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Hershberger, L. Zhang, A. Zhu, Geometric Spanner for Routing in Mobile Networks, Proceedings of the 2nd ACM International Symposium on Mobile Ad hoc Networking and Computing, pp. 45–55, 2001. 7. E. Kranakis, H. Singh, and J. Urrutia, Compass Routing on Geometric Networks, In Proceedings of llth Canadian Conference on Computational Geometry, pp. 51–54, Vancouver, August, 1999. 8. F. Kuhn, R. Wattenhofer, and A. Zollinger, Asymptotically Optimal Geometric Mobile Ad-Hoc Routing. In Proc. of the 6th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (DIALM), pp.
In such systems it is vital that route discovery uses only local information and is adaptable to the network connectivity An important technique for discovering routes between any two nodes in a ad-hoc network without the use of flooding is the face routing [2,7] algorithm The face routing algorithm succeeds in discovering a route in a networ providing that the underlying graph is planar. Since in practice, th original ad-hoc wireless network is never planar with many links crossin each other, before using the face routing we need to extract from th original network, using a local method, a planar connected network spanning the entire underlying network.
It is easy to see that T(0) is the root and is the set of all nodes. Assume that at most labels for the root and at most labels for any other node are enough. We will reach a contradiction which would mean that some node needs labels. Let be the set of leaves of the tree and let be the set of leaves of for We can prove the following claim using induction if we assume that the root has at most labels, and each other node has at most labels. Claim. For every the set contains a subset which satisfies the following properties: 1.
