Abstract not found

The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimum-weight k-connected spanning subgraph in a weighted complete graph, a NP-hard problem even if the weights satisfy the triangle inequality. We show that our algorithm gives an approximation ratio of O(k log n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0, 1] 2 and O(log n k) for a complete graph with random edge weights U(0, 1). We show that our scheme is optimal with respect to the amount of “local information ” needed to compute any connected spanning subgraph. We then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k-connected spanning subgraph (for any k ≥ 1) in a point-to-point distributed model, where the processors form a complete network. Our algorithm takes O(log n n k) time and expected O(nk log

Let K be an abelian group and G be a connected graph, both finite. Using basic properties of circulations, we show that it is easy to generate uniformly random K-circulations on G. This leads to efficient algorithms for computing the cut edges, cut edge-pairs, and cut vertices of a graph; for example, the cut edges are “usually ” the edges where a random circulation vanishes. In the distributed setting, we improve the best known time complexity of any algorithm for finding cut edge-pairs to O(Diam), and for cut vertices to O(Diam + ∆ / log |V |), where Diam is the diameter of the graph and ∆ is the maximum degree. Our algorithms are the Las Vegas kind and use messages of length O(log |V |). The distributed cut vertex algorithm can also be used to find the blocks of G. 1

Abstract—We consider a system composed of autonomous mobile robots that communicate by exchanging messages in a wireless ad-hoc network. In this setting, the failure of a single robot might result in the disconnection of the communication network. We are concerned with the problem of maintaining a fault-tolerant connected network while allowing robots to perform other tasks. Specifically we describe two local distributed algorithms to determine if a graph is k-connected. We then describe how these algorithms could be used to extend the connectivity maintenance algorithm of [2, 3] to maintain a k-connected network while allowing robots to perform other tasks. I.

Abstract. We describe a new circulation-based method to determine cuts in an undirected graph. A circulation is an oriented labeling of edges with integers so that at each vertex, the sum of the in-labels equals the sum of out-labels. For an integer k, our approach is based on simple algorithms for sampling a circulation (mod k) uniformly at random. We prove that with high probability, certain dependencies in the random circulation correspond to cuts in the graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph, and hence 2-edge-connected and 3-edge-connected components. In the model of distributed computing in a graph G = (V, E) with O(log |V |)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by D. Previously, Thurimella [J. Algorithms, 1997] gave a O(D +Ô|V |log ∗ |V |)-time algorithm to identify all cut vertices, 2-edgeconnected components, and cut edges, and Tsin [Int. J. Found. Comput. Sci., 2006] gave a O(|V |+D 2)time algorithm to identify all cut pairs and 3-edge-connected components. We obtain simple O(D)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving previous time bounds on all graphs. Under certain assumptions these new algorithms are universally optimal, due to a Ω(D)-time lower bound on every graph. These results yield the first distributed algorithms with sub-linear time for cut pairs and 3-edge-connected components. Let ∆ denote the maximum degree. We obtain a O(D + ∆/log |V |)-time distributed algorithm for finding cut vertices; this is faster than Thurimella’s algorithm on all graphs with ∆, D = O(Ô|V |). The basic distributed algorithms are Monte Carlo, but can be made Las Vegas without increasing the asymptotic complexity. 1

Vertex and edge connectivity are special cases of mixedconnectivity, in which all edges and a specified set of vertices play a similar role. Certificates of k-connectivity for a graph are obtained by removing a subset of its edges, while preserving its connectivity up to k. We unify the previous work on connectivity certificates and extend it to handle mixed connectivity and multigraphs. Our treatment contributes a new insight of the pertinent structures, yielding more general results and simpler proofs. Also, we present a communication-optimal distributed algorithm for finding mixed connectivity certificates.

The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimum-weight k-connected spanning subgraph in a weighted complete graph, a NP-hard problem even if the weights satisfy the triangle inequality. We show that our algorithm gives an approximation ratio of O(k log n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0, 1] 2 and O(log n k) for a complete graph with random edge weights U(0, 1). We show that our scheme is optimal with respect to the amount of “local information ” needed to compute any connected spanning subgraph. We then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k-connected spanning subgraph (for any k ≥ 1) in a point-to-point distributed model, where the processors form a complete network. Our algorithm takes O(log n n k) time and expected O(nk log

Abstract. Local distributed algorithms can only gather sufficient information to identify local graph traits, that is, properties that hold within the local neighborhood of each node. However, it is frequently the case that global graph properties (connectivity, diameter, girth, etc) have a large influence on the execution of a distributed algorithm. This paper studies local graph traits and their relationship with global graph properties. Specifically, we focus on graph k-connectivity. First we prove a negative result that shows there does not exist a local graph trait which perfectly captures graph k-connectivity. We then present three different local graph traits which can be used to reliably predict the k-connectivity of a graph with varying degrees of accuracy. As a simple application of these results, we present upper and lower bounds for a local distributed algorithm which determines if a graph is k-connected. As a more elaborate application of local graph traits, we describe, and prove the correctness of, a local distributed algorithm that preserves k-connectivity in mobile ad hoc networks while allowing nodes to move independently whenever possible. 1

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