions and Computing Depth of Collision

Abstract: Given two intersecting polyhedra P, Q and a direction d, find the smallest translation of Q along d that renders the interiors of P and Q disjoint. The same question can also be asked 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.

Unification in a Combination of Equational Theories: an Efficient Algorithm

Abstract: An algorithm is presented for solving equations in a combination of arbitrary theories with disjoint sets of function symbols. It is an extension of [3] in which the problem was treated for the combination of an arbitrary and a simple theory. The algorithm consists in a set of transformation rules that simplify a unification problem until a solved form is obtained. Each rule is shown to preserve solutions, and solved problems are unification problems in normal form. The rules terminate for any control that delays replacement until the end. The algorithm is more efficient than [13] because nondeterministic branching is performed only when necessary, that is when theory clashes or compound cycles are encountered.

Shortest circuit covers and postman tours in graphs with a nowhere zero 4-flow

Abstract: Let G be a graph with a nowhere zero 4-flow. It is shown that the length of a shortest circuit cover of $E(G)$ is equal to the length of a shortest postman tour of $E(G)$. Using this result an efficient algorithm for constructing shortest circuit covers for graphs that possess two disjoint spanning trees is obtained. It is also deduced that if H is a $2m$-edge connected graph, $m \geqq 2$, then there exists a circuit cover of $E(H)$ of length at most $|E(H)|+\min \{ {{| E(H) |} / {(2m + 1)}}, | V(H) | - 1 \}$ and that if G has a nowhere zero 4-flow, then there exists a circuit cover of $V(G)$ of length at most $2|V(G)| -2$. Finally, it is shown that the equivalence between shortest circuit covers and postman tours may be extended to binary matroids that possess a nowhere zero $\mathbb{Z} _2 ^2$-flow.

A minimax arc theorem for reducible flow graphs

Abstract: A conjecture of Frank and Gyarfas is established by proving that the cardinality of a minimum feedback arc set in a reducible flow graph is equal to the cardinality of a maximum collection of arc disjoint cycles.