Showing papers on "Connectivity published in 1995"

A Graph-Theoretic Game and its Application to the $k$-Server Problem

Abstract: This paper investigates a zero-sum game played on a weighted connected graph $G$ between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree $T$ and the edge player chooses an edge $e$. The payoff to the edge player is $cost(T,e)$, defined as follows: If $e$ lies in the tree $T$ then $cost(T,e)=0$; if $e$ does not lie in the tree then $cost(T,e) = cycle(T,e)/w(e)$, where $w(e)$ is the weight of edge $e$ and $cycle(T,e)$ is the weight of the unique cycle formed when edge $e$ is added to the tree $T$. The main result is that the value of the game on any $n$-vertex graph is bounded above by $\exp(O(\sqrt{\log n \log\log n}))$. It is conjectured that the value of the game is $O(\log n)$. The game arises in connection with the $k$-server problem on a road network; i.e., a metric space that can be represented as a multigraph $G$ in which each edge $e$ represents a road of length $w(e)$. It is shown that, if the value of the game on $G$ is $Val(G,w)$, then there is a randomized strategy that achieves a competitive ratio of $k(1 + Val(G,w))$ against any oblivious adversary. Thus, on any $n$-vertex road network, there is a randomized algorithm for the $k$-server problem that is $k\cdot\exp(O(\sqrt{\log n \log\log n}))$ competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any $n$-vertex weighted, connected multigraph, the algorithm constructs a spanning tree $T$ such that the average, over all edges $e$, of $cost(T,e)$ is less than or equal to $\exp(O(\sqrt{\log n \log\log n}))$. This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph.

Graph decompositions and secret sharing schemes

Abstract: In this paper we continue a study of secret sharing schemes for-access structures based on graphs. Given a graph G, we require that a subset of participants can compute a secret key if they contain an edge of G; otherwise, they can obtain no information regarding the key. We study the information rate of such schemes, which measures how much information in being distributed as shares compared with the size of the secret key, and the average information rate, which is the ratio between the secret size and the arithmetic mean of the size of the shares. We give both upper and lower bounds on the optimal information rate and average information rate that can be obtained. Upper bounds arise by applying entropy arguments due to Capocelli et al. [15]. Lower bounds come from constructions that are based on graph decompositions. Application of these constructions requires solving a particular linear programming problem. We prove some general results concerning the information rate and average information rate for paths, cycles, and trees. Also, we study the 30 (connected) graphs on at most five vertices, obtaining exact values for the optimal information rate in 26 of the 30 cases, and for the optimal average information rate in 28 of the 30 cases.

Randomized query processing in robot path planning

Abstract: : The subject of this paper is the analysis of a randomized preprocessing scheme that has been used for query processing in robot motion planning. The attractiveness of the scheme stems from its general applicability to virtually any motion planning problem, and its empirically observed success. In this paper we initiate a theoretical basis for explaining this empirical success. Under a simple assumption about the configuration space, we show that it is possible to perform a preprocessing step following which queries can be answered quickly. En route, we pose and give solutions to related problems on graph connectivity in the evasiveness model, and art gallery theorems.

Application of graph theory to the synchronization in an array of coupled nonlinear oscillators

Abstract: In this letter, we show how algebraic graph theory can be used to derive sufficient conditions for an array of resistively coupled nonlinear oscillators to synchronize. These conditions are derived from the connectivity graph, which describes how the oscillators are connected. In particular, we show how such a sufficient condition is dependent on the algebraic connectivity of the connectivity graph. Intuition tells us that if the oscillators are more "closely connected" to each other, then they are more likely to synchronize. We discuss how to quantify connectedness in graph-theoretical terms and its relation to algebraic connectivity and show that our results are in accordance with this intuition. We also give an upper bound on the coupling conductance required for synchronization for arbitrary graphs, which is in the order of n/sup 2/, where n is the number of oscillators. >

New sparseness results on graph spanners

Abstract: Let G=(V, E) be an n-vertex connected graph with positive edge weights. A subgraph G′=(V, E′) is a t-spanner of G if for all u, v∈V, the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight. For an arbitrary positive edge-weighted graph G, for any t>1, and any ∈>0, we show that a t-spanner of G with weight can be constructed in polynomial time. We also show that (log2 n)-spanners of weight O(1) · wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the vector. We show that for these graphs, t-spanners with total weight O(log n) · wt(MST) can be constructed in polynomial time.

Approximating the Minimum Equivalent Digraph

Abstract: The MEG (minimum equivalent graph) problem is the following: "Given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes." This problem is NP-hard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in polynomial time. The algorithm achieves a performance guarantee of 1.75 in the time required for transitive closure. The heart of the MEG problem is the minimum SCSS (strongly connected spanning subgraph) problem --- the MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives a practical, nearly linear-time implementation achieving a performance guarantee of 1.75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to $2$-\Exchange, a general "local improvement" algorithm, showing that its performance guarantee is 1.75.

Efficient search and hierarchical motion planning by dynamically maintaining single-source shortest paths trees

Abstract: Hierarchical approximate cell decomposition is a popular approach to the geometric robot motion planning problem. In many cases, the search effort expended at a particular iteration can be greatly reduced by exploiting the work done during previous iterations. In this paper, we describe how this exploitation of past computation can be effected by the use of a dynamically maintained single-source shortest paths tree. We embed a single-source shortest paths tree in the connectivity graph of the approximate representation of the robot configuration space. This shortest paths tree records the most promising path to each vertex in the connectivity graph from the vertex corresponding to the robot's initial configuration. At each iteration, some vertex in the connectivity graph is replaced with a new set of vertices, corresponding to a more detailed representation of the configuration space. Our new, dynamic algorithm is then used to update the single-source shortest paths tree to reflect these changes to the underlying connectivity graph. >

Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Abstract: Generalizing and unifying earlier results of W. Mader, and A. Frank and B. Jackson, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edge-connectivity prescriptions. An extension of Edmonds's theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.

On the "log rank"-conjecture in communication complexity

Abstract: We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove: Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.

Optimum CMOS stack generation with analog constraints

Abstract: An algorithm for the automatic generation of full-stacked layouts in CMOS analog circuits is described in this paper. The set of stacks obtained is optimum with respect to a cost function which accounts for critical parasitics and device area minimization. Device interleaving and common-centroid patterns are automatically introduced when possible, and all symmetry and matching constraints are enforced. The algorithm is based on operations performed on a graph representation of circuit connectivity, exploiting the equivalence between stack generation and path partitioning in the circuit graph. Path partitioning is carried out in two phases: in the first phase, all paths are generated by a dynamic programming procedure. In the second phase, the optimum partition is selected by solving a clique problem. Original heuristics have been introduced, which preserve the optimality of the solution, while effectively improving the computational efficiency of the algorithm. The algorithm has been implemented in the "C" programming language. Many test cases have been run, and the quality of results is comparable to that of hand-made circuits. Results also demonstrate the effectiveness of the heuristics employed, even for relatively complex circuits. >

A spanning tree heuristic for regional clustering

Abstract: We consider the problem of subdividing a region into districts—formed by geographically contiguous sites—which are as homogeneous as possible with respect to a certain set of characteristics. One possible mathematical formulation is the following. Given a connected graph G, in which a vector of characteristics is associated with each vertex, find a minimum inertia partition of the vertex-set of G into a prescribed number of connected clusters. We propose a heuristic approach based on the solution of a sequence of relaxed problems where G is replaced by one of its spanning trees. Numerical experiments on medium-large graphs arising from real-life applications show that our heuristic almost always yields partitions with smaller inertia than those generated by a well established ascending algorithm.

Undirected vertex-connectivity structure and smallest four-vertex-connectivity augmentation (extended abstract)

Abstract: In this paper, we give an O(n·log n+m)-time algorithm to solve the problem of finding a smallest set of edges whose addition 4-vertex-connects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. We also show a formula to compute this smallest number in O(n·α(n,n)+m) time, where α is the inverse of the Ackermann function.

New Lower Bounds for Orthogonal Graph Drawings

Abstract: An orthogonal drawing is an embedding of a graph such that edges are drawn as sequences of horizontal and vertical segments. In this paper we explore lower bounds. We find lower bounds on the number of bends when crossings are allowed, and lower bounds on both the grid-size and the number of bends for planar and plane drawings.

Triangulations of polytopes and computational algebra

Abstract: Let ${\cal A}$ be a point configuration The graph of triangulations of ${\cal A},$ G$\sb{\cal A}$, is the graph whose vertices are the triangulations of ${\cal A}$ and two triangulations are adjacent when there is a geometric bistellar operation taking one triangulation into the other Chapter one discusses algorithms and computer software developed to compute G$\sb{\cal A}$ A well-known open problem is to decide whether G$\sb{\cal A}$ is always a connected graph The main result of chapter one indicates that a disconnected example may exist We found an example of a three dimensional configuration whose graph of triangulations has low connectivity Gel'fand, Kapranov and Zelevinsky asked whether products of two simplices could have non-regular triangulations In chapter two we present an affirmative answer to this open question Chapter two also contains a detailed study of triangulations for other (0,1)-polytopes such as cubes and hypersimplices Our techniques include the use of Grobner bases theory Chapter three discusses Grobner bases for a family of arrangements of linear subspaces This family has a strong relation to coloring problems in graph theory We apply our results to the enumeration of vertex colorings In Chapter four we present an algorithmic version of a theorem of G Polya Given a real homogeneous polynomial F, strictly positive in the non-negative orthant, Polya's theorem says that for a sufficiently large exponent p the coefficients of F($x\sb1,$,$x\sb{n})\cdot(x\sb1 + {\cdot\cdot\cdot} + x\sb{n})\sp{p}$ are strictly positive As an application of Polya's theorem we present a new algorithm for the decomposition of a strictly positive polynomial as the quotient of two sums of squares of polynomials (special case of the algorithmic version of Hilbert's 17th problem) We also discuss how Hilbert's 17th problem originated in the study of geometric constructions with marked ruler

A representation of cuts within 6/5 times the edge connectivity with applications

Abstract: Let G be an undirected c-edge connected graph. In this paper we give an O(n/sup 2/)-sized planar geometric representation for all edge cuts with capacity less than 6/5c. The representation can be very efficiently built, by using a single run of the Karger-Stein algorithm for finding near-mincuts. We demonstrate that the representation provides an efficient query structure for near-mincuts, as well as a new proof technique through geometric arguments. We show that in algorithms based on edge splitting, computing our representation O(log n) times substitute for one, or sometimes even /spl Omega/(n), u-/spl nu/ mincut computations; this can lead to significant savings, since our representation can be computed /spl theta//spl tilde/(m/n) times faster than the currently best known u-/spl nu/ mincut algorithm. We also improve the running time of the edge augmentation problem, provided the initial edge weights are polynomially bounded.

Graph Connectivity, Monadic NP and Built-in Relations of Moderate Degree

Abstract: It has been conjectured [FSV93] that an existential secondoder formula, in which the second-order quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary built-in relations.

Short length versions of Menger's theorem

Abstract: Consider a simple n-vertex undirected graph and assume there are ~ edge-disjoint paths between two vertices u and V. We prove the following two results: There are K edge-disjoint paths between u and v, the average length of which is 0(7t/~) If all vertices have degree at least HI there are ~ edge-disjoint paths between u and v, each of which has length O(n/K). These bouncls are best possible. For directed graphs, the first result still holds but not the second. Some of the paths can be at least Q(n) long. We also describe how to use a minimum cost flow algorithm to find the paths irnpliecl by the above results in time O(tcrn). In a ~ edge-connected graph, we define the concept of B-distance (or bulk distance). The B-distance between u and v is the minimum over all R edge-disjoint paths between u and v of the maximum path length. We prove that B-distance forms a metric. We give NPhardness results on computing B-distances in two cases. The third remaining case is open but we give evidence to its difficulty.

The number of spanning trees in graphs with a given degree sequence

Abstract: Alon's [1] idea is slightly refined to prove that for each connected graph G with degree sequence 1

An O ( n log log n ) time algorithm for constructing a graph of maximum connectivity with prescribed degrees

Abstract: A sequence of nonnegative integers D = ( d 1 , d 2 , ..., d n ) is graphical if there is a graph with degree sequence D . The connectivity κ( D ) of a graphical sequence D is defined to be the maximum integer k such that there is a k -connected graph with degree sequence D . In this paper, we present an O ( n log log n ) time algorithm, for a given graphical sequence D , to construct a κ( D )-connected graph with degree sequence D .

Average case analysis of dynamic graph algorithms

Abstract: We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., not random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2-edge connectivity, k- edge connectivity, k-vertex connectivity, and bipartiteness. Given a random graph G with m0 edges and n vertices and a sequence of l update operations such that the graph contains mi edges after operation i , the expected time for performing the updates for any l is O.l log nC6 l iD1 n= p mi/ in the case of minimum spanning forests, connectivity, 2-edge connectivity, and bipartiteness. The expected time per update operation is O.n/ in the case of maximum matching. We also give improved bounds fork-edge and k-vertex connectivity. Additionally we give an insertions-only algorithm for maximum cardinality matching with worst-case O.n/ amortized time per insertion.

Random Walks and Undirected Graph Connectivity: A Survey

Abstract: We survey a number of algorithms that decide connectivity in undirected graphs. Our focus is on the use of random walks as a tool in reducing the space complexity of these algorithms.

Stocastic Graphs Have Short Memory: Fully Dynamic Connectivity in Poly-Log Expected Time

Abstract: This paper introduces average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions) To this end we introduce the model of stochastic graph processes, ie dynamically changing random graphs with random equiprobable edge insertions and deletions, which generalizes Erdos and Renyi's 35 year-old random graph process As the stochastic graph process continues indefinitely, all potential edge locations (in V × V) may be repeatedly inspected (and learned) by the algorithm This learning of the structure seems to imply that traditional random graph analysis methods cannot be employed (since an observed edge is not a random event anymore) However, we show that a small (logarithmic) number of dynamic random updates are enough to allow our algorithm to re-examine edges as if they were random with respect to certain events (ie the graph “forgets” its structure) This short memory property of the stochastic graph process enables us to present an algorithm for graph connectivity which admits an amortized expected cost of O(log3n) time per update In contrast, the best known deterministic worst-case algorithms for fully dynamic connectivity require n1/2 time per update

The poset on connected induced subgraphs of a graph need not be Sperner

Abstract: SupposeG is a finite connected graph. LetC(G) denote the inclusion ordering on the connected vertex-induced subgraphs ofG. Penrice asked whetherC(G) is Sperner for general graphsG. Answering Penrice's question in the negative, we present a treeT such thatC(T) is not Sperner. We also construct a related distributive lattice that is not Sperner.

Mapping molecular dynamics computations on to hypercubes

Abstract: We propose an approach for partitioning an irregular application problem in computational biology called Molecular Dynamics (MD) of Macromolecules. We model the application as a task graph which we call a compact MD graph. Such a modeling allows existing mapping heuristics to be applied to this problem. We then provide a parallel algorithm for this application, by using an efficient mapping heuristic called Allocation By Recursive Mincut (ARM) to map the compact MD graph to a hypercube connected parallel computer, the nCUBE 2S. A canonical model for executing parallel computations modeled as graphs is described. Thus, we attempt to provide the missing link between the mapping research and application implementation research, and demonstrate that the execution time can be sufficiently reduced by considering formal mapping techniques, while designing parallel programs for important applications.

Planar Strong Connectivity Helps in Parallel Depth-First Search

Abstract: This paper proves that for a strongly connected planar directed graph of size $n$, a depth-first search tree rooted at a specified vertex can be computed in $O(\log^{5}n)$ time with $n/\log{n}$ processors. Previously, for planar directed graphs that may not be strongly connected, the best depth-first search algorithm runs in $O(\log^{10}n)$ time with $n$ processors. Both algorithms run on a parallel random access machine that allows concurrent reads and concurrent writes in its shared memory, and in case of a write conflict, permits an arbitrary processor to succeed.

A 1-(S,T)-Edge-Connectivity Augmentation Algorithm

Abstract: We present a combinatorial algorithm for the $1$-$(S,T)$-edge-connectivitxy augmentation problem in digraphs. The general $k$-$(S,T)$-edge-connectivity augmentation problem was first solved by A. Frank and T. Jordan, Minimal Edge-coverings by Pairs of Sets, Journal of Combinatorial Theory Ser. B, submitted, but their proof does not yield a polynomial-time algorithm. Our algorithm generalizes an earlier result of P. Eswaran and R. E. Tarjan and relies heavily on the nature of the special case $k = 1$.