Showing papers on "Connectivity published in 1996"

On-line maintenance of triconnected components with SPQR-trees

Abstract: We consider the problem of maintaining on-line the triconnected components of a graphG. Letn be the current number of vertices ofG. We present anO(n)-space data structure that supports insertions of vertices and edges, and queries of the type "Are there three vertex-disjoint paths between verticesv1 andv2?" A sequence ofk operations takes timeO(k·ź(k, n)) ifG is biconnected(ź(k, n) denotes the well-known Ackermann's function inverse), and timeO(n logn+k) ifG is not biconnected. Note that the bounds do not depend on the number of edges ofG. We use theSPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph with respect to its triconnected components, and theBC-tree, which represents the decomposition of a connected graph with respect to its biconnected components.

Single-source unsplittable flow

Abstract: The max-flow min-cut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity Menger's theorem provides a good characterization for the following single-source disjoint paths problem: given a graph G, with a source vertex s and terminals t/sub 1/,...,t/sub k/, decide whether there exist edge-disjoint s-t/sub i/ paths for i=1,...,k. We consider a natural, NP-hard generalization of this problem, which we call the single-source unsplittable flow problem. We are given a source and terminals as before; but now each terminal t/sub i/ has a demand p/sub i//spl les/1, and each edge e of G has a capacity c/sub e//spl ges/1. The problem is to decide whether one can choose a single s-t/sub i/ path for each i, so that the resulting set of paths respects the capacity constraints-the total amount of demand routed across any edge e must be bounded by the capacity c/sub e/. The main results of this paper are constant-factor approximation algorithms for three natural optimization versions of this problem, in arbitrary directed and undirected graphs. The development of these algorithms requires a number of new techniques for rounding fractional solutions to network flow problems; for two of the three problems we consider, there were no previous techniques capable of providing an approximation in the general case, and for the third, the randomized rounding algorithm of Raghavan and Thompson provides a logarithmic approximation. Our techniques are also of interest from the perspective of a family of NP-hard load balancing and machine scheduling problems that can be reduced to the single-source unsplittable flow problem.

Method and apparatus for organizing objects of a network map

Abstract: A layout technique generates a compact connected graph of linked objects that is organized on a page basis, i.e., the page comprises a collection of related objects displayed on a single scrollable window. Objects are generally grouped near neighboring objects with whom they share links. The technique further organizes the connected graph as a hub-and-spoke arrangement to reduce the number of overlapping objects and crossing links. Collectively, these layout features facilitate comprehension of the internetwork topology by a user.

Near-optimal path apparatus and method

Abstract: A method and apparatus for dynamically providing a path through a network of nodes or granules may use a limited, advanced look at potential steps along a plurality of available paths. Given an initial position, at an initial node or granule within a network, and some destination node or granule in the network, all nodes or granules may be represented in a connected graph. An apparatus and method may evaluate current potential paths, or edges between nodes still considered to lie in potential paths, according to some cost or distance function associated therewith. Each next edge may lie ahead across the advancing "partial" wavefront, toward a new candidate node being considered for the path. With each advancement of the wavefront, one or more potential paths, previously considered, may be dropped from consideration. Thus, a "partial" wavefront, limited in size (number of nodes and connecting edges) continues to evaluate some number of the best paths "so far." The method deletes worst paths, backs out of cul-de-sacs, and penalizes turning around. The method and apparatus may be implemented to manage a computer network, a computer internetwork, parallel processors, parallel processes in a multi-processing operating system, a smart scissor for a drawing application, and other systems of nodes.

Output-Sensitive Reporting of Disjoint Paths

Abstract: A $k$-path query on a graph consists of computing $k$ vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing $k$-path queries, with $k \leq 3$, in a graph $G$ with $n$ vertices. We denote with $\ell$ the total length of the paths reported. For $k \leq 3$, we present an optimal data structure for $G$ that uses $O(n)$ space and executes $k$-path queries in output-sensitive $O(\ell)$ time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar ($st$) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

Data Security Equals Graph Connectivity

Abstract: To protect sensitive information in a cross-tabulated table, it is a common practice to suppress some of the cells in the table. This paper investigates four levels of data security of a two-dimensional table concerning the effectiveness of this practice. These four levels of data security protect the information contained in, respectively, individual cells, individual rows and columns, several rows or columns as a whole, and a table as a whole. The paper presents efficient algorithms and NP-completeness results for testing and achieving these four levels of data security. All these complexity results are obtained by means of fundamental equivalences between the four levels of data security of a table and four types of connectivity of a graph constructed from that table.

Optimal randomized EREW PRAM algorithms for finding spanning forests and for other basic graph connectivity problems

Abstract: We present the first randomized O(log n) time and O(m + n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G = (V, E) with n vertices and m edges. Our algorithm is optimal with respect to time, work and space. As a consequence we get optimal randomized EREW PRAM algorithms for other basic connectivity problems such as finding a bipartite partition, finding bridges and biconnected components, and finding Euler tours in Eulerean graphs. For other problems such as finding an ear decomposition, finding an open ear decomposition, finding a strong orientation, and finding an st-numbering we get optimal randomized CREW PRAM algorithms.

A word graph based N-best search in continuous speech recognition

Abstract: The authors introduce an efficient algorithm for the exhaustive search of N-best sentence hypotheses in a word graph. The search procedure is based on a two-pass algorithm. In the first pass, a word graph is constructed with standard time-synchronous beam search. The actual extraction of N-best word sequences from the word graph takes place during the second pass. With the implementation of a tree-organized N-best list, the search is performed directly on the resulting word graph. Therefore, the parallel bookkeeping of N hypotheses at each processing step during the search is not necessary. It is important to point out that the proposed N-best search algorithm produces an exact N-best list as defined by the word graph structure. Possible errors can only result from pruning during the construction of the word graph. In a postprocessing step, the N candidates can be rescored with a more complex language model with highly reduced computational cost. This algorithm is also applied in speech understanding to select the most likely sentence hypothesis that satisfies some additional constraints.

On certificates and lookahead in dynamic graph problems

Abstract: Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient 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. A direct outcome of our lower bounds is the demonstration that a generic technique for designing efficient dynamic graph algorithms, viz., sparsification, will not apply to the difficult problems. More generally, it is our belief that the boundary between tractable and intractable dynamic graph problems can be demarcated in terms of certificate complexity. 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,more » to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient fully-dynamic algorithms for strong connectivity and transitive closure.« less

A fast randomized LOGSPACE algorithm for graph connectivity

Abstract: We study the relationship between undirected graph reachability and graph connectivity, in the context of randomized LOGSPACE algorithms. Aleluinas et al. [2] show that graph reachability (checking whether there is a path connecting vertices S and T) can be decided in logarithmic space and polynomial time, by starting a random walk at S, and checking whether T is hit within some time limit. The random algorithm has one-sided error (with small probability, it fails to determine that S and T are connected). The reachability algorithm may be used in order to decide (with one sided error) whether a graph is connected, by running it n−1 times, each time with a different target vertex T. This increases the running time by a factor of n. In this paper we give an alternative RLOGSPACE algorithm for graph connectivity. Its running time varies between O(n2) steps and O(n3) steps, depending on the structure of the input graph. This matches the fastest known RLOGSPACE algorithm for reachability, up to a constant factor. Our algorithm has two-sided error.

Algorithms for the constrained maximum-weight connected graph problem

Abstract: Given a positive integer R and a weight for each vertex in a graph, the maximum-weight connected graph (MCG) problem is to find a connected subgraph with R vertices that maximizes the sum of the weights. The MCG problem is strongly NP-complete, and we study a special case of it: the constrained MCG (CMCG) problem, which is the MCG problem with a constraint of having a predetermined vertex included in the solution. We first show that the Steiner tree problem is a special case of the CMCG problem. Then we present three optimization algorithms for the CMCG problem. The first two algorithms deal with special graphs (tree and layered graphs) and employ different dynamic programming techniques, solving the CMCG problem in polynomial times. The third one deals with a general graph and uses a variant of the Balas additive method with an imbedded connectivity test and a pruning method. We also present a heuristic algorithm for the CMCG problem with a general graph and its bound analysis. We combine the two algorithms, heuristic and optimization, and present a practical solution method to the CMCG problem. Computational results are reported and future research issues are discussed. © 1996 John Wiley & Sons, Inc.

Non-Stanley Bounds for Network Reliability

Abstract: Suppose that each edge of a connected graph G of order n is independently operational with probability ps the reliability of G is the probability that the operational edges form a spanning connected subgraph. A useful expansion of the reliability is as p^{n-1} \sum_{i=0}^d\ H_i(1 \ - \ p)^i, and the Ball-Provan method for bounding reliability relies on Stanley's combinatorial bounds for the H-vectors of shellable complexes. We prove some new bounds here for the H-vectors arising from graphs, and the results here shed light on the problem of characterizing the H-vectors of matroids.

Optimal Bi-Level Augmentation for Selectivity Enhancing Graph Connectivity with Applications

Abstract: Given an undirected graph G and two vertex subsets H1 and H2, the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H1 and two edgedisjoint paths between every pair of vertices in H2. We solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log2n) time using n + m processors on an EREW PRAM.

Cayley Graphs with Neighbor Connectivity One

Abstract: We give an algebraic characterization of the generating sets of Cayley graphs with neighbor connectivity equal to one for a class of Cayley graphs that includes the Cayley graphs of all abelian groups. We also show that the determination of the neighbor connectivity of a graph is NP-hard.

Structural Constraints and Neutrality in RNA

Abstract: Generic properties of neutral networks of RNA secondary structures can be described by means of random graph theory. The success of this approach is dependent on details of the underlying secondary structure. Some of these dependencies are analyzed in this paper. In addition we present an algorithm, which, given a network does conform to the random graph model, allows to determine whether it is a connected graph. The algorithm is linear in time in the sequence length, this being possible because of local connectivity, a special property of graphs under the random graph model.

Embedding classical communication topologies in the scalable OPAM architecture

Abstract: The paper presents novel embeddings of various classical topologies into the OPAM multicomputer. OPAM consists of a large number of processors that are connected by a two level, crossbar based interconnection network. The network combines a large, optical circuit-switched crossbar (reconfigurable network), with many small, packet-switching crossbars. The necessary embedding is very different than classical approaches. The goal in our case is to minimize routing decisions, so that communication requests can be satisfied by passing through two small crossbars. We show how to map parallel programs to this architecture using graph contraction notations. The family of parallel programs that we consider consists of multiple processes and communication links that are represented by connected, regular graphs such as rings, trees, two dimensional grids, cube connected cycles and hypercubes. In each case we show how to partition the vertex set of the program's graph to subsets, and how to assign each subset a cluster of processors in order to realize the topology of the given problem. In some of the cases we also prove that our partition and assignment algorithms are optimal.

Graph Algorithms for Conformance Testing Using the Rural ChinesePostman Tour

Abstract: This paper presents new results and graph algorithms for the automatic testing of protocols using “unique input/output” (UIO) sequences. UIO sequences can be efficiently employed in checking confor...

Pancyclicity and extendability in strong products

Abstract: In this paper, we first prove that for any connected graph G with at least two vertices, there is an integer m for which the strong product X⌅Gm has pancyclic ordering from each vertex. After characterizing the graphs G for which GX⌅K2 is Hamiltonian, we determine a criterion for extendability of cycles. We also prove that if G is a connected, K1.3-free graph with δ ≥ 2, then GX⌅XK2 is fully cycle extendable as well as 1-edge Hamiltonian. © 1996 John Wiley & Sons, Inc.

Monadic NP and Built-in Trees

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 is cycle-free. We obtain our result by giving a winning strategy for the duplicator in the Ajtai-Fagin Ehrenfeucht-Fraisse Game. The result can be strengthened to obtain nondefinability for a larger class of graphs.

Extensions and consequences of Chvátal-Erdős' theorem

Abstract: A classical result of Chvatal and Erd?s states that a graph with independence number smaller or equal to its connectivity contains a Hamilton cycle. In this note we discuss some extensions of this theorem and show how they can be used to proof several other results in hamiltonian graph theory. Although several of the results are known, the proofs in this note are in general essentially shorter than the original proofs, and also give an indication of the relations between the results.

A coherent model for reliability of multiprocessor networks

Abstract: Graph G has perfectly reliable nodes and edges that are subject to stochastic failure. The network reliability R is the probability that the surviving edges induce a spanning connected subgraph of G. Analysis problems concern determining efficient algorithms to calculate R, which is known to be NP-hard for general graphs. Synthesis problems concern determining graphs that are, according to some definition, the most reliable in the class of all graphs having a given number of edges and nodes. In applications where the edges are perfectly reliable and the nodes are subject to failure, another measure (residual node connectedness reliability) is defined as the probability that the surviving nodes induce a connected subgraph of G. Referring to such a subset as an operating state, the measure is not coherent because a superset of an operating state need not be an operating state. This paper proposes a new definition of network reliability that handles the case of node failures; it is coherent. We determine many of its properties, and present several analysis and synthesis results.

On the Length of the Chinese Postman Tour in Regular Graphs

Abstract: The Chinese Postman problem, i.e., the problem of finding a shortest closed walk traversing all edges of a regular graph of an odd degree is equivalent to the problem of finding a minimum join, i.e., a spanning subgraph with all vertices having odd degrees. It is known that the size of a minimum join in any connected graph on n vertices is at least n/2 and at most n - 1. We establish upper bounds for the size of a minimum join in a regular graph of odd degree in terms of its connectivity and some other parameters. Most of the bounds obtained are tight.

Output-Sensitive Reporting of Disjoint Paths (Extended Abstract)

Abstract: A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k ≤ 3, in a graph G with n vertices. We denote with l the total length of the paths reported. For k ≤ 3, we present an optimal static data structure for G that uses O(n) space and executes k-path queries in output-sensitive O(l) time. We also give a semi-dynamic version of the data structure that supports a sequence of intermixed queries and insertions of vertices and edges in a planar graph, with worst-case query time O(l) and amortized insertion time O(log n). Our results make use of a new combinatorial structure for triconnected planar graphs that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex drawings of triconnected planar graphs.