Abstract
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 3-dimensional Euclidean space. Our first contribution is a local distributed algorithm that, given a UBG U and a parameter α < π/3, constructs a sparse spanner of U with stretch factor 1/(1 − 2sin(α/2)), improving the previous upper bound of 1/(1 − α) 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 of the spanners, the stretch factor and locality of the algorithms, are much better than the theoretical upper bounds proved in this paper.
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References
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete & Computational Geometry 9, 81–100 (1993)
Barrière, L., Fraigniaud, P., Narayanan, L.: Robust position-based routing in wireless Ad Hoc networks with unstable transmission ranges. In: DIALM, pp. 19–27 (2001)
Bose, P., Gudmundsson, J., Smid, M.: Constructing plane spanners of bounded degree and low weight. Algorithmica 42(3-4), 249–264 (2005)
Brinkmann, G., Dress, A.W.M.: A constructive enumeration of fullerenes. J. Algorithms 23(2), 345–358 (1997)
Chen, J., Jiang, A., Kanj, I., Xia, G., Zhang, F.: Separability and topology control of quasi unit disk graphs. In: Proceedings of INFOCOM, pp. 2225–2233 (2007)
Damian, M., Pandit, S., Pemmaraju, S.: Local approximation schemes for topology control. In: Proceedings of PODC, pp. 208–217 (2006)
Das, G., Heffernan, P., Narasimhan, G.: Optimally sparse spanners in 3-D Euclidean space. In: Proceedings of SoCG, pp. 53–62 (1993)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput. 31(5), 1479–1500 (2002)
Hales, T.C.: Sphere packings, vi. tame graphs and linear programs. Discrete & Computational Geometry 36(1), 205–265 (2006)
Kanj, I., Wiese, A., Zhang, F.: Computing the k-hop neighborhoods locally. Technique report #08-007, http://www.cdm.depaul.edu/research/Pages/TechnicalReports.aspx
Kroto, H.W., Heath, J.R., O’Brien, S.C., Curl, R.F., Smalley, R.E.: C60: buckminsterfullerene. Nature 318, 162–163
Levcopoulos, C., Lingas, A.: There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees. Algorithmica 8(3), 251–256 (1992)
Li, X.-Y., Song, W.-Z., Wang, W.: A unified energy-efficient topology for unicast and broadcast. In: MOBICOM, pp. 1–15 (2005)
Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)
Malkevitch, J.: A note on fullerenes. Fullerenes, Nanotubes and Carbon Nanostructures 2(4), 423–426 (1994)
Peleg, D.: Distributed computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematis and Applications (2000)
Wiese, A., Kranakis, E.: Local construction and coloring of spanners of location aware unit disk graphs. In: WG, pp. 372–383 (2008)
Yao, A.C.-C.: On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing 11(4), 721–736 (1982)
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Kanj, I.A., Xia, G., Zhang, F. (2009). Local Construction of Spanners in the 3-D Space. In: Krishnamachari, B., Suri, S., Heinzelman, W., Mitra, U. (eds) Distributed Computing in Sensor Systems. DCOSS 2009. Lecture Notes in Computer Science, vol 5516. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02085-8_23
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DOI: https://doi.org/10.1007/978-3-642-02085-8_23
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