We present a novel distributed algorithm for the maximal independent set (MIS) problem. On growth-bounded graphs (GBG) our deterministic algorithm finishes in O(log ∗ n) time, n being the number of nodes. In light of Linial’s Ω(log ∗ n) lower bound our algorithm is asymptotically optimal. Our algorithm answers prominent open problems in the ad hoc/sensor network domain. For instance, it solves the connected dominating set problem for unit disk graphs in O(log ∗ n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ + 1 coloring in O(log ∗ n) time, where δ is the maximum degree of the graph.

In this paper, we study the capacity of a large-scale random wireless network for multicast. Assume that n wireless nodes are randomly deployed in a square region with side-length a and all nodes have the uniform transmission range r and uniform interference range R> r. We further assume that each wireless node can transmit/receive at W bits/second over a common wireless channel. For each node vi, we randomly pick k − 1 nodes from the other n − 1 nodes as the receivers of the multicast session rooted at node vi. The aggregated multicast capacity is defined as the total data rate of all multicast sessions in the network. In this paper we derive matching asymptotic upper bounds and lower bounds on multicast capacity of random wireless networks. We show that the total multicast capacity is Θ( � n log n · W √ k) when k = O ( n log n

Abstract — We consider a wireless sensor network con-sisting of a set of sensors deployed randomly. A point in the monitored area is covered if it is within the sensing range of a sensor. In some applications, when the network is sufficiently dense, area coverage can be approximated by guaranteeing point coverage. In this case, all the points of wireless devices could be used to represent the whole area, and the working sensors are supposed to cover all the sensors. Many applications related to security and reliability require guaranteed k-coverage of the area at all times. In this paper, we formalize the k-(Connected) Coverage Set (k-CCS/k-CS) problems, develop a linear programming algorithm, and design two non-global solu-tions for them. Some theoretical analysis is also provided followed by simulation results. Index Terms — Coverage problem, linear programming, localized algorithms, reliability, wireless sensor networks.

Abstract — Multipoint relays (MPR) [1] provides a localized and optimized way of broadcasting messages in a mobile ad hoc network (MANET). Using 2-hop neighborhood information, each node determines a small set of forward neighbors to relay messages. Selected forward nodes form a connected dominating set (CDS) to ensure full coverage. Adjih, Jacquet, and Viennot [2] proposed a novel localized algorithm to construct a small CDS based on the original MPR. In this paper, we provide several extensions to generate a smaller CDS using 3-hop neighborhood information to cover each node’s 2-hop neighbor set. In addition, we extend the notion of coverage in the original MPR. We show that the extended MPR has a constant local approximation ratio compared with a logarithmic local ratio in the original MPR. The effectiveness of our approach is confirmed through a simulation study.

Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4) time using O(n 2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)-approximation algorithm with running time O(nm + n 2 (log n +1/ε 3d)) using O(n 2)space.

Given a set S of n points in R D, and an integer k such that 0 � k < n, we show that a geometric graph with vertex set S, at most n − 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n − 1 + k edges has dilation Ω(n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position. 1 Preliminaries and introduction A geometric network is an undirected graph whose vertices are points in R D. Geometric networks, especially geometric networks of points in the plane, arise in many applications. Road networks, railway networks, computer networks—any collection of objects that have some connections between them can be modeled as a geometric network. A natural and widely studied type of geometric network is the Euclidean network, where the weight of an edge is simply the Euclidean distance between its two endpoints. Such networks for points in R D form the topic of study of our paper. When designing a network for a given set S of points, several criteria have to be taken into account. In particular, in many applications it is important to ensure a short connection between every two points in S.

In this paper, we review a recently developed class of algorithms that solve global problems in unit distance wireless networks by means of local algorithms. A local algorithm is one in which any node of a network only has information on nodes at distance at most k from itself, for a constant k. For example, given a unit distance wireless network N, we want to obtain a planar subnetwork of N by means of an algorithm in which all nodes can communicate only with their neighbors in N, perform some operations, and then halt. We review algorithms for obtaining planar subnetworks, approximations to minimum weight spanning trees, Delaunay triangulations, and relative neighbor graphs. Given a unit distance wireless network N, we present new local algorithms to solve the following problems: 1. Calculate small dominating sets (not necessarily connected) of N. 2. Extract a bounded degree planar subgraph H of N and obtain a proper edge coloring of H with at most 12 colors. The second of these algorithms can be used in the channel assignment problem. 1

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We present a protocol to construct routing backbones in wireless networks composed of selfish participants. Backbones are inherently cooperative, so constructing them in selfish environments is particularly difficult; participants want a backbone to exist (so others relay their packets) but do not want to join the backbone (so they do not have to relay packets for others). We model the wireless backbone as a public good and use impatience as an incentive for cooperation. To determine if and when to donate to this public good, each participant calculates how patient it should be in obtaining the public good. We quantify patience using the Volunteer’s Timing Dilemma (VTD), which we extend to general multihop network settings. Using our generalized VTD analysis, each node individually computes as its dominant strategy the amount of time to wait before joining the backbone. We evaluate our protocol using both simulations and an implementation. Our results show that, even though participants in our system deliberately wait before volunteering, a backbone is formed quickly. Further, the quality of the backbone (such as the size and resulting network lifetime) is comparable to that of existing backbone protocols that assume altruistic behavior.

We present a strategy for organizing the communication in wireless ad hoc networks based on a cell structure. We use the unit disk graph model and assume positioning capabilities for all nodes. The cell structure is an abstract view on the network and represents regions where nodes reside (node cells), regions that can be used for the communication flow (link cells) and regions that cannot be bridged due to the restricted transmission range (barrier cells). The cell structure helps to determine local minima for greedy forwarding and improves recovery from such minima, because for recovery all edges can be used in contrast to other topology-based rules that work only on a planar subgraph. For the analysis of positionbased routing algorithms the measures time and traffic are based on the cell structure. The difficulty of exploring the network is expressed by the size of the barriers (number of cells in the perimeters).