Showing papers on "Connectivity published in 2000"

A Linear Time Implementation of SPQR-Trees

Abstract: The data structure SPQR-tree represents the decomposition of a biconnected graph with respect to its triconnected components. SPQR-trees have been introduced by Di Battista and Tamassia [8] and, since then, became quite important in the field of graph algorithms. Theoretical papers using SPQR-trees claim that they can be implemented in linear time using a modification of the algorithm by Hopcroft and Tarjan [15] for decomposing a graph into its triconnected components. So far no correct linear time implementation of either triconnectivity decomposition or SPQR-trees is known to us. Here, we show the incorrectness of the Hopcroft and Tarjan algorithm [15], and correct the faulty parts. We describe the relationship between SPQR-trees and triconnected components and apply the resulting algorithm to the computation of SPQR-trees. Our implementation is publically available in AGD [1].

Exploring Unknown Environments

Abstract: We consider exploration problems where a robot has to construct a complete map of an unknown environment. We assume that the environment is modeled by a directed, strongly connected graph. The robot's task is to visit all nodes and edges of the graph using the minimum number R of edge traversals. Deng and Papadimitriou [ Proceedings of the 31st Symposium on the Foundations of Computer Science, 1990, pp. 356--361] showed an upper bound for R of dO(d) m and Koutsoupias (reported by Deng and Papadimitriou) gave a lower bound of $\Omega(d^2 m)$, where m is the number of edges in the graph and d is the minimum number of edges that have to be added to make the graph Eulerian. We give the first subexponential algorithm for this exploration problem, which achieves an upper bound of dO(log d) m. We also show a matching lower bound of $d^{\Omega(\log d)}m$ for our algorithm. Additionally, we give lower bounds of $2^{\Omega(d)}m$, respectively, $d^{\Omega(\log d)}m$ for various other natural exploration algorithms.

Near-optimal fully-dynamic graph connectivity

Abstract: In this paper we present near-optimal bounds for fullydynamic graph connectivity which is the most basic nontrivial fully-dynamic graph problem. Connectivity queries are supported in O(log n/log log log n) time while the updates are supported in O(log n(log log n) 3) expected amortized time. The previous best update time was O((log n)2). Our new bound is only doubly-logarithmic factors from a general cell probe lower bound of f2(log n~ log log n). Our algorithm runs on a pointer machine, and uses only standard AC ° instructions. In our developments we make some comparatively trivial observations improving some deterministic bounds. The space bound of the previous O((log n) ~) connectivity algorithm is improved from O(m + n log n) to O(m). The previous time complexity of fully-dynamic 2-edge and biconnectivity is improved from O((log n) 4) to O((log n) 3 log log n).

Parameterized Complexity of Finding Subgraphs with Hereditary Properties

Abstract: We consider the parameterized complexity of the following problem under the framework introduced by Downey and Fellows: Given a graph G, an integer parameter k and a non-trivial hereditary property Π, are there k vertices of G that induce a subgraph with property Π? This problem has been proved NP-hard by Lewis and Yannakakis. We show that if Π includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class W[1] and is fixed parameter tractable otherwise. In the former case, if the forbidden set of the property is finite, we show, in fact, that the problem is W[1]-complete. Our proofs, both of the tractability as well as the hardness ones, involve clever use of Ramsey numbers.

On Identifying Strongly Connected Components in Parallel

Abstract: The standard serial algorithm for strongly connected components is based on depth first search, which is difficult to parallelize. We describe a divide-and-conquer algorithm for this problem which has significantly greater potential for parallelization. For a graph with n vertices in which degrees are bounded by a constant, we sho w the expected serial running time of our algorithm to be O(n log n).

Graphs with large total domination number

Abstract: Let G = (V,E) be a graph. A set S ⊆ V is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number of G, denoted by Υt(G), is the minimum cardinality of a total dominating set of G. We establish a property of minimum total dominating sets in graphs. If G is a connected graph of order n ≥ 3, then (see [3]) Υt(G) ≤ 2n-3. We show that if G is a connected graph of order n with minimum degree at least 2, then either Υt(G) ≤ 4n-7 or G e {C3, C5, C6, C10}. A characterization of those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G) e 2 and Υt(G) ≥ 4n-7 is obtained. We establish that if G is a connected graph of size q with minimum degree at least 2, then Υt(G) ≤(q + 2)-2. Connected graphs G of size q with minimum degree at least 2 satisfying Υt(G) > q-2 are characterized. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 2145, 2000

Improved Shortest Paths on the Word RAM

Abstract: Thorup recently showed that single-source shortest-paths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0, . . . , 2w -1} can be solved in O(n+m) time and space on a unit-cost random-access machine with a word length of w bits. His algorithm works by traversing a so-called component tree. Two new related results are provided here. First, and most importantly, Thorup's approach is generalized from undirected to directed networks. The resulting time bound, O(n + m log w), is the best deterministic linear-space bound known for sparse networks unless w is superpolynomial in log n. As an application, all-pairs shortest-paths problems in directed networks with n vertices, m edges, and edge weights in {-2w, . . . , 2w} can be solved in O(nm + n2 log log n) time and O(n + m) space (not counting the output space). Second, it is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup. Another contribution of the present paper is a greatly simplified view of the principles underlying algorithms based on component trees.

Automatic detection of conserved gene clusters in multiple genomes by graph comparison and P-quasi grouping

Abstract: We previously reported two graph algorithms for analysis of genomic information: a graph comparison algorithm to detect locally similar regions called correlated clusters and an algorithm to find a graph feature called P-quasi complete linkage. Based on these algorithms we have developed an automatic procedure to detect conserved gene clusters and align orthologous gene orders in multiple genomes. In the first step, the graph comparison is applied to pairwise genome comparisons, where the genome is considered as a one-dimensionally connected graph with genes as its nodes, and correlated clusters of genes that share sequence similarities are identified. In the next step, the P-quasi complete linkage analysis is applied to grouping of related clusters and conserved gene clusters in multiple genomes are identified. In the last step, orthologous relations of genes are established among each conserved cluster. We analyzed 17 completely sequenced microbial genomes and obtained 2313 clusters when the completeness parameter P was 40%. About one quarter contained at least two genes that appeared in the metabolic and regulatory pathways in the KEGG database. This collection of conserved gene clusters is used to refine and augment ortholog group tables in KEGG and also to define ortholog identifiers as an extension of EC numbers.

Phylogenetic k-Root and Steiner k-Root

Abstract: Given a graph G = (V, E) and a positive integer k, the PHYLOGENETIC k-ROOT PROBLEM asks for a (unrooted) tree T without degree-2 nodes such that its leaves are labeled by V and (u, v) ∈ E if and only if dT (u, v) ≤ k. If the vertices in V are also allowed to be internal nodes in T, then we have the Steiner k-ROOT PROBLEM. Moreover, if a particular subset S of V are required to be internal nodes in T, then we have the RESTRICTED STEINER k-ROOT PROBLEM. Phylogenetic k-roots and Steiner k-roots extend the standard notion of GRAPH ROOTS and are motivated by applications in computational biology. In this paper, we first present O(n + e)-time algorithms to determine if a (not necessarily connected) graph G = (V, E) has an S-restricted 1-root Steiner tree for a given subset S ⊂ V , and to determine if a connected graph G = (V, E) has an S-restricted 2-root Steiner tree for a given subset S ⊂ V, where n = |V| and e = |E|. We then use these two algorithms as subroutines to design O(n + e)-time algorithms to determine if a given (not necessarily connected) graph G = (V, E) has a 3-root phylogeny and to determine if a given connected graph G = (V, E) has a 4-root phylogeny.

A New Methodology to Computer Deadlock-Free Routing Tables for Irregular Networks

Abstract: Networks of workstations (NOWs) are being considered as a cost-effective alternative to parallel computers Many NOWs are arranged as a switch-based network with irregular topology, which makes routing and deadlock avoidance quite complicated Current proposals use the up*/down* routing algorithm to remove cyclic dependencies between channels and avoid deadlock However, routing is considerably restricted and most messages must follow non-minimal paths, increasing latency and wasting resources In this paper, we propose a new methodology to compute deadlock-free routing tables for NOWs The methodology tries to minimize the limitations of the current proposals in order to improve network performance It is based on generating an underlying acyclic connected graph from the network graph and assigning a sequence number to each switch, which is used to remove cyclic dependencies Evaluation results show that the routing algorithm based on the new methodology increases throughput by a factor of up to 2 in large networks, also reducing latency significantly

Geodetic sets in graphs

Abstract: For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in 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. If I[S] = V (G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u, v ∈ S, there exists a third vertex w of G that lies in some u− v geodesic but in no x− y geodesic for x, y ∈ S and {x, y} 6= {u, v}. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a < b the minimum order of such a graph is b + 2.

Average distance, minimum degree, and spanning trees

Abstract: The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n2)-1 Σ{x,y}‚V(G) dG(x, y), where V(G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most $${n\over \delta + 1} + 5$$. We give improved bounds for K3-free graphs, C4-free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 113, 2000

A generalized particle algorithm for high velocity impact computations

Abstract: This paper presents a Generalized Particle Algorithm (GPA) for high velocity impact and other dynamics problems. A velocity smoothing algorithm is also presented. This generalized algorithm allows for both variable nodal connectivity and fixed nodal connectivity. The variable connectivity option allows for severe distortions with a Lagrangian approach. With fixed nodal connectivity and smoothing it is possible to provide stable computations for large tensile strains. The algorithms are provided for 2D axisymmetric geometry and 3D geometry, and examples are included to demonstrate some of the capabilities. A discussion of interface problems and solutions is also included.

Improving the Up*/Down* Routing Scheme for Networks of Workstations

Abstract: Networks of workstations (NOWs) are being considered as a cost-effective alternative to parallel computers. Many NOWs are arranged as a switch-based network with irregular topology, which makes routing and deadlock avoidance quite complicated. Current proposals use the up*/down* routing algorithm to remove cyclic dependencies between channels and avoid deadlock. Recently, a simple and effective methodology to compute up*/down* routing tables has been proposed by us. The resulting up*/down* routing scheme makes use of a different link direction assignment to compute routing tables. Assignment of link direction is based on generating an underlying acyclic connected graph from the network graph. In this paper, we propose and evaluate new heuristic rules to compute the underlying graph. Moreover, we propose a traffic balancing algorithm to obtain more efficient up*/down* routing tables when source routing is used. Evaluation results show that the routing algorithm based on the new methodology increases throughput by a factor of up to 2.8 in large networks, also reducing latency significantly.

The product replacement algorithm is polynomial

Abstract: The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating /spl kappa/-tuples of the group, and then output a random component. The algorithm was designed by C.R. Leedham-Green, and further investigated by F. Cellar et al. (1995). It was found to have an outstanding performance, much better than the previously known algorithms (P. Diaconis and L. Saloff-Coste, 1996). The algorithm is now included in two major group algebra packages: GAP (M. Scheonert et al., 1995) and MAGMA (W. Bosma et al., 1997). In spite of the many serious attempts and partial results, the analysis of the algorithm remains difficult at best. For small values of /spl kappa/, even graph connectivity becomes a serious obstacle. The most general results are due to Diaconis and Saloff-Coste, who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)-exponential bounds in the general case. The main result of the paper is a polynomial upper bound for the cost of the algorithm, provided /spl kappa/ is large enough.

The Ising model on diluted graphs and strong amenability

Abstract: Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external fieldWe show that for nonamenable graphs, for Bernoulli percolation with $p$ close to 1, all the infinite clusters have persistent transitionOn the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition This extends a result of Georgii for the cubic lattice A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (ie, their anchored Cheeger constant is 0) Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter $p$, on an arbitrary bounded degree graph, is a continuous function of $p$

Improved Greedy Algorithms for Constructing Sparse Geometric Spanners

Abstract: Let G=(V, E) be a connected graph with positive weights and n vertices. A subgraph G′ is a t-spanner if for all u, v∈;V, the distance between u and v in the subgraph G′ is at most t times the corresponding distance in G. We show a O(n log n)-time algorithm which, given a set V of n points in d-dimensional space, and any constant t>1, produces a t-spanner of the complete Euclidean graph of G. The produced spanner have O(n) edges, constant degree and weight O(wt(MST)).

An O(log(n)4/3) space algorithm for (s, t) connectivity in undirected graphs

Abstract: We present a deterministic algorithm that computes st-connectivity in undirected graphs using O(log 4/3n) space. This improves the previous O(log3/2n) bound of Nisan et al. [1992].

On the cover time of planar graphs

Abstract: The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for bounded-degree planar graphs the cover time is at least c n(log n)^2, and at most 6n^2, where c is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings.

On conditional edge-connectivity of graphs

Abstract: Letk andh be two integers, 0≤h

Connectivity planning for closed-chain reconfiguration

Abstract: Modular reconfigurable robots can change their connectivity from one arrangement to another. Performing this change involves a difficult planning problem. We study this problem by representing robot configurations as graphs, and giving an algorithm that can transform any configuration of a robot into any other in O (log n) steps. Here n is the number of modules which can attach to more than two other modules. We also show that O(log n) is best possible.

Constrained length connectivity and survivable networks

Abstract: Some problems related to constrained length connectivity are addressed in this paper. Let Sl(x, y) be the minimum number of vertices that should be removed to destroy all the paths of length at most l between two vertices x and y. Let Il(x, y) be the maximum number of such node-disjoint paths. We first focus on f(l, d), defined as the supremum of (Sl(x, y))/(Il(x, y)) taken over all graphs and all pairs of x, y separated by a distance d. One of the results shown in this paper states that this supremum is exactly equal to l + 1 − d when d ≥ ⌈⅔l + 1)⌉ and is at least constant when 2 ≤ d ≤ 2 + ⌊(l + 1)/3⌋. Some classes of two connected graphs satisfying path-length constraints are defined. Most of them describe survivable telecommunication networks. Relationships between flows and constrained length connectivity are addressed. We also study the minimum edge numbers of these two connected graphs. Some of their topological properties are presented. © 2000 John Wiley & Sons, Inc.

The Geodetic Number of an Oriented Graph

Abstract: For two vertices u and v of an oriented graph D, the set I(u, v) consists of all vertices lying on a u?v geodesic or v?u geodesic in D. If S is a set of vertices of D, then I(S) is the union of all sets I(u, v) for vertices u and v in S. The geodetic number g(D) is the minimum cardinality among the subsets S of V(D) with I(S) =V(D). Several results concerning the geodetic numbers of connected oriented graphs are presented. For a nontrivial connected graph G, the lower orientable geodetic number g?(G) of G is the minimum geodetic number among the orientations of G and the upper orientable geodetic number g+(G) is the maximum such geodetic number. It is shown that g?(G)?g+(G) for every connected graph of order at least 3. The lower and upper orientable geodetic numbers of several well known classes of graphs are determined. It is shown that for every two integers n and m with 1 ?n? 1 ?m? (n 2) , there exists a connected graph G of order n and size m such that g+(G) =n.

Measuring the Survivability of a Network: Connectivity and Rest‐Connectivity

Abstract: When evaluating the survivability performance of a communication network, it is important to detect whether the graph is connected, whether there are separation nodes and separation pairs The algorithms [ I ] and [2] developed by Tarjan and Hopcroft are the adequate tools for this purpose They are able to determine the bi- and triconnected components of a graph in o(N+E) where N is the number of nodes and E the number of edges of the graph Practically however, the decomposition of the graph is a step only in the evaluation of the connectivity performance of a graph Indeed, when a separation element (node or pair) has been detected, it is crucial to know what are the consequences if this element fails down For example, which precise parts of the graph are disconnected or more simply how large are the parts which are separated by this element'? The paper introduces the notion of rest-connectivity Basically, for each separation element, the rest-connectivity value is defined as the number of node pairs which got disconnected by the failure of that separation element In this paper, algorithms [1 ] and [2] are enhanced in order to determine the rest-connectivity values for all separation nodes and pairs of a graph while keeping the complexity in o(N+E)