Showing papers on "Disjoint sets published in 2001"

Concept Decompositions for Large Sparse Text Data Using Clustering

Abstract: Unlabeled document collections are becoming increasingly common and availables mining such data sets represents a major contemporary challenge. Using words as features, text documents are often represented as high-dimensional and sparse vectors–a few thousand dimensions and a sparsity of 95 to 99% is typical. In this paper, we study a certain spherical k-means algorithm for clustering such document vectors. The algorithm outputs k disjoint clusters each with a concept vector that is the centroid of the cluster normalized to have unit Euclidean norm. As our first contribution, we empirically demonstrate that, owing to the high-dimensionality and sparsity of the text data, the clusters produced by the algorithm have a certain “fractal-like” and “self-similar” behavior. As our second contribution, we introduce concept decompositions to approximate the matrix of document vectorss these decompositions are obtained by taking the least-squares approximation onto the linear subspace spanned by all the concept vectors. We empirically establish that the approximation errors of the concept decompositions are close to the best possible, namely, to truncated singular value decompositions. As our third contribution, we show that the concept vectors are localized in the word space, are sparse, and tend towards orthonormality. In contrast, the singular vectors are global in the word space and are dense. Nonetheless, we observe the surprising fact that the linear subspaces spanned by the concept vectors and the leading singular vectors are quite close in the sense of small principal angles between them. In conclusion, the concept vectors produced by the spherical k-means algorithm constitute a powerful sparse and localized “basis” for text data sets.

Method and apparatus for measuring and reporting channel state information in a high efficiency, high performance communications system

Abstract: Channel state information (CSI) can be used by a communications system to precondition transmissions between transmitter units and receiver units. In one aspect of the invention, disjoint sub-channel sets are assigned to transmit antennas located at a transmitter unit. Pilot symbols are generated and transmitted on a subset of the disjoint sub-channels. Upon receipt of the transmitted pilot symbols, the receiver units determine the CSI for the disjoint sub-channels that carried pilot symbols. These CSI values are reported to the transmitter unit, which will use these CSI values to generate CSI estimates for the disjoint sub-channels that did not carry pilot symbols. The amount of information necessary to report CSI on the reverse link can be further minimized through compression techniques and resource allocation techniques.

Accurate and Fast Proximity Queries Between Polyhedra Using Convex Surface Decomposition

Abstract: The need to perform fast and accurate proximity queries arises frequently in physically-based modeling, simulation, animation, real-time interaction within a virtual environment, and game dynamics. The set of proximity queries include intersection detection, tolerance verification, exact and approximate minimum distance computation, and (disjoint) contact determination. Specialized data structures and algorithms have often been designed to perform each type of query separately. We present a unified approach to perform any of these queries seamlessly for general, rigid polyhedral objects with boundary representations which are orientable 2-manifolds. The proposed method involves a hierarchical data structure built upon a surface decomposition of the models. Furthermore, the incremental query algorithm takes advantage of coherence between successive frames. It has been applied to complex benchmarks and compares very favorably with earlier algorithms and systems.

Computing a canonical polygonal schema of an orientable triangulated surface

Abstract: A closed orientable surface of genus $g$ can be obtained by appropriat e identification of pairs of edges of a $4g$-gon (the polygonal schema). The identified edges form $2g$ loops on the surface, that are disjoint except for their common end-point. These loops are generators of both the fundamental group and the homology group of the surface. The inverse problem is concerned with finding a set of $2g$ loops on a triangulated surface, such that cutting the surface along these loops yields a (canonical) polygonal schema. We present two optimal algorithms for this inverse problem. Both algorithms have been implemented using the CGAL polyhedron data structure.

Residual properties and almost equicontinuity

Abstract: A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.

A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration

Abstract: In this paper we present a meshfree discretization technique based only on a set of irregularly spaced points $x_i \in \mathbb R^d$ and the partition of unity approach. In this sequel to [M. Griebel and M. A. Schweitzer, SIAM J. Sci. Comput., 22 (2000), pp. 853--890] we focus on the cover construction and its interplay with the integration problem arising in a Galerkin discretization. We present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme. Here, we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency. The use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly. Now the overall computational costs are dominated by the number of the integration cells. We present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency. In fact, the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method, yet the presented method is completely independent of a mesh. Moreover, our method is applicable to general domains and allows for the construction of approximations of any order and regularity.

The distributed boosting algorithm

Abstract: In this paper, we propose a general framework for distributed boosting intended for efficient integrating specialized classifiers learned over very large and distributed homogeneous databases that cannot be merged at a single location. Our distributed boosting algorithm can also be used as a parallel classification technique, where a massive database that cannot fit into main computer memory is partitioned into disjoint subsets for a more efficient analysis. In the proposed method, at each boosting round the classifiers are first learned from disjoint datasets and then exchanged amongst the sites. Finally the classifiers are combined into a weighted voting ensemble on each disjoint data set. The ensemble that is applied to an unseen test set represents an ensemble of ensembles built on all distributed sites. In experiments performed on four large data sets the proposed distributed boosting method achieved classification accuracy comparable or even slightly better than the standard boosting algorithm while requiring less memory and less computational time. In addition, the communication overhead of the distributed boosting algorithm is very small making it a viable alternative to the standard boosting for large-scale databases.

Approximation algorithms for TSP with neighborhoods in the plane

Abstract: In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.

A classification of topologically stable Poisson structures on a compact oriented structure

Abstract: Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. W construct a complete set of invriants classifying these structures up to an orient-preserving Poisson isomorphism. We show that there is a set of non-trivial infinitesimal deformations which generate the second Poisson cohomology and such that each of the deformations changes exactly one of the classifying invarients. As an example, we consider Poisson structures on the sphere which vanish linearly on a set of smooth closed disjoint curves.

The clique partitioning problem: Facets and patching facets

Abstract: The clique partitioning problem (CPP) can be formulated as follows: Given is a complete graph G = (V, E), with edge weights wij ∈ ℝ for all {i, j} ∈ E. A subset A ⊆ E is called a clique partition if there is a partition of V into nonempty, disjoint sets V1,…, Vk, such that each Vp (p = 1,…, k) induces a clique (i.e., a complete subgraph), and A = ∪ {{i, j}|i, j ∈ Vp, i ≠ j}. The weight of such a clique partition A is defined as Σ{i,j}∈Awij. The problem is now to find a clique partition of maximum weight. The clique partitioning polytope P is the convex hull of the incidence vectors of all clique partitions of G. In this paper, we introduce several new classes of facet-defining inequalities of P. These suffice to characterize all facet-defining inequalities with right-hand side 1 or 2. Also, we present a procedure, called patching, which is able to construct new facets by making use of already-known facet-defining inequalities. A variant of this procedure is shown to run in polynomial time. Finally, we give limited empirical evidence that the facet-defining inequalities presented here can be of use in a cutting-plane approach for the clique partitioning problem. © 2001 John Wiley & Sons, Inc.

An improved approximation scheme for the Group Steiner Problem

Abstract: We address a practical problem which arises in several areas, including network design and VLSI circuit layout Given an undirected weighted graph G = (V, E) and a family N = {N 1 , , N k } of k disjoint groups of nodes N i ⊆ V, the Group Steiner Problem asks for a minimumcost tree which contains at least one node from each group N i- In this paper, we give polynomial-time O(k e )-approximation algorithms for any fixed e > 0 This result improves the previously known O(k)-approximation We also apply our approximation algorithms to the Steiner problem in directed graphs, while guaranteeing the same performance ratio

Graph Imperfection

Abstract: We are interested in colouring a graph G=(V, E) together with an integral weight or demand vector x=(xv:v?V) in such a way that xv colours are assigned to each node v, adjacent nodes are coloured with disjoint sets of colours, and we use as few colours as possible. Such problems arise in the design of cellular communication systems, when radio channels must be assigned to transmitters to satisfy demand and avoid interference. We are particularly interested in the ratio of chromatic number to clique number when some weights are large. We introduce a relevant new graph invariant, the “imperfection ratio” imp(G) of a graph G, present alternative equivalent descriptions, and show some basic properties. For example, imp(G)=1 if and only if G is perfect, imp(G)=imp(G) where G denotes the complement of G, and imp(G)=g/(g?1) for any line graph G where g is the minimum length of an odd hole (assuming there is an odd hole).

Symmetry for exterior elliptic problems and two conjectures in potential theory

Abstract: In this paper we extend a classical result of Serrin to a class of elliptic problems Δu+f(u,|∇u|)=0 in exterior domains R N ⧹G (or Ω⧹G with Ω and G bounded). In case G is an union of a finite number of disjoint C2-domains Gi and u=ai>0, ∂u/∂n=αi⩽0 on ∂Gi, u→0 at infinity, we show that if a non-negative solution of such a problem exists, then G has only one component and it is a ball. As a consequence we establish two results in electrostatics and capillarity theory. We further obtain symmetry results for quasilinear elliptic equations in the exterior of a ball.

On reducibility and symmetry of disjoint NP-pairs

Abstract: We consider some problems about pairs of disjoint NP sets. The theory of these sets with a natural concept of reducibility is, on the one hand, closely related to the theory of proof systems for propositional calculus, and, on the other, it resembles the theory of NP completeness. Furthermore, such pairs are important in cryptography. Among others, we prove that the Broken Mosquito Screen pair of disjoint NP sets can be polynomially reduced to Clique-Coloring pair and thus is polynomially separable and we show that the pair of disjoint NP sets canonically associated with the Resolution proof system is symmetric.

Online Searching

Abstract: We consider the problem of searching m branches which, with the exception of a common source s, are disjoint hereafter called concurrent branches. A searcher, starting at s, must find a given “exit” t whose location, unknown to the searcher, is on one of the m branches. The problem is to find a strategy that minimizes the worst-case ratio between the total distance traveled and the length of the shortest path from s to t. Additional information may be available to the searcher before he begins his search. In addition to finding optimal or near optimal deterministic online algorithms for these problems, this paper addresses the “value” of getting additional information before starting the search.

An iterative rounding 2-approximation algorithm for the element connectivity problem

Abstract: In the survivable network design problem (SNDP), given an undirected graph and values r/sub ij/ for each pair of vertices i and j, we attempt to find a minimum-cost subgraph such that there are r/sub ij/ disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. K. Jain et al. (1999) propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELC-SNDP, or ELC). These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 (K. Jain, 2001), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O (log k) approximation algorithm, where k=max/sub i,j/ r/sub ij/. VC-SNDP is not known to have a non-trivial approximation algorithm; however, recently L. Fleischer (2001) has shown how to extend the technique of K. Jain ( 2001) to give a 2-approximation algorithm in the case that r/sub ij//spl isin/{0, 1, 2}. She also shows that the same techniques will not work for VC-SNDP for more general values of r/sub ij/. The authors show that these techniques can be extended to a 2-approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures.

Edge-Disjoint Paths in Expander Graphs

Abstract: Given a graph G=(V,E)and a set of $\kappa$ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of $\kappa$ paths so found is edge-disjoint. For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed. We present a polynomial time randomized algorithm for finding edge-disjoint paths in an r-regular expander graph G. We show that if G has sufficiently strong expansion properties and r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined. This is within a constant factor of best possible.

Creating ensembles of classifiers

Abstract: Ensembles of classifiers offer promise in increasing overall classification accuracy. The availability of extremely large datasets has opened avenues for application of distributed and/or parallel learning to efficiently learn models of them. In this paper, distributed learning is done by training classifiers on disjoint subsets of the data. We examine a random partitioning method to create disjoint subsets and propose a more intelligent way of partitioning into disjoint subsets using clustering. It was observed that the intelligent method of partitioning generally performs better than random partitioning for our datasets. In both methods a significant gain in accuracy may be obtained by applying bagging to each of the disjoint subsets, creating multiple diverse classifiers. The significance of our finding is that a partition strategy for even small/moderate sized datasets when combined with bagging can yield better performance than applying a single learner using the entire dataset.

Heterogeneous coupling by virtual control methods

Abstract: The virtual control method, recently introduced to approximate elliptic and parabolic problems by overlapping domain decompositions (see [7–9]), is proposed here for heterogeneous problems. Precisely, we address the coupling of an advection equation with a diffusion-advection equation, with the aim of modelling boundary layers. We investigate both overlapping and non-overlapping (disjoint) subdomain decompositions. In the latter case, several cost functions are considered and a numerical assessment of our theoretical conclusions is carried out.

A classification of topologically stable Poisson structures on a compact oriented surface

Abstract: Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. We construct a complete set of invariants classifying these structures up to an orientation-preserving Poisson isomorphism. We show that there is a set of non-trivial infinitesimal deformations which generate the second Poisson cohomology and such that each of the deformations changes exactly one of the classifying invariants. As an example, we consider Poisson structures on the sphere which vanish linearly on a set of smooth closed disjoint curves.

Separating objects in the plane by wedges and strips

Abstract: In this paper we study the separability of two disjoint sets of objects in the plane according to two criteria: wedge separability and strip separability. We give algorithms for computing all the separating wedges and strips, the wedges with the maximum and minimum angle, and the narrowest and the widest strip. The objects we consider are points, segments, polygons and circles. As applications, we improve the computation of all the largest circles separating two sets of line segments by a logn factor, and we generalize the algorithm for computing the minimum polygonal separator of two sets of points to two sets of line segments with the same running time. ? 2001 Elsevier Science B.V. All rights reserved.

Menger's Theorem

Abstract: Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t + 1 disjoint A–B-paths in D. Here a short and elementary proof of a more general theorem is given. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 35–36, 2001