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Issue No.07 - July (2012 vol.11)
pp: 1140-1150
Jonathan P. Jenkins , North Carolina State University, Raleigh
Iyad A. Kanj , Lafayette College, Easton
Ge Xia , North Carolina State University, Raleigh
Fenghui Zhang , DePaul University, Chicago and Google, Kirkland
In this paper, we present local distributed algorithms for constructing spanners in wireless sensor networks modeled as unit ball graphs (shortly UBGs) and quasi-unit ball graphs (shortly quasi-UBGs), in the 3D euclidean space. Our first contribution is a local distributed algorithm that, given a UBG U and a parameter \alpha < \pi/3, constructs a sparse spanner of U with stretch factor 1/(1-2\sin {(\alpha/2)}), improving the previous upper bound of 1/(1-\alpha ) by Althöfer et al. which is applicable only when \alpha < 1/(1+2\sqrt{2}) < \pi/3. The second contribution of this paper is in presenting the first local distributed algorithm for the construction of bounded-degree lightweight spanners of UBGs and quasi-UBGs. The simulation results we obtained show that, empirically, the weight and the stretch factor of the spanners, and the locality of the algorithms, are much better than the theoretical upper bounds proved in this paper.
3D spanners, sparse spanners, lightweight spanners, local distributed algorithms.
Jonathan P. Jenkins, Iyad A. Kanj, Ge Xia, Fenghui Zhang, "Local Construction of Spanners in the 3D Space", IEEE Transactions on Mobile Computing, vol.11, no. 7, pp. 1140-1150, July 2012, doi:10.1109/TMC.2011.142
[1] R. Alexander, "On the Sum of Distances between n Points on a Sphere," Acta Mathematica, vol. 23, pp. 443-448, 1972.
[2] I. Althöfer, G. Das, D. Dobkin, D. Joseph, and J. Soares, "On Sparse Spanners of Weighted Graphs," Discrete and Computational Geometry, vol. 9, pp. 81-100, 1993.
[3] L. Barrière, P. Fraigniaud, and L. Narayanan, "Robust Position-Based Routing in Wireless Ad Hoc Networks with Unstable Transmission Ranges," Proc. Fifth Int'l Workshop Discrete Algorithms and Methods for Mobile Computing and Comm. (DIALM '01), pp. 19-27, 2001.
[4] P. Bose, J. Gudmundsson, and M. Smid, "Constructing Plane Spanners of Bounded Degree and Low Weight," Algorithmica, vol. 42, nos. 3/4, pp. 249-264, 2005.
[5] G. Brinkmann and A.W.M. Dress, "A Constructive Enumeration of Fullerenes," J. Algorithms, vol. 23, no. 2, pp. 345-358, 1997.
[6] J. Chen, A. Jiang, I.A. Kanj, G. Xia, and F. Zhang, "Separability and Topology Control of Quasi Unit Disk Graphs," Wireless Networks, vol. 17, no. 1, pp. 53-67, 2011.
[7] R.G. Crittenden and N.G. Turok, "Exactly Azimuthal Pixelizations of the Sky," Technical Report astro-ph/9806374v1, 1998.
[8] M. Damian, S. Pandit, and S. Pemmaraju, "Local Approximation Schemes for Topology Control," Proc. 25th Ann. ACM Symp. Principles of Distributed Computing (PODC '06), pp. 208-217, 2006.
[9] G. Das, P. Heffernan, and G. Narasimhan, "Optimally Sparse Spanners in 3-D Euclidean Space," Proc. Ninth Ann. Symp. Computational Geometry (SoCG '93), pp. 53-62, 1993.
[10] K.M. Gorski, B.D. Wandelt, E. Hivon, F.K. Hansen, and A.J. Banday, "The Healpix Primer," Technical Report astro-ph/9905275, Theoretical Astrophysics Center (TAC), 2003.
[11] J. Gudmundsson, C. Levcopoulos, and G. Narasimhan, "Fast Greedy Algorithms for Constructing Sparse Geometric Spanners," SIAM J. Computing, vol. 31, no. 5, pp. 1479-1500, 2002.
[12] T.C. Hales, "Sphere Packings, VI Tame Graphs and Linear Programs," Discrete and Computational Geometry, vol. 36, no. 1, pp. 205-265, 2006.
[13] R.H. Hardin, N.J. A. Sloane, and W.D. Smith, "Spherical Coverings," , 1994.
[14] Equidistribution and Extremal Energy of n Points on the Sphere, J.W. Jerome, ed. Clarendon, 1998.
[15] I.A. Kanj, L. Perkovic, and G. Xia, "Local Construction of Near-Optimal Power Spanners for Wireless Ad Hoc Networks," IEEE Trans. Mobile Computing, vol. 8, no. 4 pp. 460-474, Apr. 2009.
[16] I.A. Kanj, L. Perkovic, and G. Xia, "On Spanners and Lightweight Spanners of Geometric Graphs," SIAM J. Computing, vol. 39, no. 6, pp. 2132-2161, 2010.
[17] I.A. Kanj, G. Xia, and F. Zhang, "Local Construction of Spanners in the 3-D Space," Proc. IEEE Fifth Int'l Conf. Distributed Computing in Sensor Systems (DCOSS '09), pp. 315-328, 2009.
[18] A.J. Kimmerling, K. Sahr, and L. Song, "Developing an Equal Area Global Grid by Smaller Circle Subdivision," Proc. Discrete Global Grids, 2002.
[19] A.J. Kimmerling, K. Sahr, D. White, and L. Song, "Comparing Geometrical Properties of Global Grids," Cartography and Geographic Information Science, vol. 26, p. 271, 1999.
[20] H.W. Kroto, J.R. Heath, S.C. O'Brien, R.F. Curl, and R.E. Smalley, "${\rm C}_{60}$ : Buckminsterfullerene," Nature, vol. 318, pp. 162-163.
[21] A.B.J. Kuijlaars and E.B. Saff, "Asymptotics for Minimal Discrete Energy on the Sphere," Trans. Am. Math. Soc., vol. 350, pp. 523-538, 1998.
[22] P. Leopardi, "A Partition of the Unit Sphere into Regions of Equal Area and Small Diameter," Electronic Trans. Numerical Analysis, vol. 25, pp. 309-327, 2006.
[23] C. Levcopoulos and A. Lingas, "There are Planar Graphs Almost as Good as the Complete Graphs and Almost as Cheap as Minimum Spanning Trees," Algorithmica, vol. 8, no. 3, pp. 251-256, 1992.
[24] X.-Y. Li, G. Calinescu, P.-J. Wan, and Y. Wang, "Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks," IEEE Trans. Parallel and Distributed Systems., vol. 14, no. 10, pp. 1035-1047, Oct. 2003.
[25] X.-Y. Li, W.-Z. Song, and W. Wang, "A Unified Energy-Efficient Topology for Unicast and Broadcast," Proc. ACM MobiCom, pp. 1-15, 2005.
[26] D. Peleg, Distributed Computing: A Locality-Sensitive Approach, Monographs on Discrete Math. and Applications, SIAM, 2000.
[27] E.A. Rakhmanov, E.B. Saff, and Y.M. Zhou, "Minimal Discrete Energy on the Sphere," Math. Research Letters, vol. 1, pp. 647-662, 1994.
[28] E.A. Rakhmanov, E.B. Saff, and Y.M. Zhou, "Minimal Discrete Energy on the Sphere," Math. Research Letters, vol. 1, pp. 647-662, 1994.
[29] E.B. Saff, "Equal-Area Partitions of Sphere," Univ. New South Wales, presentation, 2003.
[30] E.B. Saff and A.B.J. Kuijlaars, "Distributing Many Points on a Sphere," The Math. Intelligencer, vol. 19, no. 1, pp. 5-11, 1997.
[31] W.-Z. Song, X.-Y. Li, O. Frieder, and W. Wang, "Localized Topology Control for Unicast and Broadcast in Wireless Ad Hoc Networks," IEEE Trans. Parallel and Distributed Systems, vol. 17, no. 4, pp. 321-334, Apr. 2006.
[32] M. Tegmark, "An Icosahedron-Based Method for Pixelizing the Celestial Sphere," Technical Report astro-ph/9610094, 1996.
[33] K. Thomsen, "Generalized Spiral Points: Further Improvement," , 2007.
[34] B.L. van der Waerden, "Punkte Auf Der Kugel, Drei Zusätze," Math. Ann., vol. 125, pp. 213-222, 1952.
[35] Y. Wang and X.-Y. Li, "Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks," MONET, vol. 11, no. 2, pp. 161-175, 2006.
[36] A. Wiese and E. Kranakis, "Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs," Proc. Working Group, pp. 372-383, 2008.
[37] 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.
[38] Y.M. Zhou, "Arrangements of Points on the Sphere," PhD thesis, Dept. of Math., Univ. South Florida, 1995.
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