Multiple hop routing in mobile ad hoc networks can minimize energy consumption and increase data throughput. Yet, the problem of radio interferences remains. However if the routes are restricted to a basic network based on local neighborhoods, these interferences can be reduced such that standard routing algorithms can be applied. We compare different network topologies for these basic networks with respect to degree, spanner-properties, radio interferences, energy, and congestion, i.e. the Yao-graph (aka. Θ-graph) and some also known related models, which will be called the SymmY-graph (aka. YS-graph), the SparsY-graph (aka.YY-graph) and the BoundY-graph. Further, we present a promising network topology called the HL-graph (based on Hierarchical Layers). Further, we compare the ability of these topologies to handle dynamic changes of the network when radio stations appear and disappear. For this we measure the number of involved radio stations and present distributed algorithms for repairing the network structure. MotivationOur research aims at the implementation of a mobile ad hoc network based on distributed robust communication protocols. Besides the traditional use of omni-directional transmitters, we want to investigate the effect of space multiplexing techniques and variable transmission powers on the efficiency and capacity of ad hoc networks. Therefore our radios can send and receive radio signals independently in k sectors of angle θ using one frequency. Furthermore, our radio stations can regulate its transmission power for each transmitted signal. To show that this approach is also suitable in practical situations, we are currently developing a communication module for the mini robot Khepera [11,8] that can transmit and receive in eight sectors using infrared light with variable transmission distances up to one meter, see Fig. 1. A colony of Khepera robots will be equipped with
In this paper we investigate the relations between spanners, weak spanners, and power spanners in R D for any dimension D and apply our results to topology control in wireless networks. For c ∈ R, a c-spanner is a subgraph of the complete Euclidean graph satisfying the condition that between any two vertices there exists a path of length at most c-times their Euclidean distance. Based on this ability to approximate the complete Euclidean graph, sparse spanners have found many applications, e.g., in FPTAS, geometric searching, and radio networks. In a weak c-spanner, this path may be arbitrarily long, but must remain within a disk or sphere of radius c-times the Euclidean distance between the vertices. Finally in a c-power spanner, the total energy consumed on such a path, where the energy is given by the sum of the squares of the edge lengths on this path, must be at most c-times the square of the Euclidean distance of the direct edge or communication link. While it is known that any c-spanner is also both a weak C1-spanner and a C2-power spanner for appropriate C1, C2 depending only on c but not on the graph under consideration, we show that the converse is not true: there exists a family of c1-power spanners that are not weak C-spanners and also a family of weak c2-spanners that are not C-spanners for any fixed C. However a main result of this paper reveals that any weak c-spanner is also a C-power spanner for an appropriate constant C. We further generalize the latter notion by considering (c, δ)-power spanners where the sum of the δ-th powers of the lengths has to be bounded; so (c, 2)-power spanners coincide with the usual power spanners and (c, 1)-power spanners are classical spanners. Interestingly, these (c, δ)-power spanners form a strict hierarchy where the above results still hold for any δ ≥ D; some even hold for δ > 1 while counterexamples exist for δ < D. We show that every self-similar curve of fractal dimension D f > δ is not a (C, δ)-power spanner for any fixed C, in general. Finally, we consider the sparsified Yao-graph (SparsY-graph or YY) that is a well-known sparse topology for wireless networks. We prove that all SparsY-graphs are weak c-spanners for a constant c and hence they allow us to approximate energy-optimal wireless networks by a constant factor.
We investigate the problem of path selection in radio networks for a given set of sites in two-dimensional space. For some given static point-to-point communication demand we define measures for congestion, energy consumption and dilation that take interferences between communication links into account.We show that energy optimal path selection for radio networks can be computed in polynomial time. Then, we introduce the diversity g(V ) of a set V ⊆ Ê 2 . It can be used to upperbound the number of interfering edges. For real-world applications it can be regarded as Θ(log n). A main result of the paper is that a weak c-spanner construction as a communication network allows to approximate the congestionoptimal communication network by a factor of O(g(V ) 2 ).Furthermore, we show that there are vertex sets where only one of the performance parameters congestion, energy, and dilation can be optimized at a time. We show trade-offs lower bounding congestion × dilation and dilation × energy. For congestion and energy the situation is even worse. It is only possible to find a reasonable approximation for either congestion or energy minimization, while the other parameter is at least a polynomial factor worse than in the optimal network.
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