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 higher-dimensional 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 greedy-face-greedy approach for 2D networks. I.

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/energy-constrained devices; however, it is generally difficult to predict the end-to-end 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 distance-regular networks with varying numbers of geographical neighbors, and obtain simple estimate-formulas of hitting times by numerical analysis. Third, we study the more sophisticated settings of random geographical locations and distance-dependent sojourn times through simulations. Finally, we discuss the implications of this on the optimization of latency-overhead trade-off. Index Terms—Opportunistic forwarding, Wireless sensor networks, Delay tolerant networks, Random walks on finite graphs,

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 bi-quorum 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 bi-quorum systems in ad hoc networks. Such an asymmetric construction is also useful for other types of networks with non uniform access costs (e.g, peer-to-peer 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;

Opportunistic forwarding, by which data is randomly relayed to a neighbor based on local network information, is a fault-tolerant 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 end-to-end 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 finite-sized networks, and hence, are practically more useful than other asymptotic analyses in the literature.

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

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.

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 multi-scale analysis. We also give an application of our results to analyze the time to broadcast a message in a mobile network.

We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e.g., wireless networks, the Internet, etc.) need to be studied by using a “richer ” stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph’s clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the ln n chromatic number is identical: χ = (1 + o(1)). Finally, ln ln n we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n) 2, and specify the constant C. 1

In opportunistic forwarding, a node randomly relays packets to one of its neighbors based on local information, without the knowledge of global topology. Each intermediate node continues this process until the packet arrives at its destination. This is particularly attractive in certain types of wireless ad hoc and sensor networks where obtaining accurate knowledge of global topology may be infeasible. However, opportunistic forwarding is hampered by the difficulty to control its performance, particularly, the end-to-end latency. This paper presents new analytical results that shed light on the latency of “random walk”, the simplest type of opportunistic forwarding, for several useful regular network topologies, such as r-nearest cycle that can model variable wireless transmission distance in one dimensional scenario, and a 2D regular torus-type graph that can approximate grid-like deployments. We give new exact and approximation formulas for the maximum expected hitting time of random walk on such topologies.

In recent years the popularity of multi-hop wireless networks has been growing. Its flexible topology and abundant routing path enables many types of applications. However, the lack of a centralized controller often makes it difficult to design a reliable service in multi-hop wireless networks. While packet routing has been the center of attention for decades, recent research focuses on data discovery such as file sharing in multi-hop wireless networks. Although there are many peer-to-peer lookup (P2P-lookup) schemes for wired networks, they have inherent limitations for multi-hop wireless networks. First, a wired P2P-lookup builds a search structure on the overlay network and disregards the underlying topology. Second, the performance guarantee often relies on specific topology models such as random graphs, which do not apply to multi-hop wireless networks. Past studies on wireless P2P-lookup either combined existing solutions with known routing algorithms or proposed tree-based routing, which is prone to traffic congestion. In this paper, we present two wireless P2P-lookup schemes that strictly build a topology-dependent structure. We first propose the Ring Interval Graph Search (RIGS) that constructs a DHT only through direct connections between the