Localization and routing in sensor networks by local angle information

ABSTRACT

Location information is very useful in the design of sensor network infrastructures. In this paper, we study the anchor-free 2D localization problem by using local angle measurements in a sensor network. We prove that given a unit disk graph and the angles between adjacent edges, it is NP-hard to find a valid embedding in the plane such that neighboring nodes are within distance 1 from each other and non-neighboring nodes are at least distance 1 away. Despite the negative results, however, one can find a planar spanner of a unit disk graph by using only local angles. The planar spanner can be used to generate a set of virtual coordinates that enable efficient and local routing schemes such as geographical routing or approximate shortest path routing. We also proposed a practical anchor-free embedding scheme by solving a linear program. We show by simulation that not only does it give very good local embedding, i.e., neighboring nodes are close and non-neighboring nodes are far away, but it also gives a quite accurate global view such that geographical routing and approximate shortest path routing on the embedded graph are almost identical to those on the original (true) embedding. The embedding algorithm can be adapted to other models of wireless sensor networks and is robust to measurement noise.

REFERENCES

• 1: J. Aspnes, D. Goldenberg, and Y. R. Yang. On the computational complexity of sensor network localization. In The 1st International Workshop on Algorithmic Aspects of Wireless Sensor Networks (ALGOSENSORS), pages 32--44, 2004.
• 2: Pratik Biswas , Yinyu Ye, Semidefinite programming for ad hoc wireless sensor network localization, Proceedings of the third international symposium on Information processing in sensor networks, April 26-27, 2004, Berkeley, California, USA [doi>10.1145/984622.984630]
• 3: I. Borg and P. Groenen. Modern Multidimensional Scaling: Theory and Applications. Springer-Verlag, 1997.
• 4: Prosenjit Bose , Pat Morin , Ivan Stojmenović , Jorge Urrutia, Routing with guaranteed delivery in ad hoc wireless networks, Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications, p.48-55, August 20-20, 1999, Seattle, Washington, United States [doi>10.1145/313239.313282]
• 5: Heinz Breu , David G. Kirkpatrick, Unit disk graph recognition is NP-hard, Computational Geometry: Theory and Applications, v.9 n.1-2, p.3-24, Jan. 1998 [doi>10.1016/S0925-7721(97)00014-X]
• 6: V. W. Bryant. Straight line representation of planar graphs. Elementary Mathematics, 44:64--66, 1989.
• 7: Mihai Bǎdoiu , Erik D. Demaine , Mohammad Taghi Hajiaghayi , Piotr Indyk, Low-dimensional embedding with extra information, Proceedings of the twentieth annual symposium on Computational geometry, June 08-11, 2004, Brooklyn, New York, USA [doi>10.1145/997817.997866]
• 8: L. Doherty, L. E. Ghaoui, and S. J. Pister. Convex position estimation in wireless sensor networks. In IEEE Infocom, volume 3, pages 1655--1663, April 2001.
• 9: I. Fáry. On straight line representations of planar graphs. Acta Scientiarum Mathematicarum (Szeged), 11:229--233, 1948.
• 10: D. Ganesan, B. Krishnamachari, A. Woo, D. Culler, D. Estrin, and S. Wicker. Complex behavior at scale: An experimental study of low-power wireless sensor networks. Technical Report UCLA/CSD-TR 02-0013, UCLA, 2002.
• 11: Jie Gao , Leonidas J. Guibas , John Hershberger , Li Zhang , An Zhu, Geometric spanner for routing in mobile networks, Proceedings of the 2nd ACM international symposium on Mobile ad hoc networking & computing, October 04-05, 2001, Long Beach, CA, USA [doi>10.1145/501422.501424]
• 12: Cyril Gavoille , David Peleg , Stéphane Pérennes , Ran Raz, Distance labeling in graphs, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.210-219, January 07-09, 2001, Washington, D.C., United States

INDEX TERMS

Primary Classification:
  E. Data
  E.1 DATA STRUCTURES
      Subjects: Graphs and networks

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems

General Terms:
Algorithms, Design, Theory

Keywords:
embedding, geographical routing, localization, planar spanner subgraph, sensor networks, wireless networks