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Randomized 3D Geographic Routing
"... Abstract—We reconsider the problem of geographic routing in wireless ad hoc networks. We are interested in local, memoryless routing algorithms, i.e. each network node bases its routing decision solely on its local view of the network, nodes do not store any message state, and the message itself can ..."
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Abstract—We reconsider the problem of geographic routing in wireless ad hoc networks. We are interested in local, memoryless routing algorithms, i.e. each network node bases its routing decision solely on its local view of the network, nodes do not store any message state, and the message itself can only carry information about O(1) nodes. In geographic routing schemes, each network node is assumed to know the coordinates of itself and all adjacent nodes, and each message carries the coordinates of its target. Whereas many of the aspects of geographic routing have already been solved for 2D networks, little is known about higherdimensional networks. It has been shown only recently that there is in fact no local memoryless routing algorithm for 3D networks that delivers messages deterministically. In this paper, we show that a cubic routing stretch constitutes a lower bound for any local memoryless routing algorithm, and propose and analyze several randomized geographic routing algorithms which work well for 3D network topologies. For unit ball graphs, we present a technique to locally capture the surface of holes in the network, which leads to 3D routing algorithms similar to the greedyfacegreedy approach for 2D networks. I.
Mobile Geometric Graphs: Detection, Coverage and Percolation
"... Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks. In this paper we consider a natural extension of the random geometric graph ..."
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Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks. In this paper we consider a natural extension of the random geometric graph model to the mobile setting by allowing nodes to move in space according to Brownian motion. We study three fundamental questionsinthismodel: detection (the time until a given target point—which may be either fixed or moving—is detected by the network), coverage (the time until all points inside a finite box are detected by the network), and percolation (the time until a given node is able to communicate with the giant component of the network). We derive precise asymptotics for these problems by combining ideas from stochastic geometry, coupling and multiscale analysis. We also give an application of our results to analyze the time to broadcast a message in a mobile network.
Opportunistic forwarding in wireless networks with duty cycling
 in Proc. of ACM Workshop on Challenged Networks (CHANTS
, 2008
"... Opportunistic forwarding, by which data is randomly relayed to a neighbor based on local network information, is a faulttolerant distributed algorithm particularly useful for challenged ad hoc and sensor networks where it is difficult to obtain global topology information because of frequent disrup ..."
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Cited by 18 (7 self)
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Opportunistic forwarding, by which data is randomly relayed to a neighbor based on local network information, is a faulttolerant distributed algorithm particularly useful for challenged ad hoc and sensor networks where it is difficult to obtain global topology information because of frequent disruptions. Also, duty cycling is a common technique that constrains the RF operations of wireless devices for saving the battery energy and thus extending the longevity of the network. The combination of opportunistic forwarding and duty cycling is a useful approach for wireless ad hoc and sensor networks that are plagued with energy constraints and poor connectivity. However, such a design is hampered by the difficulty of analyzing and controlling its performance, particularly, the endtoend latency. This paper presents analytical results that shed light on the latency of opportunistic forwarding in wireless networks with duty cycling. In particular, we give approximation formulas and bounds for the expected latency of opportunistic forwarding in presence of duty cycling for general finite network topologies, and an exact formula for a specific regular network topology that captures some common sensor network deployment scenarios. Moreover, our results concern finitesized networks, and hence, are practically more useful than other asymptotic analyses in the literature.
Exact analysis of latency of stateless opportunistic forwarding
 In The 28th IEEE Conference on Computer Communications (IEEE INFOCOM
, 2009
"... Abstract—Stateless opportunistic forwarding is a simple faulttolerant distributed approach for data delivery and information querying in wireless ad hoc networks, where packets are forwarded to the next available neighbors in a “random walk” fashion, until they reach the destinations or expire. This ..."
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Abstract—Stateless opportunistic forwarding is a simple faulttolerant distributed approach for data delivery and information querying in wireless ad hoc networks, where packets are forwarded to the next available neighbors in a “random walk” fashion, until they reach the destinations or expire. This approach is robust against ad hoc topology changes and is amenable to computation/bandwidth/energyconstrained devices; however, it is generally difficult to predict the endtoend latency suffered by such a random walk in a given network. In this paper, we make several contributions on this topic. First, by using spectral graph theory we derive a general formula for computing the exact hitting and commute times of weighted random walks on a finite graph with heterogeneous sojourn times at relaying nodes. Such sojourn times can model heterogeneous duty cycling rates in sensor networks, or heterogeneous delivery times in delay tolerant networks. Second, we study a common class of distanceregular networks with varying numbers of geographical neighbors, and obtain simple estimateformulas of hitting times by numerical analysis. Third, we study the more sophisticated settings of random geographical locations and distancedependent sojourn times through simulations. Finally, we discuss the implications of this on the optimization of latencyoverhead tradeoff. Index Terms—Opportunistic forwarding, Wireless sensor networks, Delay tolerant networks, Random walks on finite graphs,
The impact of mobility on gossip algorithms
 IN PROC. 28TH CONF. COMPUTER COMMUNICATIONS (INFOCOM), RIO DE JANEIRO
, 2009
"... We analyze how node mobility can influence the convergence time of averaging gossip algorithms on networks. Our main result is that even a small number of fully mobile nodes can yield a significant decrease in convergence time. We develop a method for deriving lower bounds on the convergence time b ..."
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Cited by 17 (2 self)
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We analyze how node mobility can influence the convergence time of averaging gossip algorithms on networks. Our main result is that even a small number of fully mobile nodes can yield a significant decrease in convergence time. We develop a method for deriving lower bounds on the convergence time by merging nodes according to their mobility pattern. We use this method to show that if the agents have onedimensional mobility in the same direction the convergence time is improved by at most a constant. We also obtain upper bounds on the convergence time using the Poincaré inequality and show that simple models of mobility can dramatically accelerate gossip as long as the mobility paths significantly overlap. We use simulations to show that our bounds are still valid for more general mobility models that seem analytically intractable, and further illustrate that different mobility patterns can have significantly different effects on the convergence of distributed algorithms.
The cover time of random geometric graphs
, 2009
"... We study the cover time of random geometric graphs. Let I(d) = [0,1] d denote the unit torus in d dimensions. Let D(x,r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0,1) in d dimensions. A random geometric graph G = G(d,r,n) in d dimensions is defined as follows: Sam ..."
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Cited by 15 (2 self)
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We study the cover time of random geometric graphs. Let I(d) = [0,1] d denote the unit torus in d dimensions. Let D(x,r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0,1) in d dimensions. A random geometric graph G = G(d,r,n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x,r) of radius r about x. The vertex set V (G) = V and the edge set E(G) = {{v,w} : w ̸ = v, w ∈ D(v,r)}. Let G(d,r,n), d ≥ 3 be a random geometric graph. Let c> 1 be constant, and let r = (clog n/(Υdn)) 1/d. Then whp c CG ∼ clog n log n. c − 1 1
Efficient Broadcast on Random Geometric Graphs
"... A Random Geometric Graph (RGG) in two dimensions is constructed by distributing n nodes independently and uniformly at random in [0, √ n] 2 and creating edges between every pair of nodes having Euclidean distance at most r, for some prescribed r. We analyze the following randomized broadcast algori ..."
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Cited by 11 (2 self)
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A Random Geometric Graph (RGG) in two dimensions is constructed by distributing n nodes independently and uniformly at random in [0, √ n] 2 and creating edges between every pair of nodes having Euclidean distance at most r, for some prescribed r. We analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that with probability 1−O(n −1) this algorithm informs every node in the largest connected component of an RGG within O ( √ n/r+logn) rounds. This holds for any value of r larger than the critical value for the emergence of a connected component with Ω(n) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in [0, √ n] 2, the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is Θ ( √ n/r), which surprisingly has been an open problem so far.
Locationaided fast distributed consensus
 IEEE Transactions on Information Theory
, 2007
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Probabilistic quorum systems in wireless ad hoc networks
 In Proceedings of the 38th IEEE International Conference on Dependable Systems and Networks (DSNDCCS
, 2008
"... Quorums are a basic construct in solving many fundamental distributed computing problems. One of the known ways of making quorums scalable and efficient is by weakening their intersection guarantee to being probabilistic. This paper explores several access strategies for implementing probabilistic q ..."
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Cited by 10 (5 self)
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Quorums are a basic construct in solving many fundamental distributed computing problems. One of the known ways of making quorums scalable and efficient is by weakening their intersection guarantee to being probabilistic. This paper explores several access strategies for implementing probabilistic quorums in ad hoc networks. In particular, we present the first detailed study of asymmetric probabilistic biquorum systems, that allow to mix different access strategies and different quorums sizes, while guaranteeing the desired intersection probability. We show the advantages of asymmetric probabilistic biquorum systems in ad hoc networks. Such an asymmetric construction is also useful for other types of networks with non uniform access costs (e.g, peertopeer networks). The paper includes both a formal analysis of these approaches backed up by an extensive simulation based study. In particular, we show that one of the strategies that uses Random Walks, exhibits the smallest communication overhead, thus being very attractive for ad hoc networks. Categories and Subject Descriptors: C.2.1 [Comp.Communication Networks]: Network Architecture and Design—Wireless communication;
ERROR SCALING LAWS FOR LINEAR OPTIMAL ESTIMATION FROM RELATIVE MEASUREMENTS
, 2009
"... We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the diffe ..."
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Cited by 8 (1 self)
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We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables. We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in R d, 1 ≤ d ≤ 3, that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable’s estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithmindependent limits on the estimation accuracy achievable in large graphs.