Showing papers on "Disjoint sets published in 2000"

Efficient algorithms for mining outliers from large data sets

Abstract: In this paper, we propose a novel formulation for distance-based outliers that is based on the distance of a point from its kth nearest neighbor. We rank each point on the basis of its distance to its kth nearest neighbor and declare the top n points in this ranking to be outliers. In addition to developing relatively straightforward solutions to finding such outliers based on the classical nested-loop join and index join algorithms, we develop a highly efficient partition-based algorithm for mining outliers. This algorithm first partitions the input data set into disjoint subsets, and then prunes entire partitions as soon as it is determined that they cannot contain outliers. This results in substantial savings in computation. We present the results of an extensive experimental study on real-life and synthetic data sets. The results from a real-life NBA database highlight and reveal several expected and unexpected aspects of the database. The results from a study on synthetic data sets demonstrate that the partition-based algorithm scales well with respect to both data set size and data set dimensionality.

Blind separation of disjoint orthogonal signals: demixing N sources from 2 mixtures

Abstract: We present a novel method for blind separation of any number of sources using only two mixtures. The method applies when sources are (W-)disjoint orthogonal, that is, when the supports of the (windowed) Fourier transform of any two signals in the mixture are disjoint sets. We show that, for anechoic mixtures of attenuated and delayed sources, the method allows one to estimate the mixing parameters by clustering ratios of the time-frequency representations of the mixtures. The estimates of the mixing parameters are then used to partition the time-frequency representation of one mixture to recover the original sources. The technique is valid even in the case when the number of sources is larger than the number of mixtures. The general results are verified on both speech and wireless signals.

Classification ability of single hidden layer feedforward neural networks

Abstract: Multilayer perceptrons with hard-limiting (signum) activation functions can form complex decision regions. It is well known that a three-layer perceptron (two hidden layers) can form arbitrary disjoint decision regions and a two-layer perceptron (one hidden layer) can form single convex decision regions. This paper further proves that single hidden layer feedforward neural networks (SLFN) with any continuous bounded nonconstant activation function or any arbitrary bounded (continuous or not continuous) activation function which has unequal limits at infinities (not just perceptrons) can form disjoint decision regions with arbitrary shapes in multidimensional cases, SLFN with some unbounded activation function can also form disjoint decision regions with arbitrary shapes.

Approximating the domatic number

Abstract: A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, δ the minimum degree, and ∆ the maximum degree. We show that every graph has a domatic partition with (1−o(1))(δ+1)/ lnn dominating sets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) lnn approximation algorithm for domatic number, since the domatic number is always at most δ + 1. We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multi-prover protocols with zero-knowledge techniques, we show that for every ǫ > 0, a (1−ǫ) lnn-approximation implies that NP ⊆ DTIME(n log ). This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with (1 − o(1))(δ + 1)/ ln∆ dominating sets, where the “o(1)” term goes to zero as ∆ increases. This can be turned into an efficient algorithm that produces a domatic partition of Ω(δ/ ln∆) sets.

Channel Assignment and Weighted Coloring

Abstract: In cellular telephone networks, sets of radio channels (colors) must be assigned to transmitters (vertices) while avoiding interterence. Often, the transmitters are laid out like vertices of a triangular lattice in the plane. We investigated the corresponding weighted coloring problem of assigning sets of colors to vertices of the triangular lattice so that the sets of colors assigned to adjacent vertices are disjoint. We present a hardness result and an efficient algorithm yielding an approximate solution.

Approximation Algorithms for Disjoint Paths and Related Routing and Packing Problems

Abstract: Given a network and a set of connection requests on it, we consider the maximum edge-disjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multicommodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented.

Rough Set Processing of Vague Information Using Fuzzy Similarity Relations

Abstract: The rough sets theory has proved to be a very useful tool for analysis of information tables describing objects by means of disjoint subsets of condition and decision attributes. The key idea of rough sets is approximation of knowledge expressed by decision attributes using knowledge expressed by condition attributes. From a formal point of view, the rough sets theory was originally founded on the idea of approximating a given set represented by objects having the same description in terms of decision attributes, by means of an indiscernibility binary relation linking pairs of objects having the same description by condition attributes. The indiscernibility relation is an equivalence binary relation (reflexive, symmetric and transitive) and implies an impossibility to distinguish two objects having the same description in terms of the condition attributes. It produces crisp granules of knowledge that are used to build approximations. In reality, due to vagueness of the available information about objects, small differences are not considered significant. This situation may be formally modelled by similarity or tolerance relations instead of the indiscernibility relation. We are using a similarity relation which is only reflexive, relaxing therefore the properties of symmetry and transitivity.

Analytic Combinatorics of Chord Diagrams

Abstract: In this paper we study the enumeration of diagrams of n chords joining 2n points on a circle in disjoint pairs. We establish limit laws for the following three parameters: number of components, size of the largest component, and number of crossings. We also find exact formulas for the moments of the distribution of number of components and number of crossings.

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

Abstract: We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the $B_{i,j}$ allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum $#$ for simple $n$-dimensional polytopes; when $P^n$ is a product of simplices, we describe $P^n# Q^n$ by applying an appropriate sequence of {\it pruning operators}, or hyperplane cuts, to $Q^n$.

Quantum secret sharing for general access structures

Abstract: We explore the conversion of classical secret-sharing schemes to quantum ones, and how this can be used to give efficient QSS schemes for general adversary structures. Our first result is that quantum secret-sharing is possible for any structure for which no two disjoint sets can reconstruct the secret (this was also proved, somewhat differently, by D. Gottesman). To obtain this we show that a large class of linear classical SS schemes can be converted into quantum schemes of the same efficiency. We also give a necessary and sufficient condiion for the direct conversion of classical schemes into quantum ones, and show that all group homomorphic schemes satisfy it.

Higher order intersection numbers of 2-spheres in 4-manifolds

Abstract: This is the beginning of an obstruction theory for deciding whether a map f : S 2 ! X 4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The rst obstruction is Wall’s self-intersection number (f ) which tells the whole story in higher dimensions. Our second order obstruction (f ) is dened if (f ) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of 1X modulo S3 -symmetry (rather then just one copy modulo S2 -symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant dened in [2] and [12] which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sucient conditions for moving three maps f1;f 2;f 3 :S 2 !X 4 to a position in which they have disjoint images. Again the obstruction (f1;f 2;f 3) generalizes Wall’s intersection number (f1;f 2) which answers the same question for two spheres but is not sufcient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant (1; 2; 3), generalizing the Matsumoto triple [10] to the non simply-connected setting.

Algorithms for maximum independent set applied to map labelling

Abstract: We consider the following map labelling problem: given distinct points p,p, . . . ,pn in the plane, and given σ, find a maximum cardinality set of pairwise disjoint axis-parallel σ× σ squares Q1, Q2, . . . , Qr. This problem reduces to that of finding a maximum cardinality independent set in an associated graph called the conflict graph. We describe several heuristics for the maximum cardinality independent set problem, some of which use an LP solution as input. Also, we describe a branch-and-cut algorithm to solve it to optimality. The standard independent set formulation has an inequality for each edge in the conflict graph which ensures that only one of its endpoints can belong to an independent set. To obtain good starting points for our LP-based heuristics and good upper bounds on the optimal value for our branch-and-cut algorithm we replace this set of inequalities by the set of inequalities describing all maximal cliques in the conflict graph. For this strengthened formulation we also generate lifted odd hole inequalities and mod-k inequalities. We present a comprehensive computational study of solving map labelling instances for sizes up to n = 950 to optimality. Previously, optimal solutions to instances of size n ≤ 300 have been reported on in the literature. By comparing against these optimal solutions we show that our heuristics are capable of producing near-optimal solutions for large-scale instances.

Vertex-Disjoint Cycles Containing Specified Edges

Abstract: Dirac and Ore-type degree conditions are given for a graph to contain vertex disjoint cycles each of which contains a previously specified edge. One set of conditions is given that imply vertex disjoint cycles of length at most 4, and another set of conditions are given that imply the existence of cycles that span all of the vertices of the graph (i.e. a 2-factor). The conditions are shown to be sharp and give positive answers to conjectures of Enomoto in [3] and Wang in [5].

Polynomial time approximation schemes for geometric k-clustering

Abstract: We deal with the problem of clustering data points. Given n points in a larger set (for example, R/sup d/) endowed with a distance function (for example, L/sup 2/ distance), we would like to partition the data set into k disjoint clusters, each with a "cluster center", so as to minimize the sum over all data points of the distance between the point and the center of the cluster containing the point. The problem is provably NP-hard in some high dimensional geometric settings, even for k=2. We give polynomial time approximation schemes for this problem in several settings, including the binary cube (0, 1)/sup d/ with Hamming distance, and R/sup d/ either with L/sup 1/ distance, or with L/sup 2/ distance, or with the square of L/sup 2/ distance. In all these settings, the best previous results were constant factor approximation guarantees. We note that our problem is similar in flavor to the k-median problem (and the related facility location problem), which has been considered in graph-theoretic and fixed dimensional geometric settings, where it becomes hard when k is part of the input. In contrast, we study the problem when k is fixed, but the dimension is part of the input. Our algorithms are based on a dimension reduction construction for the Hamming cube, which may be of independent interest.

Set Theory for Verification: II. Induction and Recursion

Abstract: A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski Theorem over a suitable set. Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the Knaster-Tarski Theorem to a set, such as V[omega], that is closed under Cartesian product and disjoint sum. Worked examples include the transitive closure of a relation, lists, variable-branching trees and mutually recursive trees and forests. The Schroder-Bernstein Theorem and the soundness of propositional logic are proved in Isabelle sessions.

Orbits of permutation groups on the power set

Abstract: Let G be a permutation group on a finite set $\Omega $ . If G does not involve A n for $n \geqq 5 $ , then there exist two disjoint subsets of $\Omega $ such that no Sylow subgroup of G stabilizes both and four disjoint subsets of $\Omega $ whose stabilizers in G intersect trivially.

Splittings of groups and intersection numbers

Abstract: We prove algebraic analogues of the facts that a curve on a surface with selfintersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.

The Packing Property

Abstract: A clutter (V, E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. An m×n 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices.

Block-iterative interior point optimization methods for image reconstruction from limited data

Abstract: Iterative algorithms for image reconstruction often involve minimizing some cost function h(x) that measures the degree of agreement between the measured data and a theoretical parametrized model In addition, one may wish to have x satisfy certain constraints It is usually the case that the cost function is the sum of simpler functions: h(x) = ∑i = 1Ihi(x) Partitioning the set {i = 1,,I} as the union of the disjoint sets Bn,n = 1,,N, we let hn(x) = ∑iBnhi(x) The method presented here is block iterative, in the sense that at each step only the gradient of a single hn(x) is employed Convergence can be significantly accelerated, compared to that of the single-block (N = 1) method, through the use of appropriately chosen scaling factors The algorithm is an interior point method, in the sense that the images xk + 1 obtained at each step of the iteration satisfy the desired constraints Here the constraints are imposed by having the next iterate xk + 1 satisfy the gradient equation ∇F(xk + 1) = ∇F(xk)-tn∇hn(xk), for appropriate scalars tn, where the convex function F is defined and differentiable only on vectors satisfying the constraints Special cases of the algorithm that apply to tomographic image reconstruction, and permit inclusion of upper and lower bounds on individual pixels, are presented The focus here is on the development of the underlying convergence theory of the algorithm Behaviour of special cases has been considered elsewhere

Stochastic optimal control under Poisson-distributed observations

Abstract: Optimal control problems for linear, stochastic continuous-time systems are considered, in which the time domain is decomposed into a finite set of N disjoint random intervals of the form [t/sub i/, t/sub i+1/), in which a complete state observation is taken at each instant t/sub i/, 0/spl les/i/spl les/N-1. Two optimal control problems termed, respectively, the (piecewise) time-invariant control and time-variant control are considered in this framework. Concerning the observation point process, we first consider the general situation in which the increment intervals are i.i.d.r.v.s with unspecified probabilistic distributions. The (piecewise) time-invariant solution is thoroughly developed in this general case, and computations are illustrated using Erlang as the observations interarrival distribution. Next, the problem is specialized so increments are exponentially distributed, and the particular optimal control structure that results from this assumption is presented. Finally, and still under the Poisson assumption and for the time-variant case, we show that the control problem is closely related to linear quadratic Gaussian regulation with an exponentially discounted cost. The optimal control is made again of a sequence of piecewise open-loop controls corresponding, in this case, to linear feedback of the state predictor based on the most recent information on each interval. The feedback gains are time-varying matrices obtained from a sequence of algebraic Riccati equations, which are also computed off-line.

Data mapping method and apparatus with multi-party capability

Abstract: Mapping or encryption of a message made up of characters from a character set is provided. Each character in the character set is associated with one or more disjoint real number intervals. Each character in the message is represented by a real number randomly selected from one of the intervals with which the character of the character set is associated. In some aspects, the resulting set of real numbers is translated to another set of real numbers by the application of a translation function or successive application of two or more translation functions. The translation functions never result in two different real numbers being translated to a single number.

Sharp Bounds on Geometric Permutations of Pairwise Disjoint Balls in R d

Abstract: We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in \R d , is Θ (n d-1 ) . This improves substantially the upper bound of O(n 2d-2 ) known for general convex sets [9]. We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks in the plane is two, improving the previous upper bound of three given in [5].

Approximation Algorithms for Minimum K -Cut

Abstract: Let G=(V,E) be a complete undirected graph, with node set V={v 1 , . . ., v n } and edge set E . The edges (v i ,v j ) ∈ E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = { k i } i=1 p $(\sum_{i=1}^p k_i \leq |V|$) , the minimum K-cut problem is to compute disjoint subsets with sizes { k i } i=1 p , minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts.

Horseshoes and the Conley index spectrum

Abstract: Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semi-conjugacy onto the full shift dynamics on two symbols.