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Distributed verification and hardness of distributed approximation
 CoRR
"... We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows in t ..."
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Cited by 19 (6 self)
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We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows in the end of the process whether H has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many A full version of this paper is available as [5] at
A Simple Randomized Scheme for Constructing LowWeight kConnected Spanning Subgraphs with Applications to Distributed Algorithms
"... The main focus of this paper is the analysis of a simple randomized scheme for constructing lowweight kconnected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimumweight kconnected spanning subgraph in a weighted complete graph, a NPh ..."
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Cited by 3 (1 self)
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The main focus of this paper is the analysis of a simple randomized scheme for constructing lowweight kconnected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimumweight kconnected spanning subgraph in a weighted complete graph, a NPhard 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 kconnected spanning subgraph (for any k ≥ 1) in a pointtopoint distributed model, where the processors form a complete network. Our algorithm takes O(log n n k) time and expected O(nk log
How to use random circulations to find small cuts
, 2007
"... 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 Kcirculations on G. This leads to efficient algorithms for computing the cut edges, cut edgepairs, and cut vertices of a graph; for exampl ..."
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Cited by 1 (1 self)
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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 Kcirculations on G. This leads to efficient algorithms for computing the cut edges, cut edgepairs, 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 edgepairs 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
FaultTolerance Through kConnectivity
"... Abstract—We consider a system composed of autonomous mobile robots that communicate by exchanging messages in a wireless adhoc 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 f ..."
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Cited by 1 (1 self)
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Abstract—We consider a system composed of autonomous mobile robots that communicate by exchanging messages in a wireless adhoc 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 faulttolerant connected network while allowing robots to perform other tasks. Specifically we describe two local distributed algorithms to determine if a graph is kconnected. We then describe how these algorithms could be used to extend the connectivity maintenance algorithm of [2, 3] to maintain a kconnected network while allowing robots to perform other tasks. I.
Reliably Detecting Connectivity using Local Graph Traits
"... 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 in ..."
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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 kconnectivity. First we prove a negative result that shows there does not exist a local graph trait which perfectly captures graph kconnectivity. We then present three different local graph traits which can be used to reliably predict the kconnectivity 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 kconnected. As a more elaborate application of local graph traits, we describe, and prove the correctness of, a local distributed algorithm that preserves kconnectivity in mobile ad hoc networks while allowing nodes to move independently whenever possible. 1
Fast Computation of Small Cuts via Cycle Space Sampling ∗
, 2009
"... We describe a new samplingbased method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to si ..."
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Cited by 1 (0 self)
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We describe a new samplingbased method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new lineartime sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. 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, and the maximum degree by ∆. We obtain simple O(D)time distributed algorithms to find all cut edges, 2edgeconnected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal — i.e. a Ω(D)time lower bound holds on every graph. We obtain a O(D + ∆ / log V )time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when ∆, D = O ( √ V ). A simple extension of our work yields the first distributed algorithm with sublinear time for 3edgeconnected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3edgeconnected components. 1
Fast Distributed Computation of Cuts via Random Circulations
"... Abstract. We describe a new circulationbased 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 inlabels equals the sum of outlabels. For an integer k, our approach is based on simple algorithms for ..."
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Abstract. We describe a new circulationbased 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 inlabels equals the sum of outlabels. 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 lineartime sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph, and hence 2edgeconnected and 3edgeconnected 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 +ÔV log ∗ V )time algorithm to identify all cut vertices, 2edgeconnected 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 3edgeconnected components. We obtain simple O(D)time distributed algorithms to find all cut edges, 2edgeconnected 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 sublinear time for cut pairs and 3edgeconnected 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(ÔV ). The basic distributed algorithms are Monte Carlo, but can be made Las Vegas without increasing the asymptotic complexity. 1
On Mixed Connectivity Certificates (Extended Abstract)
, 1995
"... 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 kconnectivity for a graph are obtained by removing a subset of its edges, while preserving its connectivity up to k. We unify the previous wor ..."
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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 kconnectivity 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 communicationoptimal distributed algorithm for finding mixed connectivity certificates.