Showing papers on "Connectivity published in 1999"

The web as a graph: measurements, models, and methods

Abstract: The pages and hyperlinks of the World-Wide Web may be viewed as nodes and edges in a directed graph. This graph is a fascinating object of study: it has several hundred million nodes today, over a billion links, and appears to grow exponentially with time. There are many reasons -- mathematical, sociological, and commercial -- for studying the evolution of this graph. In this paper we begin by describing two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery. We then report a number of measurements and properties of this graph that manifested themselves as we ran these algorithms on the Web. Finally, we observe that traditional random graph models do not explain these observations, and we propose a new family of random graph models. These models point to a rich new sub-field of the study of random graphs, and raise questions about the analysis of graph algorithms on the Web.

An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree

Abstract: Given a graph G=(V,E) and a tree T =(V,F) with E ∩ F = φ such that G + T =(V,F ∪ E) is 2-edge-connected, we consider the problem of finding a smallest 2-edge-connected spanning subgraph (V,F ∪ E') of G + T containing T The problem, which is known to be NP-hard, admits a 2-approximation algorithm However, obtaining a factor better than 2 for this problem has been one of the main open problems in the graph augmentation problem In this paper, we show that the problem is (1875 + e)-approximable in O(n½m + n2) time for any constant e < 0, where n = |V| and m = |E ∪ F|

Graph Clustering Using Distance-k Cliques

Abstract: Identifying the natural clusters of nodes in a graph and treating them as supernodes or metanodes for a higher level graph (or an abstract graph) is a technique used for the reduction of visual complexity of graphs with a large number of nodes. In this paper we report on the implementation of a clustering algorithm based on the idea of distance-k cliques, a generalization of the idea of the cliques in graphs. The performance of the clustering algorithm on some large graphs obtained from the archives of Bell Laboratories is presented.

Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings

Abstract: Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using 4/3m-1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most $(2.5+2\log{3})\min\{n,f\}-7$ bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.

On the Nature of Structure and Its Identification

Abstract: When working on systems of the real world, abstractions in the form of graphs have proven a superior modeling and representation approach. This paper is on the analysis of such graphs. Based on the paradigm that a graph of a system contains information about the system's structure, the paper contributes within the following respects: 1. It introduces a new and lucid structure measure, the so-called weighted partial connectivity, Λ, whose maximization defines a graph's structure (Section 2). 2. It presents a fast algorithm that approximates a graph's optimum Λ-value (Section 3). Moreover, the proposed structure definition is compared to existing clustering approaches (Section 4), resulting in a new splitting theorem concerning the well-known minimum cut splitting measure. A key concept of the proposed structure definition is its implicit determination of an optimum number of clusters. Different applications, which illustrate the usability of the measure and the algorithm, round off the paper (Section 5).

An O ( n 2 ) self-stabilizing algorithm for computing bridge-connected components

Abstract: The basic idea of our new approach is to determine in a first step for each node those pairs of nodes which allow a good interpolation of the unknowns located at this node. These pairs of neighbor nodes (in some cases only one node) are called parent nodes. This is done by solving a local minimization problem which, in addition, yields the interpolation and restriction coefficients. The construction scheme has been generalized to systems of convection-diffusion-reaction equations using a point-block approach. After these suitable pairs of parent nodes have been determined, the nodes are labeled as C- and F-nodes such that each F-node can be interpolated using one of these suitable pairs of parent nodes and the already computed coefficients. Additionally, a simple heuristic algorithm tries to minimize the number of C-nodes and the number of non-zero entries in the coarse grid matrix. The algorithm has been parallelized and shows mesh size independent convergence for standard model problems. Realistic numerical experiments confirm the efficiency of the presented algorithm.

Increasing the rooted-connectivity of a digraph by one

Abstract: In order to unify these approaches, here we describe a two-phase greedy algorithm working on a slight extension of lattice polyhedra. This framework includes the rooted node-connectivity augmentation problem, as well, and hence the resulting algorithm, when appropriately specialized, finds a minimum cost of new edges whose addition to a digraph increases its rooted connectivity by one. The only known algorithm for this problem used submodular flows. Actually, the specialized algorithm solves an extension of the rooted edge-connectivity and node-connectivity augmentation problem.

Level Planar Embedding in linear Time

Abstract: In a level 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 ∈ Vi and v ∈; Vj 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 Vi, all v ∈ Vi are drawn on the line li = {(x, k - i) | x ∈ ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each Vi (1 ≤ i ≤ k). We present an O(|V |) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Junger, Leipert, and Mutzel [6].

On some balanced, totally balanced and submodular delivery games

Abstract: This paper studies a class of delivery problems associated with the Chinese postman problem and a corresponding class of delivery games. A delivery problem in this class is determined by a connected graph, a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office. It is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server, who starts at the post office and is expected to traverse all edges in the graph before returning to the post office. A graph G is called Chinese postman-submodular, or, for short, CP-submodular (CP-totally balanced, CP-balanced, respectively) if for each delivery problem in which G is the underlying graph the associated delivery game is submodular (totally balanced, balanced, respectively). For undirected graphs we prove that CP-submodular graphs and CP-totally balanced graphs are weakly cyclic graphs and conversely. An undirected graph is shown to be CP-balanced if and only if it is a weakly Euler graph. For directed graphs, CP-submodular graphs can be characterized by directed weakly cyclic graphs. Further, it is proven that any strongly connected directed graph is CP-balanced. For mixed graphs it is shown that a graph is CP-submodular if and only if it is a mixed weakly cyclic graph. Finally, we note that undirected, directed and mixed weakly cyclic graphs can be recognized in linear time.

Rectangle of influence drawings of graphs without filled 3-cycles

Abstract: In this paper, we study rectangle of influence drawings, i.e., drawings of graphs such that for any edge the axis-parallel rectangle defined by the two endpoints of the edge is empty. Specifically, we show that if G is a planar graph without filled 3-cycles, i.e., a planar graph that can be drawn such that the interior of every 3-cycle is empty, then G has a rectangle of influence drawing.

Covering symmetric supermodular functions by graphs

Abstract: The minimum number of edges of an undirected graph covering a symmetric, supermodular set-function is determined. As a special case, we derive an extension of a theorem of J. Bang-Jensen and B. Jackson on hypergraph connectivity augmentation.

The Vertex-Exchange Graph: A New Concept for Multi-level Crossing Minimisation

Abstract: In this paper we consider the problems of testing a multi- level graph for planarity and laying out a multi-level graph.We introduce a new abstraction that we call a vertex-exchange graph. We demonstrate how this concept can be used to solve these problems by providing clear and simple algorithms for testing a multi-level graph for planarity and laying out a multi-level graph when planar.We also show how the concept can be used to solve other problems relating to multi-level graph layout.

A Graph Based Method for Generating the Fiedler Vector of Irregular Problems

Abstract: In this paper we present new algorithms for spectral graph partitioning. Previously, the best partitioning methods were based on a combination of Combinatorial algorithms and application of the Lanczos method when the graph allows this method to be cheap enough. Our new algorithms are purely spectral. They calculate the Fiedler vector of the original graph and use the information about the problem in the form of a preconditioner for the graph Laplacian. In addition, we use a favorable subspace for starting the Davidson algorithm and reordering of variables for locality of memory references.

A uniform framework for approximating weighted connectivity problems

Abstract: We introduce a new algorithmic technique that applies to several graph connectivity problems. Its power is demonstrated by experimental studies of the minimum-weight strongly-connected spanning subgraph problem and the minimum-weight augmentation problem. Even though we are unable to improve the approximation ratios for these problems, OUT studies indicate that the new method generates significantly better solutions than the current known approximation algorithms, and yields solutions very close to optimal. We believe that our technique will eventually lead to algorithms that improve the performance ratios as well.

Augmenting hypergraphs by edges of size two

Abstract: We give a good characterization for the minimum number of edges of size two whose addition to a given hypergraph H makes it k-edge-connected. Our result extends a previous theorem by E. Cheng [4] for the case when H is already (k - 1)-edge-connected. We also describe a strongly polynomial algorithm to find both a minimum cardinality augmentation, and a minimum cost augmentation where the cost of an edge is equal to the sum of the weights of its endvertices, for some given weight function on V(H). Our proof is based on a new ‘splitting’ theorem for hypergraphs for the case when the special vertex s is only incident with edges of size two. This theorem generalizes a ‘splitting’ result of Lovasz for graphs.

Center, median, and centroid subgraphs

Abstract: The median and centroid of an arbitrary graph G are two different generalizations of the branch weight centroid of a tree. As such, they are closely related, but they can actually be disjoint. On the one hand, they are, for example, always contained in the same block of any connected graph G. However, they can be arbitrarily far apart. Specifically, given any three graphs H, J, and K, and a positive integer k ≥ 4, there exists a graph G with center, median, and centroid subgraphs isomorphic to H, J, and K, respectively, and the distance between any two of these subgraphs is at least k. © 1999 John Wiley & Sons, Inc. Networks 34: 303–311, 1999

Broadcasting in grid graphs

Abstract: This work consists of two separate parts. The first part deals with the problem of multiple message broadcasting, and the topic of the second part is line broadcasting. Broadcasting is a process in which one vertex in a graph knows one or more messages. The goal is to inform all remaining vertices as fast as possible. In this work we consider a special kind of graphs, grids. In 1980 A. M. Farley showed that the minimum time required to broadcast a set of M messages in any connected graph with diameter d is d + 2(M − 1). This work presents an approach to broadcasting multiple messages from the corner vertex of a 2-dimensional grid. This approach gives us a broadcasting scheme that differs only by 2 (and in the case of an even x even grid by only 1) from the above lower bound. Line broadcasting describes a different variant of the broadcasting process. A. M. Farley showed that line broadcasting can always be completed in [log n] time units in any connected graph on n vertices. He defined three different cost measures for line broadcasting. This work presents strategies for minimizing those costs for various grid sizes.

A Randomized Time-Work Optimal Parallel Algorithm for Finding a Minimum Spanning Forest

Abstract: We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an EREW PRAM. This result is optimal with respect to both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures.

Successive edge-connectivity augmentation problems

Abstract: ,G1,G2,... of supergraphs of G such that Gi is a subgraph of Gj for any i

On the connectivity and the conditional diameter of graphs and digraphs

Abstract: Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D ≤ 2l- 1, then G has maximum connectivity (κ = δ), and if D ≤ 2l, then it attains maximum edge-connectivity (λ = δ), where l is a parameter which can be thought of as a generalization of the girth of a graph. In this paper, we study some similar conditions for a digraph to attain high connectivities, which are given in terms of what we call the conditional diameter or P-diameter of G. This parameter measures how far apart can be a pair of subdigraphs satisfying a given property P, and, hence, it generalizes the standard concept of diameter. As a corollary, some new sufficient conditions to attain maximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bipartite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since l ≥ 1, some known results of Plesnik and Znam are either reobtained or improved. For instance, it is shown that any graph whose line graph has diameter D = 2 (respectively, D ≤ 3) has maximum connectivity (respectively, edge-connectivity.) Moreover, for graphs with even girth and minimum degree large enough, we obtain a lower bound on their connectivities.

Directed vertex-connectivity augmentation

Abstract: The problem of finding a smallest set of new edges to be added to a given directed graph to make it k-vertex-connected was shown to be polynomially solvable recently in [6] for any target connectivity k ≤ 1. However, the algorithm given there relied on the ellipsoid method. Here we refine some results of [6] which makes it possible to construct a combinatorial algorithm which is polynomial for any fixed k. Short proofs for (extensions of) some earlier results related to this problem will also be given.

Hypergraph connectivity augmentation

Abstract: The hypergraph augmentation problem is to augment a hypergraph by hyperedges to meet prescribed local connectivity requirements. We provide here a minimax theorem for this problem. The result is derived from the degree constrained version of the problem by a standard method. We shall construct the required hypergraph for the latter problem by a greedy type algorithm. A similar minimax result will be given for the problem of augmenting a hypergraph by weighted edges (hyperedges of size two with weights) to meet prescribed local connectivity requirements. Moreover, a special case of an earlier result of Schrijver on supermodular colourings shall be derived from our theorem.