Abstract: In this paper, we presented a fully distributed algorithm to compute a planar subgraph of the underlying wireless connectivity graph. We considered the idealized unit disk graph model in which nodes are assumed to be connected if and only if nodes are within their transmission range. The main contribution of this work is a fully distributed algorithm to extract the connected, planar graph for routing in the wireless networks. The communication cost of the proposed algorithm is O(d log d) bits, where d is the degree of anode. In addition, this paper also presented a geometric routing algorithm. The algorithm is fully distributed and nodes know only the position of other nodes and can communicate with neighboring nodes in their transmission range

Abstract: The quality of an embedding Φ : V → R2 of a graph G = (V,E) into the Euclidean plane is the ratio of max{u,v}∈E ∥Φ(u)-Φ(v)∥2 to min{u,v}∉E ∥Φ(u)-Φ(v)∥2. Given a unit disk graph G = (V,E), we seek algorithms to compute an embedding Φ : V → R2 of best (smallest) quality. This paper presents a simple, combinatorial algorithm for computing a O(log2.5 n)-quality 2-dimensional embedding of a given unit disk graph. Note that G comes with no associated geometric information. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in RO(1). Our results extend to unit ball graphs (UBGs) in fixed dimensional Euclidean space. Constructing a "growth-restricted approximation" of the given unit disk graph lies at the core of our algorithm. This approach allows us to bypass the standard and costly technique of solving a linear program with exponentially many "spreading constraints". As a side effect of our construction, we get a constant-factor approximation to the minimum clique cover problem on UBGs, described without geometry. Our problem is a version of the well known localization problem in wireless networks.

Abstract: Unit disk graphs are the intersection graphs of equal sized disks in the plane, they are widely used as a mathematical model for wireless ad-hoc networks and some problems in computational geometry. In this paper we first show that the Roman domination problem in unit disk graphs is NP-hard, and then present a simple linear time approximation algorithm and a polynomial-time approximation scheme for this problem, respectively.

Abstract: Flooding in wireless ad hoc networks is a fundamental and critical operation in supporting various applications and protocols. However, the traditional flooding scheme generates excessive redundant packet retransmissions, causing contention, packet collisions and ultimately wasting precious limited bandwidth and energy. In this paper, we propose an efficient flooding protocol called vertex forwarding,which minimizes the flooding traffic by leveraging location information of 1-hop neighbor nodes. Our scheme works as if there were existing a hexagonal grid in the network field to guide the flooding procedure, only the vertex nodes which are located at or nearest to the vertices of the grid should be nominated to forward the message. We also provide a distributed algorithm for finding the vertex nodes. Simulation results show that our scheme is so efficient that it is almost able to reduce the number of forward nodes to the lower bound.

Abstract: The unreachability problem (i.e. the so-called void problem) which exists in the greedy routing algorithms has been studied for the wireless sensor networks. However, most of the current research work can not fully resolve the problem (i.e. to ensure the delivery of packets) within their formulation. In this letter, the greedy anti-void routing (GAR) protocol is proposed, which solves the void problem by exploiting the boundary finding technique for the unit disk graph (UDG). The proposed rolling-ball UDG boundary traversal (RUT) is employed to completely guarantee the delivery of packets from the source to the destination node. The proofs of correctness for the proposed GAR protocol are also given at the end of this letter.

Abstract: Many applications in sensor networks require positional information of the sensors. Recovering node positions is closely related to graph realization problems for geometric graphs. Here, we address the case where nodes have angular information. Whereas Bruck et al. proved that the corresponding realization problem together with unit-disk-graph-constraints is $\mathcal{NP}$-hard [2], we focus on rigid components which allow both efficient identification and fast, unique realizations. Our technique allows to identify maximum rigid components in graphs with partially known rigid components using a reduction to maximum flow problems. This approach is analyzed for the two-dimensional case, but can easily be extended to higher dimensions.

Abstract: We present the first location oblivious distributed unit disk graph coloring algorithm having a provable performance ratio of three (ie the number of colors used by the algorithm is at most three times the chromatic number of the graph) This is an improvement over the standard sequential coloring algorithm since we present a new lower bound of 10/3 for the worst-case performance ratio of the sequential coloring algorithm The previous greatest lower bound on the performance ratio of the sequential coloring algorithm was 5/2 Using simulation, we also compare our algorithm with other existing unit disk graph coloring algorithms

Abstract: Interference due to transmissions by adjacent nodes in a multi-hop wireless network can be modeled using a unit disc graph (UDG). We investigate the reliability associated with using the clique number instead of the chromatic number of the UDG while computing the interference. In our extensive simulations with UDGs of random networks, we observed that the clique number and the chromatic number values were typically very close to one another and the maximum deviation was much less than the theoretical bounds. This implies very high reliability in the proposed approximation.

Abstract: We present an approach to estimating distances in sensor networks. It works by counting common neighbors, high values indicating closeness. Such distance estimates are needed in many self-localization algorithms. Other than many other approaches, ours does not rely on special equipment in the devices.

Abstract: This paper presented a fully distributed algorithm to compute a planar subgraph for geo-routing in ad hoc wireless networks. We considered the idealized unit disk graph model in which nodes are assumed to be connected if and only if nodes are within their transmission range. The main contribution of this work is a fully distributed algorithm to extract the connected, planar graph for routing in the wireless networks. The communication cost of the proposed algorithm is O(d log d) bits, where d is the degree of a node.

Abstract: The focus of the present paper is on providing a local deterministic algorithm for colouring the edges of Yao-like subgraphs of Unit Disc Graphs. These are geometric graphs such that for some positive integers l, k the following property holds at each node v: if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v. We assume that the nodes are location aware, i.e. they know their Cartesian coordinates in the plane. The algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant (depending on k and l, but not on the size of the graph) number of hops away from it, and hence the algorithm terminates in a constant number of steps. The number of colours used is 2kl + 1 and this is optimal for local algorithms (since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm), thus showing that in this class of graphs the price for locality is only one additional colour.

Abstract: We introduce a generalization of the Yao graph where the cones used are adaptively centered on a set of nearest neighbors for each node, thus creating a directed or undirected spanning subgraph of a given unit disk graph (UDG). We also permit the apex of the cones to be positioned anywhere along the line segment between the node and its nearest neighbor, leading to a class of Yao-type subgraphs. We show that these locally constructed spanning subgraphs are strongly connected, have bounded out-degree, are t-spanners with bounded stretch factor, contain the Euclidean minimum spanning tree as a subgraph, and are orientationinvariant. Since a continuous set of cone angles are possible, these subgraphs also permit control over the degree of the graph. We demonstrate through simulations that these subgraphs of the UDG combines the desirable properties of the Yao and the Half Space Proximal subgraphs of the UDG.

Abstract: In this paper, we presented a fully distributed algorithm to compute a planar subgraph of the underlying wireless connectivity graph. We considered the idealized unit disk graph model in which nodes are assumed to be connected if and only if nodes are within their transmission range. The main contribution of this work is a fully distributed algorithm to extract the connected, planar graph for routing in the wireless networks. The communication cost of the proposed algorithm is O(d log d) bits, where d is the degree of a node. In addition, this paper also presented a geometric routing algorithm and established its lower bound. The algorithm is fully distributed and nodes know only the position of other nodes and can communicate with neighboring nodes in their transmission range.

Abstract: We introduce a family of directed geometric graphs, denoted $\paz$, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the $\paz$ graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the $\paz$ graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on $\paz$. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of $\paz$ is better than the proven theoretical bounds.

Abstract: Session 1. Invited Talks.- Fast Distributed Algorithms Via Primal-Dual (Extended Abstract).- Time Optimal Gathering in Sensor Networks.- Treewidth: Structure and Algorithms.- Session 2. Autonomous Systems: Graph Exploration.- Fast Periodic Graph Exploration with Constant Memory.- Why Robots Need Maps.- Graph Searching with Advice.- Session 3. Distributed Algorithms: Fault Tolerance.- From Renaming to Set Agreement.- A Self-stabilizing Algorithm for the Median Problem in Partial Rectangular Grids and Their Relatives.- A New Self-stabilizing Maximal Matching Algorithm.- Session 4. Distributed Algorithms and Data Structures.- Labeling Schemes with Queries.- A Simple Optimistic Skiplist Algorithm.- Data Aggregation in Sensor Networks: Balancing Communication and Delay Costs.- Session 5. Autonomous Systems: Location Problems.- Optimal Moves for Gossiping Among Mobile Agents.- Swing Words to Make Circle Formation Quiescent.- Distributed Algorithms for Partitioning a Swarm of Autonomous Mobile Robots.- Session 6. Wireless Networks.- Local Edge Colouring of Yao-Like Subgraphs of Unit Disk Graphs.- Proxy Assignments for Filling Gaps in Wireless Ad-Hoc Lattice Computers.- Location Oblivious Distributed Unit Disk Graph Coloring.- Session 7. Communication Networks: Fault Tolerance.- Edge Fault-Diameter of Cartesian Product of Graphs.- Rapid Almost-Complete Broadcasting in Faulty Networks.- Design of Minimal Fault Tolerant On-Board Networks: Practical Constructions.- Session 8. Autonomous Systems: Fault Tolerance.- Dynamic Compass Models and Gathering Algorithms for Autonomous Mobile Robots.- Fault-Tolerant Simulation of Message-Passing Algorithms by Mobile Agents.- Session 9. Communication Networks: Parallel Computing and Selfish Routing.- Optimal Conclusive Sets for Comparator Networks.- Selfish Routing with Oblivious Users.- Upper Bounds and Algorithms for Parallel Knock-Out Numbers.

Abstract: We present a strictly-localized distributed algorithm that, given a wireless ad-hoc network modeled as a unit disk graph in the plane, constructs a planar power spanner of whose degree is bounded by and whose stretch factor is bounded by , where is an integer parameter and ! is the power exponent constant. For the same degree bound , the stretch factor of our algorithm significantly improves the previous best bounds by Song et al. and Kanj and Perkovic. We show that this bound is near-optimal by proving that the slightly smaller stretch factor of " # $ % &(') is unattainable for the same degree bound . In contrast to previous algorithms by Song et al. and by Kanj and Perkovic, the presented algorithm is strictly localized: the construction of the power spanner depends solely on the local structure and does not require information propagation. As a consequence, the algorithm is highly scalable and robust. Moreover, on a graph with * points and + edges the algorithm exchanges no more than ,+messages and has a local processing time of ,. / 01.2 435,*6/ 01* at a node of degree . . Finally, while the algorithm is efficient and easy to implement in practice, it relies on deep insights on the geometry of unit disk graphs and novel techniques that are of independent interest.

Abstract: In the last decade there has been an increasing interest in graphs whose nodes are placed in the plane. In particular when modeling the communication pattern in wireless ad-hoc networks. The different communication ways or protocol implementations have directed the interest of the commnity to study and use different intersection graph families as basic models of communication. In this talk we review those models when the graph nodes are placed at random in the plane. In general we assume that the set of vertices is a random set of points generated by placing n points uniformly at random in the unit square. The basic distance model, the random geometric graph connects two points if they are at distance at most r where r is a parameter of the model [2]. A second model is the k-neighbor graph in which each node selects as neighbors the k-nearest neighbors in P [3]. Another variation, inspired by the communication pattern of directional radio frequency and optical networks, is the random sector graph, a generalization of the random geometric graph introduced in [1]. In the setting under consideration, each node has a fixed angle α (0 < α ≤ 2π) defining a sector Si of transmission determined by a random angle between the sector and the horizontal axis. Every node that falls inside of Si can potentially receive the signal emitted by i. In this talk we survey the main properties and parameters of such random graph families, and their use in the modelling and experimental evaluation of algorithms and protocols for several network problems.

Abstract: In the last decade there has been an increasing interest in graphs whose nodes are placed in the plane In particular when modeling the communication pattern in wireless ad-hoc networks The different communication ways or protocol implementations have directed the interest of the commnity to study and use different intersection graph families as basic models of communication In this talk we review those models when the graph nodes are placed at random in the plane In general we assume that the set of vertices is a random set of points generated by placing n points uniformly at random in the unit square The basic distance model, the random geometric graph connects two points if they are at distance at most r where r is a parameter of the model [2] A second model is the k-neighbor graph in which each node selects as neighbors the k-nearest neighbors in P [3] Another variation, inspired by the communication pattern of directional radio frequency and optical networks, is the random sector graph, a generalization of the random geometric graph introduced in [1] In the setting under consideration, each node has a fixed angle α (0 < α ≤ 2π) defining a sector S i of transmission determined by a random angle between the sector and the horizontal axis Every node that falls inside of S i can potentially receive the signal emitted by i In this talk we survey the main properties and parameters of such random graph families, and their use in the modelling and experimental evaluation of algorithms and protocols for several network problems

Abstract: Unlike a cellular or wired network, there is no base station or network infrastructure in a wireless ad-hoc network, in which nodes communicate with each other via peer communications. In order to make routing and flooding efficient in such an infrastructureless network, Connected Dominating Set (CDS) as a virtual backbone has been extensively studied. Most of the existing studies on the CDS problem have focused on unit disk graphs, where every node in a network has the same transmission range. However, nodes may have different powers due to difference in functionalities, power control, topology control, and so on. In this case, it is desirable to model such a network as a disk graph where each node has different transmission range. In this paper, we define Minimum Strongly Connected Dominating and Absorbent Set (MSCDAS) in a disk graph, which is the counterpart of minimum CDS in unit disk graph. We propose a constant approximation algorithm when the ratio of the maximum to the minimum in transmission range is bounded. We also present two heuristics and compare the performances of the proposed schemes through simulation.

Abstract: We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth which are used to model wireless communication networks. The approach presented yields a robust algorithm, that is, it accepts any undirected graph as input, and returns a (1+@e)-approximate minimum independent dominating set, or a certificate showing that the input graph does not satisfy the bounded growth property.

Abstract: We consider the problem of activity scheduling and area coverage in sensor networks, and especially focus on problems that arise when using a more realistic physical layer. Indeed, most of the previous work in this area has been studied within an ideal environment, where messages are always correctly received. In this paper, we argue that protocols developed with such an assumption can hardly provide satisfying results in a more realistic world. To show this, we replace the classic unit disk graph model by the lognormal shadowing one. The results show that either the resulting area coverage is not sufficient or the percentage of active nodes is very high. We thus present an original method, where a node decides to turn off when there exists in its vicinity a sufficiently reliable covering set of neighbors. We show that our solution is very efficient as it preserves area coverage while minimizing the quantity of active nodes.

Abstract: This paper presented a fully distributed algorithm to compute a planar subgraph of the underlying wireless connectivity graph. This work considered the idealized unit disk graph model in which nodes are assumed to be connected if, and only if, nodes are within their transmission range. The main contribution of this work is a fully distributed algorithm to extract the connected, planar graph for routing in the wireless networks. The communication cost of the proposed algorithm is O(d log d) bits, where d is the degree of a node. In addition, this paper also presented a geometric routing algorithm and established its lower bound. The algorithm is fully distributed and nodes know only the position of other nodes and can communicate with neighboring nodes in their transmission range.