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Topology control in constant rate mobile ad hoc networks

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Abstract

Topology control is the problem of assigning power levels to the nodes of an ad hoc network so as to create a specified network topology while minimizing the energy consumed by the network nodes. While considerable theoretical attention has been given to the issue of topology control in wireless ad hoc networks, all of that prior work has concerned stationary networks. When the nodes are mobile, there is no algorithm that can guarantee a graph property (such as network connectivity) throughout the node movement. In this paper we study topology control in mobile wireless ad hoc networks (MANETs). We define a mobility model, namely the constant rate mobile network (CRMN) model, in which we assume that the speed and direction of each moving node are known. The goal of topology control under this model is to minimize the maximum power used by any network node in maintaining a specified monotone graph property. Network connectivity is one of the most fundamental monotone properties. Under the CRMN model, we develop general frameworks for solving both the decision version (i.e. for a given value p > 0, will a specified monotone property hold for the network induced by assigning the power value p to every node?) and the optimization version (i.e. find the minimum value p such that the specified monotone property holds for the network induced by assigning the power value p to every node) of the topology control problems. Efficient algorithms for specific monotone properties can be derived from these frameworks. For example, when the monotone property is network connectivity, our algorithms for the decision and optimization versions have running times of O(n 2 log2 n) and O(n 4 log2 n), respectively. Our results represent a step towards the development of efficient and provably good distributed algorithms for topology control problems for MANETs.

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Notes

  1. In the context of this paper, a framework is a collection of concrete algorithms with each algorithm solving the topology control problem for a particular graph property.

  2. We let the node name represent both itself and the Euclidean position of the node.

  3. In particular, this assumption can be made without loss of generality for monotone properties, as will become clear from the next section.

  4. Henceforth, when the meaning is clear, we omit the word “Euclidean”.

  5. For an optimization problem, its decision version is: Given any proposed solution to the optimization problem, determine if that solution satisfies the objective (without optimization) of the problem. Specifically, the decision version of \(\langle{\sc CRMN}, {\mathcal{P}}, {\sc MaxP}\rangle\) is to determine whether for a given a power p, the CRMN \({\mathcal{N}}\) achieves monotone property \({\mathcal{P}}\) under p.

  6. \(\vec v_i\) represents the vector \(\overrightarrow{V_iV_i^{\prime}}\) starting at V i and ending at \(V_i^{\prime}\).

  7. Imaginary solutions do not have any physical significance in this context.

  8. This is assuming that there is only one slicing point at \(t_{k+1}\). When there are multiple slicing points at \(t_{k+1}\), the update operations described in this proof are applied to each of those slicing points.

  9. In [10], there are only two graph properties (related to this paper) handled by their dynamic graph algorithms. These properties are Connected and 2-Node/Edge-Connected. However, it is not hard to see that Minimum Degree Constraint can be processed incrementally in constant time as follows. Let every node be associated with a counter that records the number of incident edges on that node. The update operation is that when an edge comes into or goes out of existence, the degree of each node is accounted for by increasing or decreasing the counter associated with each endpoint of that edge.

  10. Note that when there are identical threshold functions, multiple pairs of edges might define the same slicing points. In that case, the procedure described in the remainder of the proof is carried out for each such pair of edges.

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Acknowledgments

We thank the reviewers for reading the manuscript and providing helpful suggestions.

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The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.

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Authors and Affiliations

  1. Department of Computer and Information Sciences, University of Delaware, Newark, DE, 19716, USA

    Liang Zhao & Errol L. Lloyd

  2. Department of Computer Science, University at Albany – SUNY, Albany, NY, 12222, USA

    S. S. Ravi

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  1. Liang Zhao
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  2. Errol L. Lloyd
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  3. S. S. Ravi
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Corresponding author

Correspondence to Errol L. Lloyd.

Additional information

Preliminary results regarding connectivity only were presented at the GLOBECOM’06-WirelessComm [22] conference in San Francisco, California, November 2006.

This article was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes not withstanding any copyright notation thereon. (This statement applies only to the first and second authors).

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Zhao, L., Lloyd, E.L. & Ravi, S.S. Topology control in constant rate mobile ad hoc networks. Wireless Netw 16, 467–480 (2010). https://doi.org/10.1007/s11276-008-0147-9

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