Showing papers on "Connectivity published in 1998"
Patent•
09 Apr 1998
TL;DR: In this article, a set of documents are ranked according to their content and their connectivity by using topic distillation, and a relevance weight is correspondingly assigned to each node, and nodes in the second subset having relevance weight less than a predetermined threshold are pruned from the graph.
Abstract: In a computerized method, a set of documents is ranked according to their content and their connectivity by using topic distillation. The documents include links that connect the documents to each other, either directly, or indirectly. A graph is constructed in a memory of a computer system. In the graph, nodes represent the documents, and directed edges represent the links. Based on the number of links connecting the various nodes, a subset of documents is selected to form a topic. A second subset of the documents is chosen based on the number of directed edges connecting the nodes. Nodes in the second subset are compared with the topic to determine similarity to the topic, and a relevance weight is correspondingly assigned to each node. Nodes in the second subset having a relevance weight less than a predetermined threshold are pruned from the graph. The documents represented by the remaining nodes in the graph are ranked by connectivity based ranking scheme.
243 citations
23 May 1998
TL;DR: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity.
Abstract: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity. Assuming that we start with no edges in a graph with n vertices, the amortized operation costs are O(log 2 n) for connectivity, O(log 4 n) for minimum spanning forest, 2-edge connectivity, and O(log 5 n) biconnectivity.
210 citations
TL;DR: The main theorem asserts that H is minimizable if and only if his bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense that H can be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit.
Abstract: LetH=(T,U) be a connected graph,V?Ta set, andca non-negative function on the unordered pairs of elements ofV. In theminimum0-extension problem(*), one is asked to minimize the inner productcmover all metricsmonVsuch that (i)mcoincides with the distance function ofHwithinT; and (ii) eachv?Vis at zero distance from somes?T, i.e.m(v,s)=0. This problem is known to be NP-hard ifH=K3(as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable ifH=K2(the minimum cut problem) orH=K2,r(the minimum (2,r)-metric problem). We study problem (*) for all fixedH. More precisely, we consider the linear programming relaxation (**) of (*) that is obtained by dropping condition (ii) above, and callHminimizableif the minima in (*) and (**) coincide for allVandc. Note that for such anHproblem (*) is solvable in strongly polynomial time. Our main theorem asserts thatHis minimizable if and only ifHis bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense thatHcan be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics. Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in 4, we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphsH. Other results are also presented.
108 citations
TL;DR: In this paper, the algebraic connectivity of a weighted connected graph is investigated when the graph is perturbed by removing one or more connected components at a vertex and replacing this collection by a single connected component.
Abstract: The main problem of interest is to investigate how the algebraic connectivity o f a weighted connected graph behaves when the graph is perturbed by removing one or more connected components at a xed vertex and replacing this collection by a single connected component. This analysis leads to exhibiting the unique up to isomorphismtrees on n vertices with speciied diameter that maximize and minimize the algebraic connectivity o ver all such trees. When the radius of a graph is the speciied constraint the unique minimizer of the algebraic connectivity o ver all such graphs is also determined. Analogous results are proved for unicyclic graphs with xed girth. In particular, the unique minimizer and maximizer of the algebraic connectivity is given over all such graphs with girth 3.
105 citations
TL;DR: The first polynomial-time approximation algorithm was given by Williamson et al. as mentioned in this paper, which is based on a combinatorial characterization of the "redundant" edges.
Abstract: The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph ofn vertices and connectivity requirements that are at mostk. Both algorithms find a solution that is within a factor 2k − 1 of optimal fork ⩾ 2 and a factor 2 of optimal fork = 1. Our algorithm improves the time from O(k
3n4) to O
$$(k^2 n^2 + kn^2 \sqrt {\log \log n} )$$
). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the “redundant” edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (s, t) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
83 citations
TL;DR: Under a simple assumption about the configuration space, it is shown that it is possible to perform preprocessing following which queries can be answered quickly, and a theoretical basis for explaining this empirical success is initiated.
74 citations
20 Apr 1998
TL;DR: A linear time algorithm is developed for the following problem: Given a graph G and a hierarchical clustering of the vertices such that all clusters induce connected subgraphs, determine whether G may be embedded into the plane such that no cluster has a hole.
Abstract: We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices such that all clusters induce connected subgraphs, determine whether G may be embedded into the plane such that no cluster has a hole.
70 citations
TL;DR: The aim of this paper is to investigate the kind of properties that a dependency model must verify in order to be equivalent to a singly connected graph structure, as a way of driving automated discovery and construction of singlyconnected networks in data.
Abstract: Graphical structures such as Bayesian networks or Markov networks are very useful tools for representing irrelevance or independency relationships, and they may be used to ec ciently perform reasoning tasks. Singly connected networks are important speci® c cases where there is no more than one undirected path connecting each pair of variables. The aim of this paper is to investigate the kind of properties that a dependency model must verify in order to be equivalent to a singly connected graph structure, as a way of driving automated discovery and construction of singly connected networks in data. The main results are the characterizations of those dependency models which are isomorphic to singly connected graphs (either via the d-separation criterion for directed acyclic graphs or via the separation criterion for undirected graphs), as well as the development of ec cient algorithms for learning singly connected graph representations of dependency models.
68 citations
01 Aug 1998
TL;DR: Self-organizing graphs are presented, a novel approach to graph layout based on a competitive learning algorithm that is very flexibly adaptable to arbitrary types of visualization spaces, for it is explicitly parameterized by a metric model of the layout space.
Abstract: The paper presents self-organizing graphs, a novel approach to graph layout based on a competitive learning algorithm. This method is an extension of self-organization strategies known from unsupervised neural networks, namely from Kohonen's self-organizing map. Its main advantage is that it is very flexibly adaptable to arbitrary types of visualization spaces, for it is explicitly parameterized by a metric model of the layout space. Yet the method consumes comparatively little computational resources and does not need any heavy-duty preprocessing. Unlike with other stochastic layout algorithms, not even the costly repeated evaluation of an objective function is required. To our knowledge this is the first connectionist approach to graph layout. The paper presents applications to 2D-layout as well as to 3D-layout and to layout in arbitrary metric spaces, such as networks on spherical surfaces.
67 citations
TL;DR: This work presents the first known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity, which can use as little as n/2^{\Theta(\sqrt{\log n})}$ space while still running in polynometric time.
Abstract: Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the first known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For $n$-vertex graphs, our algorithm can use as little as $n/2^{\Theta(\sqrt{\log n})}$ space while still running in polynomial time.
53 citations
01 Aug 1998
TL;DR: This work converts the problem of visualizing interconnections in railroad systems into a graph layout problem and exploits the generality of random field layout models for its solution.
Abstract: We are concerned with the problem of visualizing interconnections in railroad systems. The real-world systems we have to deal with contain connections of thousands of trains. To visualize such a system from a given set of time tables a so-called train graph is used. It contains a vertex for each station met by any train, and one edge between every pair of vertices connected by some train running from one station to the other without halting in between.In visualizations of train graphs, positions of vertices are predetermined, since each station has a given geographical location. If all edges are represented by straight-lines, the result is visual clutter with many overlaps and small angles between pairs of lines. We here present a non-uniform approach using different representations for edges of distinct meaning in the exploration of the data. Only edges of certain type are represented by straight-lines, whereas so-called transitive edges are rendered using Bezier curves. The layout problem then consists of placing control points for these curves. We transform it into a graph layout problem and exploit the generality of random field layout models for its solution.
01 Aug 1998
TL;DR: In this paper, the authors presented a correct linear time level planarity testing algorithm that is based on two main new techniques that replace the incorrect crucial parts of the algorithm of Heath and Pemmaraju (1996a,b).
Abstract: In a leveled directed acyclic graph G = (V,E) the vertex set V is partitioned into k ? |V| levels V1; V2,...,Vk such that for each edge (u, v) ? E with u ? i and v ? j we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level i, all v ? i are drawn on the line li = {(x, k-i) | x ? R}, the edges are drawn monotone with respect to the vertical direction, and no edges intersect except at their end vertices. If G has a single source, the test can be performed in O(|V|) time by an algorithm of Di Battista and Nardelli (1988) that uses the PQ-tree data structure introduced by Booth and Lueker (1976). PQ-trees have also been proposed by Heath and Pemmaraju (1996a,b) to test level planarity of leveled directed acyclic graphs with several sources and sinks. It has been shown in Junger, Leipert, and Mutzel (1997) that this algorithm is not correct in the sense that it does not state correctly level planarity of every level planar graph. In this paper, we present a correct linear time level planarity testing algorithm that is based on two main new techniques that replace the incorrect crucial parts of the algorithm of Heath and Pemmaraju (1996a,b).
25 Feb 1998
TL;DR: In this paper, it was shown that searching a width k maze is complete for Π k, i.e., for the k'th level of the AC 0 hierarchy.
Abstract: We show that searching a width k maze is complete for Π k, ie, for the k'th level of the AC 0 hierarchy Equivalently, st-connectivity for width k grid graphs is complete for Π k As an application, we show that there is a data structure solving dynamic st-connectivity for con stant width grid graphs with time bound O(log log n) per operation on a random access machine The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools
TL;DR: The algebraic connectivity of a connected graph is the second smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigen value as discussed by the authors.
Abstract: The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedler's original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure.
TL;DR: In this article, graph theoretical algorithms are presented for finite element nodal ordering to optimize the bandwidth of stiffness matrices of two-and three-dimensional finite element models, and the efficiency of these methods are compared.
01 Aug 1998
TL;DR: JIGGLE is a Java-based platform for experimenting with numerical optimization approaches to general graph layout that includes an implementation of the Barnes-Hut tree code to quickly compute internode repulsion forces for large graphs and an optimization procedure based on the conjugate gradient method.
Abstract: JIGGLE is a Java-based platform for experimenting with numerical optimization approaches to general graph layout. It can draw graphs with undirected edges, directed edges, or a mix of both. Its features include an implementation of the Barnes-Hut tree code to quickly compute internode repulsion forces for large graphs and an optimization procedure based on the conjugate gradient method. JIGGLE can be accessed on the World Wide Web at http://www.cs.cmu.edu/~quixote.
TL;DR: The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max-min problem) is considered and it is shown that the problem is NP-hard for rectangles with at least three rows.
Abstract: The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max-min problem) is considered. It is shown that the problem is NP-hard for rectangles with at least three rows. A shifting algorithm is given which approximates the optimal solution. Bounds for the relative error are determined under a posteriori hypotheses. A further shifting algorithm is also given which allows for error estimates under a priori hypotheses and for asymptotic error estimates. A similar approach can be taken with the problem of finding the partition whose largest component is as small as possible (the min-max problem). The case of rectangles with two rows has a polynomial algorithm and is dealt with in another paper.
01 Aug 1998
TL;DR: This paper introduces a framework for defining and validating metrics to measure the difference between two drawings of the same graph.
Abstract: Preserving the "mental map" is major goal of interactive graph drawing algorithms. Several models have been proposed for formalizing the notion of mental map. Additional work needs to be done to formulate and validate "difference" metrics which can be used in practice. This paper introduces a framework for defining and validating metrics to measure the difference between two drawings of the same graph.
TL;DR: In this paper, the hardness of finding spanners with the number of edges close to the optimum was shown to be at least as hard as the set cover problem, for every fixed k approximating the spanner problem.
Abstract: A k-spanner of a connected graph G = (V, E) is a subgraph G' consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G' is larger than that distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with the number of edges close to the optimum. It is proved that for every fixed k approximating the spanner problem is at least as hard as approximating the set cover problem
Journal Article•
TL;DR: For even n 4 and odd n 5, it was shown in this paper that oe(W n ) = n 2 + 2 for wheels W n of order n+1 and size m = 2n, where m is the size of the graph.
Abstract: A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) +L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number oe(H) of a connected graph H is the least number r of isolated vertices K r such that G = H+K r is a sum graph. It is clear that if H is of size m, then oe(H) m. Recently Hartsfield and Smyth showed that for wheels W n of order n+1 and size m = 2n, oe(W n ) 2 Theta(m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we refine these results to show that for even n 4, oe(W n ) = n=2 + 2, while for odd n 5 we disprove a conjecture of Hartsfield and Smyth by showing that oe(W n ) = n. Labellings are given that achieve these minima.
01 Aug 1998
TL;DR: This paper proposes a multidrawing approach to graph drawing, and presents a proof-of-concept implementation with which it is shown that diverse selections of symmetric-looking drawings for small graphs can be produced.
Abstract: This paper proposes a multidrawing approach to graph drawing. Current graph-drawing systems typically produce only one drawing of a graph. By contrast, the multidrawing approach calls for systematically producing many drawings of the same graph, where the drawings presented to the user represent a balance between aesthetics and diversity. This addresses a fundamental problem in graph drawing, namely, how to avoid requiring the user to specify formally and precisely all the characteristics of a single "nice" drawing. We present a proof-of-concept implementation with which we produce diverse selections of symmetric-looking drawings for small graphs.
TL;DR: It is explained why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications, and how to exploit lookahead to circumvent their inherent complexity.
Abstract: Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms. In many applications of dynamic (di)graph problems, a certain form of lookahead is available. Specifically, we consider the problems of assembly planning in robotics and the maintenance of relations in databases. These give rise to dynamic strong connectivity and dynamic transitive closure problems, respectively. We explain why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient dynamic algorithms for strong connectivity and transitive closure.
TL;DR: Efficient O(n) algorithms are given to compute a possible swap for e that minimizes the diameter of the new spanning tree of G that arises in high-speed networks, particularly in optical networks.
Abstract: Given a graph G with m edges and n nodes, a spanning tree T of G , and an edge e that is being deleted from or inserted into G , we give efficient O(n) algorithms to compute a possible swap for e that minimizes the diameter of the new spanning tree. This problem arises in high-speed networks, particularly in optical networks.
TL;DR: A class of decomposition algorithms for MCG are introduced that decompose MCG into a number of small CMCGs by adding vertices one at a time and building a partial graph and results show that these heuristics are very effective in reducing computation.
Abstract: Given a positive integer R and a weight for each vertex in a graph, the maximum-weight connected graph problem (MCG) is to find a connected subgraph with R vertices that maximizes the sum of their weights. MCG has applications to communication network design and facility expansion. The constrained MCG (CMCG) is MCG with a constraint that one predetermined vertex must be included in the solution. In this paper, we introduce a class of decomposition algorithms for MCG. These algorithms decompose MCG into a number of small CMCGs by adding vertices one at a time and building a partial graph. They differ in the ordering of adding vertices. Proving that finding an ordering that gives the minimum number of CMCGs is NP-complete, we present three heuristic algorithms. Experimental results show that these heuristics are very effective in reducing computation and that different orderings can significantly affect the number of CMCGs to be solved. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 817–837, 1998
24 Aug 1998
TL;DR: It is proved that Graph Connectivity is not in Monadic NP even in the presence of a built-in relation of arbitrary degree that does not have for an arbitrary, but fixed k ≥ 2 ∈ IN the complete graph K k as a minor.
Abstract: In our paper, we prove that Graph Connectivity is not in Monadic NP even in the presence of a built-in relation of arbitrary degree that does not have for an arbitrary, but fixed k ≥ 2 ∈ IN the complete graph K k as a minor. We obtain our result by using the method of indiscernibles and giving a winning strategy for the duplicator in the Ajtai-Fagin Ehrenfeucht-Fraisse Game .
TL;DR: It is shown that every two longest cycles of a connected graph meet in at leastck3/5vertices, where c?0.2615 is the minimum number of vertices.
TL;DR: In this article, it was proved that a connected graph G is geodetic if and only if there exists a binary operation associated with G which fulfils a certain set of four axioms.
Abstract: We say that a binary operation * is associated with a (finite undirected) graph G (without loops and multiple edges) if * is defined on V(G) and uv ∈ E(G) if and only if u ≠ v, u * v = v and v * u = u for any u, v ∈ V(G). In the paper it is proved that a connected graph G is geodetic if and only if there exists a binary operation associated with G which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).
TL;DR: A cost-effective method for the recursive bisection of two-dimensional unstructured grids into 2n subdomains is introduced and is capable of generating evenly loaded disjoint mesh subsets with small interface length, that can be efficiently processed in parallel, on distributed memory platforms.
01 Aug 1998
TL;DR: The algorithms that are presented can be differentiated as resulting from three different approaches to creating 3D-drawings; these approaches edge-lifting, half-edge- lifting, and three-phase-method are called.
Abstract: In this paper, we study orthogonal graph drawings in three dimensions with nodes drawn as boxes. The algorithms that we present can be differentiated as resulting from three different approaches to creating 3D-drawings; we call these approaches edge-lifting, half-edge-lifting, and three-phase-method.Let G be a graph with n vertices, m edges, and maximum degree ?. We obtain a drawing of G in an n ? n ? ?-grid where the surface area of the box of a node v is O(deg(v)); this improves significantly on previous results. We also consider drawings with at most one node per grid-plane, and exhibit constructions in an n ? n ? m-grid and a lower bound of ?(m2); hence upper and lower bounds match for graphs with ?(n2) edges.