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Showing papers on "Connectivity published in 2002"


Proceedings ArticleDOI
09 Jun 2002
TL;DR: The main contribution of this work is a completely distributed algorithm for finding small WCDS's and the performance of this algorithm is shown to be very close to that of the centralized approach.
Abstract: We present a series of approximation algorithms for finding a small weakly-connected dominating set (WCDS) in a given graph to be used in clustering mobile ad hoc networks. The structure of a graph can be simplified using WCDS's and made more succinct for routing in ad hoc networks. The theoretical performance ratio of these algorithms is O(ln Δ) compared to the minimum size WCDS, where Δ is the maximum degree of the input graph. The first two algorithms are based on the centralized approximation algorithms of Guha and Khuller cite guha-khuller-1998 for finding small connected dominating sets (CDS's). The main contribution of this work is a completely distributed algorithm for finding small WCDS's and the performance of this algorithm is shown to be very close to that of the centralized approach. Comparisons between our work and some previous work (CDS-based) are also given in terms of the size of resultant dominating sets and graph connectivity degradation.

286 citations


Journal ArticleDOI
01 Jan 2002-Networks
TL;DR: It is shown that if G is a graph of order n and diameter d then g(G) ≤ n − d + 1 and this bound is sharp and the minimum cardinality among the subsets S of V( G) with I(S) = V(G).
Abstract: For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. The geodetic number g(G) is the minimum cardinality among the subsets S of V(G) with I(S) = V(G). It is shown that if G is a graph of order n and diameter d then g(G) ≤ n − d + 1 and this bound is sharp. For positive integers r, d, and k ≥ 2 with r ≤ d ≤ 2r, there exists a connected graph G of radius r, diameter d, and g(G) = k. Also, for integers n, d, and k with 2 ≤ d < n, 2 ≤ k < n, and n − d − k + 1 ≥ 0, there exists a graph G of order n, diameter d, and g(G) = k. It is shown, for every nontrivial connected graph G, that g(G) ≤ g(G × K2). A sufficient condition for the equality of g(G) and g(G × K2) is presented. A graph F is a minimum geodetic subgraph if there exists a graph G containing F as an induced subgraph such that V(F) is a minimum geodetic set for G. Minimum geodetic subgraphs are characterized. © 2002 John Wiley & Sons, Inc.

170 citations


Book ChapterDOI
25 Aug 2002
TL;DR: This work studies the problem of assigning transmission ranges to the nodes of a multi-hop packet radio network so as to minimize the total power consumed under the constraint that enough power is provided to the node to ensure that the network is connected.
Abstract: We study the problem of assigning transmission ranges to the nodes of a multi-hop packet radio network (also known as static ad hoc wireless network) so as to minimize the total power consumed under the constraint that enough power is provided to the nodes to ensure that the network is connected. Precisely, we require that the bidirectional links established by the transmission range of every node form a connected graph. We call this problem Min-Power Symmetric Connectivity.

129 citations


Journal ArticleDOI
TL;DR: Sharp bounds are made on the value of this parameter, and a construction of graphs whose average connectivity is the same as the connectivity is established, to establish some new results on connectivity.

107 citations


Proceedings ArticleDOI
23 Jun 2002
TL;DR: The mobile version of the problem when d=2 is investigated through extensive simulations, which give insight on how mobility affects connectivity and reveal a useful trade-off between communication capability and energy consumption.
Abstract: We consider the following problem for wireless ad hoc networks: assume n nodes, each capable of communicating with nodes within a radius of r, are distributed in a d-dimensional region of side l; how large must the transmitting range r be to ensure that the resulting network is connected? We also consider the mobile version of the problem, in which nodes are allowed to move during a time interval and the value of r ensuring connectedness for a given fraction of the interval must be determined. For the stationary case, we give tight bounds on the relative magnitude of r, n and l yielding a connected graph with high probability in l-dimensional networks, thus solving an open problem. The mobile version of the problem when d=2 is investigated through extensive simulations, which give insight on how mobility affects connectivity and reveal a useful trade-off between communication capability and energy consumption.

105 citations


Book ChapterDOI
03 Apr 2002
TL;DR: The spanning ratio for Gabriel graphs and relative neighborhood graphs, and for points drawn independently from the uniform distribution on the unit square, it is shown that the spanning ratio of the (random) Gabriel graph and all s-skeletons with s ?
Abstract: The spanning ratio of a graph defined on n points in the Euclidean plane is the maximal ratio over all pairs of data points (u, v), of the minimum graph distance between u and v, over the Euclidean distance between u and v. A connected graph is said to be a k-spanner if the spanning ratio does not exceed k. For example, for any k, there exists a point set whose minimum spanning tree is not a k-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42- spanner[11]. For proximity graphs inbetween these two extremes, such as Gabriel graphs[8], relative neighborhood graphs[16] and s-skeletons[12] with s ? [0, 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are s-skeletons with s = 1) is ?(?n) in the worst case. For all s-skeletons with s ? [0, 1], we prove that the spanning ratio is at most O(n?) where ? = (1 - log2(1 +?1 - s2))/2. For all s-skeletons with s ? [1, 2), we prove that there exist point sets whose spanning ratio is at least (1/2- o(1) ?n. For relative neighborhood graphs[16] (skeletons with s = 2), we show that there exist point sets where the spanning ratio is ?(n). For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all s-skeletons with s ? [1, 2] tends to ? in probability as ?log n/ log log n.

97 citations


Journal ArticleDOI
TL;DR: In this paper, the connectivity distance cn, the smallest x for which Gn(x) is connected, is shown to satisfy ======(1)limn→∞cndnlogn=1,d= 1,12d,d⩾2,a.s.

90 citations


Journal ArticleDOI
TL;DR: The algorithms for Problems (1)—(7) are the first practically relevant parallel algorithms for these standard graph problems, and the number of communication rounds/ supersteps obtained in this paper is independent of the problem size, and grows only logarithmically with respect to p.
Abstract: In this paper we present deterministic parallel algorithms for the coarse-grained multicomputer (CGM) and bulk synchronous parallel (BSP) models for solving the following well-known graph problems: (1) list ranking, (2) Euler tour construction in a tree, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, and (7) 2-edge connectivity and biconnectivity (testing and component computation). The algorithms require O(log p) communication rounds with linear sequential work per round (p = no. processors, N = total input size). Each processor creates, during the entire algorithm, messages of total size O(log (p) (N/p)) . The algorithms assume that the local memory per processor (i.e., N/p ) is larger than p e , for some fixed e > 0 . Our results imply BSP algorithms with O(log p) supersteps, O(g log (p) (N/p)) communication time, and O(log (p) (N/p)) local computation time. It is important to observe that the number of communication rounds/ supersteps obtained in this paper is independent of the problem size, and grows only logarithmically with respect to p . With growing problem size, only the sizes of the messages grow but the total number of messages remains unchanged. Due to the considerable protocol overhead associated with each message transmission, this is an important property. The result for Problem (1) is a considerable improvement over those previously reported. The algorithms for Problems (2)—(7) are the first practically relevant parallel algorithms for these standard graph problems.

85 citations


Book ChapterDOI
17 Sep 2002
TL;DR: In this paper, a robot has to visit all nodes and traverse all edges of an unknown undirected connected graph, using as few edge traversals as possible, and the quality of an exploration algorithm A is measured by comparing its cost compared to that of the optimal algorithm having full knowledge of the graph.
Abstract: A robot has to visit all nodes and traverse all edges of an unknown undirected connected graph, using as few edge traversals as possible. The quality of an exploration algorithm A is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, maximized over all starting nodes in the graph and over all graphs in a given class U, is called the overhead of algorithm A for the class U of graphs. We construct natural exploration algorithms, for various classes of graphs, that have smallest, or - in one case - close to smallest, overhead. An important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms.

68 citations


Book ChapterDOI
08 Jul 2002
TL;DR: The following combinatorial optimization problem is introduced: Given a connected graph G, find a spanning tree T of G with the smallest number of branchv ertices (vertices of degree 3 or more in T).
Abstract: We introduce the following combinatorial optimization problem: Given a connected graph G, find a spanning tree T of G with the smallest number of branchv ertices (vertices of degree 3 or more in T). The problem is motivated by new technologies in the realm of optical networks. We investigate algorithmic and combinatorial aspects of the problem.

63 citations


Journal ArticleDOI
TL;DR: It is proved that the problem of finding a minimum number of new edges E' such that the augmented graph G' is biconnected and has diameter no greater than D is NP-hard for all fixed D, by employing a reduction from the DOMINATING SET problem.
Abstract: Given a graph G=(V,E) and a positive integer D , we consider the problem of finding a minimum number of new edges E' such that the augmented graph G'=(V,E\cup E') is biconnected and has diameter no greater than D. In this note we show that this problem is NP-hard for all fixed D , by employing a reduction from the DOMINATING SET problem. We prove that the problem remains NP-hard even for forests and trees, but in this case we present approximation algorithms with worst-case bounds 3 (for even D ) and 6 (for odd D ). A closely related problem of finding a minimum number of edges such that the augmented graph has diameter no greater than D has been shown to be NP-hard by Schoone et al. [21] when D=3 , and by Li et al. [17] when D=2.

Journal ArticleDOI
TL;DR: It is shown that every pair k,n of integers with 2⩽k ⩽n is realizable as the Steiner number and order of some connected graph.

Journal ArticleDOI
TL;DR: The multi-hierarchical model is formalized, the techniques that have been designed for taking advantage of multiple hierarchies in a hierarchical path search are described, and some experiments and results on the performance of these techniques are presented.
Abstract: The use of hierarchical graph searching for finding paths in graphs is well known in the literature, providing better results than plain graph searching, with respect to computational costs, in many cases. This paper offers a step forward by including multiple hierarchies in a graph-based model. Such a multi-hierarchical model has the following advantages: First, a multiple hierarchy permits us to choose the best hierarchy to solve each search problem; second, when several search problems have to be solved, a multiple hierarchy provides the possibility of solving some of them simultaneously; and third, solutions to the search problems can be expressed in any of the hierarchies of the multiple hierarchy, which allows us to represent the information in the most suitable way for each specific purpose. In general, multiple hierarchies have proven to be a more adaptable model than single-hierarchy or non-hierarchical models. This paper formalizes the multi-hierarchical model, describes the techniques that have been designed for taking advantage of multiple hierarchies in a hierarchical path search, and presents some experiments and results on the performance of these techniques.

Journal ArticleDOI
TL;DR: It is proved that if n ⩾3 g −1, then the graph which uniquely minimizes the algebraic connectivity over G n, g is the unicyclic “lollipop” graph C n , g obtained by appending a g cycle to a pendant vertex of a path on n − g vertices.

Journal ArticleDOI
Karen I. Trovato1, Leo Dorst
TL;DR: The algorithm is developed and an application to robot path planning in configuration space, where the graph topology, transition costs, or start/goals may change simultaneously, is discussed.
Abstract: A* graph search effectively computes the optimal solution path from start nodes to goal nodes in a graph, using a heuristic function. In some applications, the graph may change slightly in the course of its use and the solution path then needs to be updated. Very often, the new solution will differ only slightly from the old. Rather than perform the full A* on the new graph, we compute the necessary OPEN nodes from which the revised solution can be obtained by A*. In this "Differential A*" algorithm, the graph topology, transition costs, or start/goals may change simultaneously. We develop the algorithm and discuss when it gives an improvement over simply reapplying A*. We briefly discuss an application to robot path planning in configuration space, where such graph changes naturally arise.

Journal ArticleDOI
TL;DR: The element connectivity problem is of independent interest, since it models a realistic situation and achieves an approximation guarantee of factor 2Hk, where k is the largest requirement and Hn = 1 + ½ +... + 1/n.

Book ChapterDOI
19 Aug 2002
TL;DR: A new statistical approach for characterizing the class separability degree in Rp is proposed based on a nonparametric statistic called "the Cut Edge Weight", which shows that the classes to predict are non-separable.
Abstract: We propose a new statistical approach for characterizing the class separability degree in Rp. This approach is based on a nonparametric statistic called "the Cut Edge Weight". We show in this paper the principle and the experimental applications of this statistic. First, we build a geometrical connected graph like the Relative Neighborhood Graph of Toussaint on all examples of the learning set. Second, we cut all edges between two examples of a different class. Third, we calculate the relative weight of these cut edges. If the relative weight of the cut edges is in the expected interval of a random distribution of the labels on all the neighborhood graph's vertices, then no neighborhood-based method will give a reliable prediction model. We will say then that the classes to predict are non-separable.

Book ChapterDOI
27 Aug 2002
TL;DR: A parallel GRASP heuristic with path-relinking with parallel implementation for the 2-path network design problem obtains linear speedups on a cluster with 32 machines.
Abstract: We propose a parallel GRASP heuristic with path-relinking for the 2-path network design problem. A parallel strategy for its implementation is described. Computational results illustrating the effectiveness of the new heuristic are reported. The parallel implementation obtains linear speedups on a cluster with 32 machines.

Journal ArticleDOI
TL;DR: In this paper, an approximation algorithm for minimum-cost vertex-connectivity problems was proposed, which relies on a primal-dual approach, which has led to approximation algorithms for many edge connectivity problems.
Abstract: There is an error in our paper ``An Approximation Algorithm for Minimum-Cost Vertex- Connectivity Problems'' (Algorithmica (1997), 18:21—43). In that paper we considered the following problem: given an undirected graph and values r ij for each pair of vertices i and j , find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j . We gave approximation algorithms for two special cases of this problem. Our algorithms rely on a primal—dual approach which has led to approximation algorithms for many edge-connectivity problems. The algorithms work in a series of stages; in each stage an augmentation subroutine augments the connectivity of the current solution. The error is in a lemma for the proof of the performance guarantee of the augmentation subroutine. In the case r ij = k for all i,j , we described a polynomial-time algorithm that claimed to output a solution of cost no more than 2 H (k) times optimal, where H = 1 + 1/2 + · · · + 1/n . This result is erroneous. We describe an example where our primal—dual augmentation subroutine, when augmenting a k -vertex connected graph to a (k+1) -vertex connected graph, gives solutions that are a factor Ω(k) away from the minimum. In the case r ij ∈ {0,1,2} for all i,j , we gave a polynomial-time algorithm which outputs a solution of cost no more than three times the optimal. In this case we prove that the statement in the lemma that was erroneous for the k -vertex connected case does hold, and that the algorithm performs as claimed.

Book ChapterDOI
14 Mar 2002
TL;DR: In this article, the authors present O(n log n) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. And they also generalize these algorithms to O(log n log 2 n) for planar trees and cycles.
Abstract: The maximum detour and spanning ratio of an embedded graph G are values that measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works [5,10]. We also generalize these algorithms to obtain O(n log2 n) time algorithms for computing the maximum detour and spanning ratio of planar trees and cycles.

Journal ArticleDOI
TL;DR: A characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even and it is obtained that every (2k) edge-connected graph with |V| +|E| even has a (k-1)-edge- connected orientation in which the in-degree of every node is odd.
Abstract: Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge- connected graph with |V| + |E| even has a (k-1)-edge- connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge- disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.

Journal ArticleDOI
TL;DR: Borders on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths are given.

Proceedings ArticleDOI
06 Jan 2002
TL;DR: A new combinatorial polynomial-time approximation algorithm to the k-cut problem, with worst-case performance ratio 2.1, is provided, which is provably at least as good as the Naor-Rabani LP lower bound.
Abstract: Let G = (V, E) be an undirected connected graph with edge weights s: E → IR+. An edge set A ⊆ E is called a k-cut if G′ = (V,E\A) has k connected components. The k-cut problem is to compute a minimum weight k-cut in G.The k-cut problem is known to be NP-hard. Saran and Vazirani [3] gave a combinatorial (2-2/k)-approximation which successively finds minimum cuts until the graph is partitioned into k components. Recently, Naor and Rabani [2] gave an integer program formulation of the problem, and showed that its integrality gap is 2.1.1 Our Contributions. We provide a new combinatorial polynomial-time approximation algorithm to the k-cut problem, with worst-case performance ratio 2. We also provide a new combinatorial lower bound, which is provably at least as good as the Naor-Rabani LP lower bound.1.2 Network Strength and Attack. For an edge set A, let k (A) denote the number of connected components in G′ = (V,E\A). Define s(A) = ∑eeAs (e). The strength of the edge set A is defined to be σ(A) := s(A)/(k(A) - 1), and the strength of the graph G is σ(G) :=minA⊆Eσ(A). The strength of a singleton node is defined to be infinity.For an edge set A and a real number b > 0, define gA(b) := s(A) - b(k(A) - 1). Let g(b) := minA⊆EgA(b) be the attack value of the network. We use e(b) to denote the edge set which achieves this minimum. Define k(b) := k(e(b)). Note that g(b) is always non-positive, since letting A = O achieves g(b) = O.Cunningham [1] provided polynomial-time algorithms to compute both the strength and the attack value of a network. We use Cunningham's algorithms as a subroutine in our algorithm.

Journal ArticleDOI
TL;DR: It is proved that the problem of computing the scattering number of a graph is NP-complete and the scattering numbers of grids and those of Cartesian products of two complete graphs are determined.
Abstract: The scattering number of a noncomplete connected graph G is defined by s ( G )= max{ y ( G m X ) m X X ² V ( G ), y ( G m X ) S 2}, where y ( G m X ) denotes the number of components of G m X . Thi...

Journal ArticleDOI
TL;DR: In this paper, the authors study the dynamics of the world-wide web from the growth rules recently proposed in Tadic (Physica A 293 (2001) 273) with appropriate parameters.
Abstract: Using numerical simulations and scaling theory we study the dynamics of the world-wide Web from the growth rules recently proposed in Tadic (Physica A 293 (2001) 273) with appropriate parameters. We demonstrate that the emergence of power-law behavior of the out- and in-degree distributions in the Web involves occurrence of temporal fractal structures that are manifested in the scale-free growth of the local connectivity and in first-return time statistics. We also show how the scale-free behavior occurs in the statistics of random walks on the Web, where the walkers use information on the local graph connectivity.

Journal ArticleDOI
TL;DR: The key to the implementation is the use of graph theoretic techniques to rapidly enumerate this set of closures for this relation and Computational results are presented to suggest the efficiency of this approach.

Journal ArticleDOI
TL;DR: A self-stabilizing algorithm for finding biconnected components of a connected undirected graph on a distributed or network model of computation that is resilient to transient faults, therefore, it does not require initialization.

Book ChapterDOI
08 Jul 2002
TL;DR: This paper presents a polynomial-time algorithm that finds a path of length L, where L denotes the length of the longest simple path in the graph, and establishes the performance ratio O|V|(log log |V |/log |V|)2) for the Longest Path problem, where V denotes the graph's vertices.
Abstract: We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length ? (log L/log log L)2), where L denotes the length of the longest simple path in the graph.This establishes the performance ratio O|V|(log log |V|/log |V|)2) for the Longest Path problem, where V denotes the graph's vertices.

Proceedings ArticleDOI
17 Nov 2002
TL;DR: It is shown that very large nodes form a spontaneously arising "core network", which plays a crucial role in the connectivity, although its proportional size goes to zero as N /spl rarr/ /spl infin/.
Abstract: We analyse a random graph where the node degrees are (almost) independent and have a distribution with finite mean but infinite variance - a region observed in empirical studies of the Internet. We show that the existence of very large nodes has a great influence on the connectivity. If N denotes the number of nodes it seems that the distance between two randomly chosen nodes of the giant component grows as slowly as log log(N). The essential observation is that very large nodes form a spontaneously arising "core network", which plays a crucial role in the connectivity, although its proportional size goes to zero as N /spl rarr/ /spl infin/. Several results related to the core are proven rigorously, and a sketch of a full proof is given. Some simulations providing illustration of the findings are presented. Consequences of the results are discussed.

Book ChapterDOI
26 Aug 2002
TL;DR: The paper provides a brief review of problems and results concerning low stretch and low communication spanning trees for graphs.
Abstract: The paper provides a brief review of problems and results concerning low stretch and low communication spanning trees for graphs.