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81
Modeling sensor networks
, 2008
"... In order to develop algorithms for sensor networks and in order to give mathematical correctness and performance proofs, models for various aspects of sensor networks are needed. This chapter presents and discusses currently used models for sensor networks. Generally, finding good models is a challe ..."
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Cited by 42 (5 self)
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In order to develop algorithms for sensor networks and in order to give mathematical correctness and performance proofs, models for various aspects of sensor networks are needed. This chapter presents and discusses currently used models for sensor networks. Generally, finding good models is a challenging task. On the one hand, a
Local approximation schemes for ad hoc and sensor networks
 In Proc. 3rd Joint Workshop on Foundations of Mobile Computing (DialMPOMC
, 2005
"... We present two local approaches that yield polynomialtime approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1 + ε)approximation to the problems at hand for any given ε> 0. ..."
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Cited by 39 (9 self)
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We present two local approaches that yield polynomialtime approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1 + ε)approximation to the problems at hand for any given ε> 0. The time complexity of both algorithms is O(TMIS + log ∗n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pairwise independent nodes in every rneighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs.
Leveraging Linial’s Locality Limit
"... www.dcg.ethz.ch Abstract. In this paper we extend the lower bound technique by Linial for local coloring and maximal independent sets. We show that constant approximations to maximum independent sets on a ring require at least logstar time. More generally, the product of approximation quality and r ..."
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Cited by 30 (8 self)
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www.dcg.ethz.ch Abstract. In this paper we extend the lower bound technique by Linial for local coloring and maximal independent sets. We show that constant approximations to maximum independent sets on a ring require at least logstar time. More generally, the product of approximation quality and running time cannot be less than logstar. Using a generalized ring topology, we gain identical lower bounds for approximations to minimum dominating sets. Since our generalized ring topology is contained in a number of geometric graphs such as the unit disk graph, our bounds directly apply as lower bounds for quite a few algorithmic problems in wireless networking. Having in mind these and other results about local approximations of maximum independent sets and minimum dominating sets, one might think that the former are always at least as difficult to obtain as the latter. Conversely, we show that graphs exist, where a maximum independent set can be determined without any communication, while finding even an approximation to a minimum dominating set is as hard as in general graphs. 1
Distributed approximate matching
, 2007
"... We consider distributed algorithms for approximate maximum matching on general graphs. Our main result is a randomized (4 + ɛ)approximation distributed algorithm for weighted maximum matching, whose running time is O(log n) for any constant ɛ> 0, where n is the number of nodes in the graph. In a ..."
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Cited by 22 (2 self)
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We consider distributed algorithms for approximate maximum matching on general graphs. Our main result is a randomized (4 + ɛ)approximation distributed algorithm for weighted maximum matching, whose running time is O(log n) for any constant ɛ> 0, where n is the number of nodes in the graph. In addition, we consider the dynamic case, where nodes are inserted and deleted one at a time. For unweighted dynamic graphs, we give an algorithm that maintains a (1 + ɛ)approximation in O(1/ɛ) time for each node insertion or deletion. For weighted dynamic graphs we give a constantfactor approximation algorithm that runs in constant time for each insertion or deletion.
Improved Distributed Approximate Matching
"... We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Isr ..."
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Cited by 20 (3 self)
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We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Israeli and Itai. As a byproduct, we also provide an improved algorithm for unweighted matchings in bipartite graphs. In the context of weighted graphs, we give another algorithm which provides ( 1 − ɛ) approximation in general 2 graphs in O(log n) time. The latter result improves on the − ɛ)approximation in O(log n) time. known ( 1 4
Networks Cannot Compute Their Diameter in Sublinear Time preliminary version please check for updates
, 2011
"... We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of commun ..."
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Cited by 18 (2 self)
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We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)approximation of the diameter requires ˜ Ω ( √ n) rounds. Furthermore we use our new technique to prove an ˜ Ω ( √ n) lower bound on approximating the girth of a graph by a factor 2 − ε. Contact author:
Efficient distributed approximation algorithms via probabilistic tree embeddings
 IN: PROC. OF THE 27TH SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING
, 2008
"... We present a uniform approach to design efficient distributed approximation algorithms for various network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol, Rao, and Talwar [10] (FRT embedding). We show how to efficiently compute an ..."
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Cited by 18 (3 self)
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We present a uniform approach to design efficient distributed approximation algorithms for various network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol, Rao, and Talwar [10] (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain expected O(log n)approximate distributed algorithms for the generalized Steiner forest problem, the minimum routing cost spanning tree problem, and the ksource shortest paths problem in arbitrary networks. The time complexities of our algorithms are within a polylogarithmic factor of the optimum. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that might be of independent interest. Assuming a global order on the nodes of a network, the LE list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen [3], Cohen and Kaplan [4]). Assuming a random order on the nodes, we give an almostoptimal distributed algorithm for computing LE lists on weighted graphs. For unweighted graphs, our LE lists computation has asymptotically optimal time complexity O(D), where D is the diameter of the network. As a byproduct, we get an improved synchronous leader election algorithm for general networks.
Almost stable matchings by truncating the Gale–Shapley algorithm
 Algorithmica
, 2010
"... We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about ..."
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Cited by 18 (5 self)
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We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributedsystems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + )approximation algorithm for maximumweight matching in bicoloured graphs and a centralised randomised constanttime approximation scheme for estimating the size of a stable matching. 1
FaultTolerant Clustering in Ad Hoc and Sensor Networks
 In 26th International Conference on Distributed Computing Systems (ICDCS
, 2006
"... In this paper, we study distributed approximation algorithms for faulttolerant clustering in wireless ad hoc and sensor networks. A kfold dominating set of a graph G =(V,E) is a subset S of V such that every node v ∈ V \ S has at least k neighbors in S. We study the problem in two network models. ..."
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Cited by 17 (1 self)
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In this paper, we study distributed approximation algorithms for faulttolerant clustering in wireless ad hoc and sensor networks. A kfold dominating set of a graph G =(V,E) is a subset S of V such that every node v ∈ V \ S has at least k neighbors in S. We study the problem in two network models. In general graphs, for arbitrary parameter t, we propose a distributed algorithm that runs in time O(t 2) and achieves an approximation ratio of O(t ∆ 2/t log ∆), wheren and ∆ denote the number of nodes in the network and the maximal degree, respectively. When the network is modeled as a unit disk graph, we give a probabilistic algorithm that runs in time O(log log n) and achieves an O(1) approximation in expectation. Both algorithms require only small messages of size O(log n) bits. 1
Faulttolerant spanners: Better and simpler
 In PODC
, 2011
"... A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to verte ..."
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Cited by 16 (3 self)
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A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults r for all stretch bounds. For stretch k ≥ 3 we design a simple transformation that converts every kspanner construction with at most f(n) edges into an rfaulttolerant kspanner construction with at most O(r 3 log n) · f(2n/r) edges. Applying this to standard greedy spanner constructions gives rfault tolerant kspanners with Õ(r2 1+ 2 n k+1) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [CLPR09] depends similarly on n but exponentially on r (approximately like k r). For the case of k = 2 and unit edgelengths, an O(r log n)approximation is known from recent work of Dinitz and Krauthgamer [DK11], in which several spanner results are obtained using a common approach of rounding a natural flowbased linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log n), which is, notably, independent of the number of faults r. We further strengthen this bound in terms of the maximum degree by using the Lovász Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.