Showing papers on "Connectivity published in 1997"
TL;DR: Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges.
Abstract: Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximize the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite programming relaxation.
400 citations
TL;DR: In this article, the authors provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in timeO(n 1/2) per change; 3-edge connections, in time O(n 2/3) per insertion; 4-edge connection, in O(na(n)) per insertion.
Abstract: We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in timeO(n1/2) per change; 3-edge connectivity, in time O(n2/3) per change; 4-edge connectivity, in time O(na(n)) per change; k-edge connectivity for constant k, in time O(nlogn) per change;2-vertex connectivity, and 3-vertex connectivity, in the O(n) per change; and 4-vertex connectivity, in time O(na(n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms.All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.
239 citations
TL;DR: This paper gives anO(kε)-approximation algorithm for any ε>0.1, which improves the previously knownk-approximating.
Abstract: Given an acyclic directed network, a subsetS of nodes (terminals), and a rootr, theacyclic directed Steiner tree problem requires a minimum-cost subnetwork which contains paths fromr to each terminal. It is known that unlessNP⊆DTIME[npolylogn] no polynomial-time algorithm can guarantee better than (lnk)/4-approximation, wherek is the number of terminals. In this paper we give anO(kź)-approximation algorithm for any ź>0. This result improves the previously knownk-approximation.
141 citations
TL;DR: Ambivalent data structures are presented for several problems on undirected graphs and used to dynamically maintain 2-edge-connectivity information and are extended to find the smallest spanning trees in an embedded planar graph in time.
Abstract: Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the $k$ smallest spanning trees of a weighted undirected graph in $O(m \log \beta (m,n) + \min \{ k^{3/2}, km^{1/2} \} )$ time, where $m$ is the number of edges and $n$ the number of vertices in the graph. The techniques are extended to find the $k$ smallest spanning trees in an embedded planar graph in $O(n + k (\log n)^3 )$ time. Ambivalent data structures are also used to dynamically maintain 2-edge-connectivity information. Edges and vertices can be inserted or deleted in $O(m^{1/2})$ time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in $O(\log n)$ time, where $m$ and $n$ are understood to be the current number of edges and vertices, respectively.
132 citations
TL;DR: An O(n log2 n) time algorithm which, given a set V of n points in k-dimensional space (for any fixed k), and any real constant t > 1, produces a t-spanner of the complete Euclidean graph of V, which is similar to the size and weight of spanners constructed by the greedy algorithm.
Abstract: Let G = (V,E) be a n-vertex connected graph with positive edge weights, and let t > 1 be a real constant. A subgraph G' is a t-spanner if for all u,v ∊ V, the weight of the shortest path between u and v in G' is at most t times the weight of the corresponding shortest path in G. We design an O(n log2 n) time algorithm which, given a set V of n points in k-dimensional space (for any fixed k), and any real constant t > 1, produces a t-spanner of the complete Euclidean graph of V. This algorithm retains the spirit of a recent O(n3logn) time greedy algorithm which produces t-spanners; we use graph clustering techniques to achieve a more efficient implementation. Our spanners have O(n) edges and weight O(1)· wt(MST), which is similar to the size and weight of spanners constructed by the greedy algorithm. The constants implicit in the O-notation depend upon t and k.
121 citations
04 May 1997
TL;DR: The rst subexponential algorithm for this exploration problem, which achieves an upper bound of d O(logd) m, and lower bounds of 2 ›(d)m, respectively, d ›(logD) m for various other natural exploration algorithms are given.
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 d O(d) m and Koutsoupias (reported by Deng and Papadimitriou) gave a lower bound of ›(d2m), 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 rst subexponential algorithm for this exploration problem, which achieves an upper bound of d O(logd) m. We also show a matching lower bound of d ›(logd) m for our algorithm. Additionally, we give lower bounds of 2 ›(d) m, respectively, d ›(logd) m for various other natural exploration algorithms.
111 citations
TL;DR: In this paper, it was shown that for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the cheeger constant.
Abstract: It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant \( h(G) \geq k \) has a spanning forest in which each component is isomorphic to T(k).
84 citations
TL;DR: It was speculated that the connectivity of the n-dimensional crossed cube is n and this paper proves that the result is true.
78 citations
TL;DR: In this paper, a primal-dual approach was used to solve the survivable network design problem, where a minimum cost set of edges such that there are vertex-disjoint paths between verticesi andj must be found.
Abstract: We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, a valuerij for each pair of verticesi andj is given, and a minimum-cost set of edges such that there arerij vertex-disjoint paths between verticesi andj must be found. In the case for whichrijź{0, 1, 2} for alli, j, we can find a solution of cost no more than three times the optimal cost in polynomial time. In the case in whichrij=k for alli, j, we can find a solution of cost no more than 2H(k) times optimal, where $$\mathcal{H}(n) = 1 + \tfrac{1}{2} + \cdot \cdot \cdot + \tfrac{1}{n}$$ . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems.
75 citations
Book•
01 Jan 1997
TL;DR: The Sum Rule and the Product Rule as mentioned in this paper are the product rule and the Sum Rule is a product rule that is related to the Inclusion-Exclusion Principle of the Pigeonhole Principle.
Abstract: The Sum Rule and the Product Rule. -- Permutations and Combinations. -- The Pigeonhole Principle. -- Generalized Permutations and Combinations. -- Sequences and Selections. -- The Inclusion-Exclusion Principle. -- Generating Functions and Partitions of Integers. -- The Distribution Problem in Combinatorics. -- Recurrence Relations. -- Group Theory in Combinatorics -- Including The Burnside-Froberius Theorem. -- Permutation Groups and Their Cycles Indices and Polya's Enumeration Theorems.
49 citations
TL;DR: Algorithms are presented, along with worst case complexity bounds and experimental results for representative classes of graphs, for the validation of topological maps of an environment by an active agent (such as a mobile robot), and the localization of an agent in a given map.
TL;DR: In this article, the Grubler mobility formula is used to determine linkages which have a connectivity of six between at least two of the links, which are useful in the design of mechanical arms.
15 Sep 1997
TL;DR: In this paper, the problem of determining whether a connected graph has a ΔT-spanning tree is shown to be NP-complete for 2-and 3-connected graphs, for planar graphs, and for ΔT = 2.
Abstract: Given a connected graph G, let a ΔT-spanning tree of G be a spanning tree of G of maximum degree bounded by ΔT. It is well known that for each ΔT ≥ 2 the problem of deciding whether a connected graph has a ΔT-spanning tree is NP-complete. In this paper we investigate this problem when additionally connectivity and maximum degree of the graph are given. A complete characterization of this problem for 2- and 3-connected graphs, for planar graphs, and for ΔT = 2 is provided.
TL;DR: This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree δ of a graph, to assure maximum extraconnectivity.
TL;DR: The diameter of CCC is calculated and the same calculation works for somewhat more general graphs than just CCC, which makes them possible candidates for switching patterns of multiprocessor computers.
TL;DR: The development of a satisfiability algorithm for CDOL conditions and conditions and the extension of the RTG method with knowledge of the rule execution semantics are addressed and the effectiveness of the approach within the context of the sample application is addressed.
Abstract: This paper describes the implementation of the Refined Triggering Graph (RTG) method for active rule termination analysis. The RTG method has been defined in the context of an active, deductive, object-oriented database language known as CDOL (Comprehensive, Declarative, Object Language). The RTG method studies the contents of rule pairs and rule cycles in a triggering graph and tests for: 1) the successful unification of one rule's action with another rule's triggering event, and 2) the satisfiability of active rule conditions, asking if it is possible for the condition of a triggered rule to evaluate to true in the context of the triggering rule's condition. If the analysis can provably demonstrate that one rule cannot trigger another rule, the directed vector connecting the two rules in a basic triggering graph can be removed, thus refining the triggering graph. Two important aspects in the implementation of the method include the development of a satisfiability algorithm for CDOL conditions and conditions and the extension of the RTG method with knowledge of the rule execution semantics. The effectiveness of the approach within the context of the sample application is also addressed.
TL;DR: It is shown that any algorithm that maintains the connected components of a digital image must take Ω( log n log log log n ) time per change to the image, and the problem can be solved in O(log n) time perchange using dynamic planar graph techniques.
TL;DR: It is proved that the maximum genus, denoted by γ M ( G ), of G is equal to z ( G ).
04 Jun 1997
TL;DR: The LINK system is introduced as an educational tool which can be used to visualize and experiment with discrete algorithms and demonstrates the flexibility of the system in the context of a fundamental graph algorithm: finding the strongly connected components of a directed graph.
Abstract: This paper introduces the LINK system as an educational tool which can be used to visualize and experiment with discrete algorithms An extended example demonstrates the flexibility of the system in the context of a fundamental graph algorithm: finding the strongly connected components of a directed graph
TL;DR: This paper presents a new algorithm which automatically selects a subminimal cycle base in any planar graph based on a shortest path technique between two points of a connected graph, and makes possible to use the same input data as in the displacement method.
TL;DR: The Wiener number of an n x m hexagonal rectangle was found by the authors and the Wiener numbers of three types of II x M hexagonal jagged-rectangles are obtained.
TL;DR: A distributed algorithm is presented for generating all maximal cliques in a network graph, based on the sequential version of Tsukiyama et al.
Abstract: A distributed algorithm is presented for generating all maximal cliques in a network graph, based on the sequential version of Tsukiyama et al. [TIAS77]. The time complexity of the proposed approach is restricted to the induced neighborhood of a node, and the communication complexity is O(md) where m is the number of connections, and d is the maximum degree in the graph. Messages are O(log n) bits long, where n is the number of nodes (processors) in the system. As an application, a distributed algorithm for constructing the clique graph K (G) from a given network graph G is developed within the scope of dynamic transformations of topologies.
01 Jan 1997
TL;DR: In this article, the first subexponential algorithm for the exploration problem has been presented, which achieves an upper bound of dO(log d) m for the problem and a matching lower bound of $d^{Omega(d)m$ for the algorithm.
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
TL;DR: This paper discusses the case of i = X + 1 and shows that any max λ-min m λ+1 graph is max δ-min £1, and generalizes the notion of a max ε ≥ λ, graph to a max €‚¬1, graph (i ≥ ε), which is then shown to give at least a max â‚£1 graph for each pair of positive integers n and e.
Abstract: It was proved that the design problem of the reliable networks for small edge failure probability is equivalent to finding a max λ-min m λ graph for given numbers of nodes n and edges e with λ = [2e/n] by Bauer et al. Furthermore, at least a max λ-min m λ graph was given for each pair of n and e(≥n) in another paper by Bauer et al. In the present paper, we first generalize the notion of a max λ-min m λ graph to a max λ-min m, graph (i ≥ λ); then we discuss the case of i = X + 1 and show that any max λ-min m λ+1 graph is max λ-min m λ and give at least a max λ-min m λ+1 graph for each pair of positive integers n and e.
22 Nov 1997
TL;DR: This work presents some approximation algorithms that run in polynomial time and lead to very good (but not necessarily optimal) colorings of connected graphs.
Abstract: To guarantee the optimal bipartite vertex coloring (bipartization) of a connected graph requires a coloring algorithm that is NP-complete, effectively preventing bipartization of even modest sized graphs. We present some approximation algorithms that run in polynomial time and lead to very good (but not necessarily optimal) colorings.
18 Sep 1997
TL;DR: This paper presents non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time algorithms that take as input any graph from one of these classes and produce as output such a drawing.
Abstract: A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. In this paper we consider the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time (real RAM) algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but none of which is Delaunay drawable—that is, drawable as a Delaunay triangulation.
18 Sep 1997
TL;DR: It is proved that testing whether a graph admits a rectilinear (without bends) embedding essentially equivalent to a given embedding, and that given a graph, testing if there exists a surface such that the graph admitsA rectILinear embedding in that surface are NP-complete problems and hence the corresponding optimization problems are NP -hard.
Abstract: This paper is devoted to the study of graph embeddings in the grid of non-planar surfaces. We provide an adequate model for those embeddings and we study the complexity of minimizing the number of bends. In particular, we prove that testing whether a graph admits a rectilinear (without bends) embedding essentially equivalent to a given embedding, and that given a graph, testing if there exists a surface such that the graph admits a rectilinear embedding in that surface are NP-complete problems and hence the corresponding optimization problems are NP-hard.
TL;DR: In this paper, the general elliptic matrices are investigated and the signs of coefficients of the eigenpolynomials are discussed, and Smith's result that a simple connected graph is completely multipartite iff it has exactly one positive eigenvalue is reproved.
01 Jan 1997
TL;DR: In this paper, the problem of describing the classes of connected, those of connectivity n, and of n-connected graphs with he MPP is reduced to describing the n-partite graphs with the MPP.
Abstract: In [1] the metric prolongation property (MPP) is introduced for discrete metric spaces, and connections are pointed out of the MPP with metric properties of the locally-isometric embeddings. In the present paper, we continue the study of the MPP for finite graphs [1 – 5]. Some operations invariant with respect to the MPP are considered whose application simplifies the study of the MPP for certain classes of graphs. A theorem is proved on the isometric embedding preserving the MPP of an arbitrary graph to a graph of given connectivity, and it is proved that the problems of describing the classes of connected, those of connectivity n, and of n-connected graphs with he MPP are equivalent. For graphs of small connectivity, in [3,5], and also in the ]present paper, there is obtained a complete characterization of the natural classes of such graphs with the MPP. Conditions are given of the invariance of the MPP for the join operation, and the complete n-partite graphs with the MPP are described. An isometric embedding of an arbitrary graph of diameter 2 to an appropriate graph with the MPP is constructed. Cactii with the MPP are described, which extends the description of trees and unicyclic graphs with the MPP obtained by the author in [3,4].