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Local Construction of Near-Optimal Power Spanners for Wireless Ad Hoc Networks
April 2009 (vol. 8 no. 4)
pp. 460-474
Iyad A. Kanj, DePaul University, Chicago
Ljubomir Perković, DePaul University, Chicago
Ge Xia, Lafayette College, Easton
We present a local distributed algorithm that, given a wireless ad hoc network modeled as a unit disk graph U in the plane, constructs a planar power spanner of U whose degree is bounded by k and whose stretch factor is bounded by 1 + (2\sin{\frac{\pi}{k}})^{p}, where k \geq 10 is an integer parameter and p \in [2, 5] is the power exponent constant. For the same degree bound k, the stretch factor of our algorithm significantly improves the previous best bounds by Song et al. We show that this bound is near-optimal by proving that the slightly smaller stretch factor of 1 + (2\sin{\frac{\pi}{k + 1}})^{p} is unattainable for the same degree bound k. In contrast to previous algorithms for the problem, the presented algorithm is local. As a consequence, the algorithm is highly scalable and robust. 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.

[1] P. Bose and P. Morin, “Online Routing in Triangulations,” SIAM J.Computing, vol. 33, no. 4, pp. 937-951, 2004.
[2] N. Linial, “Locality in Distributed Graph Algorithms,” SIAM J.Computing, vol. 21, no. 1, pp. 193-201, 1992.
[3] D. Peleg, “Distributed Computing: A Locality-Sensitive Approach,” SIAM Monographs on Discrete Mathematics and Applications, 2000.
[4] R. Wattenhofer, “Sensor Networks: Distributed Algorithms Reloaded—or Revolutions?” Proc. 13th Colloquium on Structural Information and Comm. Complexity (SIROCCO '06), pp. 24-28, 2006.
[5] W.-Z. Song, X.-Y. Li, Y. Wang, and O. Frieder, “Localized Algorithms for Energy Efficient Topology in Wireless Ad Hoc Networks,” Mobile Networks and Applications, vol. 10, no. 6, pp.911-923, 2005.
[6] Y. Wang and X.-Y. Li, “Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks,” Proc. Joint Workshop Foundations of Mobile Computing, pp. 59-68, 2003.
[7] L. Li, J.Y. Halpern, P. Bahl, Y.-M. Wang, and R. Wattenhofer, “Analysis of a Cone-Based Distributed Topology Control Algorithm for Wireless Multi-Hop Networks,” Proc. 20th Ann. ACM Symp. Principles of Distributed Computing (PODC '01), pp. 264-273, 2001.
[8] X.-Y. Li, P.-J. Wan, and Y. Wang, “Power Efficient and Sparse Spanner for Wireless Ad Hoc Networks,” Proc. Int'l Conf. Computer Comm. (ICCC '01), pp. 564-567, 2001.
[9] R. Rajaraman, “Topology Control and Routing in Ad Hoc Networks: A Survey,” SIGACT News, vol. 33, no. 2, pp. 60-73, 2002.
[10] X.-Y. Li, P.-J. Wan, Y. Wang, and O. Frieder, “Sparse Power Efficient Topology for Wireless Networks,” Proc. 35th Hawaii Int'l Conf. System Sciences (HICSS '02), p. 296, 2002.
[11] R. Ramanathan and R. Hain, “Topology Control of Multihop Wireless Networks Using Transmit Power Adjustment,” Proc. IEEE INFOCOM, pp. 404-413, 2000.
[12] Y. Wang, X.-Y. Li, and O. Frieder, “Distributed Spanner with Bounded Degree for Wireless Networks,” Int'l J. Foundations of Computer Science, vol. 14, no. 2, pp. 183-200, 2003.
[13] R. Wattenhofer, L. Li, P. Bahl, and Y.-M. Wang, “Distributed Topology Control for Power Efficient Operation in Multihop Wireless Ad Hoc Networks,” Proc. IEEE INFOCOM, pp.1388-1397, 2001.
[14] C. Scheideler, Overlay Networks for Wireless Ad Hoc Networks, http://www14.in.tum.de/personen/scheideler papers, 2005.
[15] K.R. Gabriel and R.R. Sokal, “A New Statistical Approach to Geographic Variation Analysis,” Systematic Zoology, vol. 18, no. 3, pp. 259-278, 1969.
[16] A.C.-C. Yao, “On Constructing Minimum Spanning Trees in $k$ -Dimensional Spaces and Related Problems,” SIAM J.Computing, vol. 11, no. 4, pp. 721-736, 1982.
[17] P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia, “Routing with Guaranteed Delivery in Ad Hoc Wireless Networks,” Wireless Networks, vol. 7, no. 6, pp. 609-616, 2001.

Index Terms:
Spanners, unit disk graphs, Gabriel graphs, Yao graphs, local distributed algorithms.
Citation:
Iyad A. Kanj, Ljubomir Perković, Ge Xia, "Local Construction of Near-Optimal Power Spanners for Wireless Ad Hoc Networks," IEEE Transactions on Mobile Computing, vol. 8, no. 4, pp. 460-474, April 2009, doi:10.1109/TMC.2008.132
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