Geometric spanner

Abstract: We propose a new routing graph, the Restricted Delaunay Graph (RDG), for ad hoc networks. Combined with a node clustering algorithm RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: (1) it is a planar graph; (2) between any two nodes there exists a path in the RDG whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the optimum length possible; and (3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We also show by simulation that the RDG outperforms the previously proposed routing graphs under the Greedy Perimeter Stateless Routing (GPSR) protocol. In addition, we investigate theoretical bounds on the quality of paths discovered using GPSR

Abstract: Several localized routing protocols (see Bose, P. and Morin, P., Proc. 10th Annual Int. Symp. on Algorithms and Computation ISAAC, 1999) guarantee the delivery of packets when the underlying network topology is the Delaunay triangulation of all wireless nodes. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we more accurately model the network as a unit-disk graph, UDG, in which a link between two nodes exists only if the distance between them is at most the maximum transmission range. Given a graph H, a spanning subgraph G of H is a t-spanner if the length of the shortest path connecting any two points in G is no more than t times the length of the shortest path connecting the two points in H. We present a novel localized networking protocol that constructs a planar 2.5-spanner of UDG, called the localized Delaunay triangulation, as network topology. It contains all edges that are in both the UDG and the Delaunay triangulation of all wireless nodes. Our experiments show that the delivery rates of existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed topologies. The total communication cost of our networking protocol is O(n log n) bits. Moreover, the computation cost of each node u is O(d/sub u/ log d/sub u/), where d/sub u/ is the number of 1-hop neighbors of u in UDG.

Abstract: We propose a new geometric spanner for static wireless ad hoc networks, which can be constructed efficiently in a localized manner. It integrates the connected dominating set and the local Delaunay graph to form a backbone of the wireless network. Priori arts showed that both structures can be constructed locally with bounded communication costs. This new spanner has these following attractive properties: 1) the backbone is a planar graph, 2) the node degree of the backbone is bounded from above by a positive constant, 3) it is a spanner for both hops and length, 4) it can be constructed locally and is easy to maintain when the nodes move around, and 5) moreover, the communication cost of each node is bounded by a constant. Simulation results are also presented for studying its practical performance.

Abstract: We propose a new geometric spanner, for wireless ad hoc networks, which can be constructed efficiently in a distributed manner. It combines the connected dominating set and the local Delaunay graph to form the backbone of a wireless network. This new spanner has the following attractive properties: (1) the backbone is a planar graph; (2) the node degree of the backbone is bounded from above by a positive constant; (3) it is a spanner both for hops and length; moreover, we show that, given any two nodes u and /spl upsi/, there is a path connecting them in the backbone such that its length is no more than 6 times that of the shortest path and the number of links is no more than 3 times that of the shortest path; (4) it can be constructed locally and is easy to maintain when the nodes move around; and (5) we show that the computation cost of each node is at most O(d log d), where d is its l-hop neighbors in the original unit disk graph, and the communication cost of each node is bounded by a constant. Simulation results are also presented for studying its practical performance.