Abstract
Clustering can play a critical role in increasing the performance and lifetime of wireless networks. The facility location problem is a general abstraction of the clustering problem and this paper presents the first constant-factor approximation algorithm for the facility location problem on unit disk graphs (UDGs), a commonly used model for wireless networks. In this version of the problem, connection costs are not metric, i.e., they do not satisfy the triangle inequality, because connecting to a non-neighbor costs ∞. In non-metric settings the best approximation algorithms guarantee an O(logn)-factor approximation, but we are able to use structural properties of UDGs to obtain a constant-factor approximation. Our approach combines ideas from the primal-dual algorithm for facility location due to Jain and Vazirani (JACM, 2001) with recent results on the weighted minimum dominating set problem for UDGs (Huang et al., J. Comb. Opt., 2008). We then show that the facility location problem on UDGs is inherently local and one can solve local subproblems independently and combine the solutions in a simple way to obtain a good solution to the overall problem. This leads to a distributed version of our algorithm in the \(\mathcal{LOCAL}\) model that runs in constant rounds and still yields a constant-factor approximation. Even if the UDG is specified without geometry, we are able to combine recent results on maximal independent sets and clique partitioning of UDGs, to obtain an O(logn)-approximation that runs in O(log* n) rounds.
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References
Heinzelman, W.R., Chandrakasan, A., Balakrishnan, H.: Energy-efficient communication protocol for wireless microsensor networks. In: HICSS 2000: Proceedings of the 33rd Hawaii International Conference on System Sciences, vol. 8, p. 8020 (2000)
Wu, T., Biswas, S.: Minimizing inter-cluster interference by self-reorganizing mac allocation in sensor networks. Wireless Networks 13(5), 691–703 (2007)
Wan, P.J., Alzoubi, K.M., Frieder, O.: Distributed construction of connected dominating set in wireless ad hoc networks. Mob. Netw. Appl. 9(2), 141–149 (2004)
Wang, Y., Li, X.Y.: Localized construction of bounded degree and planar spanner for wireless ad hoc networks. In: DIALM-POMC 2003: Proceedings of the 2003 joint workshop on Foundations of mobile computing, pp. 59–68 (2003)
Wang, Y., Wang, W., Li, X.Y.: Distributed low-cost backbone formation for wireless ad hoc networks. In: MobiHoc., pp. 2–13 (2005)
Deb, B., Nath, B.: On the node-scheduling approach to topology control in ad hoc networks. In: MobiHoc 2005: Proceedings of the 6th ACM international symposium on Mobile ad hoc networking and computing, pp. 14–26 (2005)
Kang, J., Zhang, Y., Nath, B.: Analysis of resource increase and decrease algorithm in wireless sensor networks. In: ISCC 2006: Proceedings of the 11th IEEE Symposium on Computers and Communications, pp. 585–590 (2006)
Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: STOC 2004: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pp. 281–290 (2004)
Bilò, V., Caragiannis, I., Kaklamanis, C., Kanellopoulos, P.: Geometric clustering to minimize the sum of cluster sizes. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 460–471. Springer, Heidelberg (2005)
Cheng, X., Huang, X., Li, D., Wu, W., Du, D.Z.: A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks 42(4), 202–208 (2003)
Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In: APPROX-RANDOM, pp. 3–14 (2006)
Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 747–758. Springer, Heidelberg (2008)
Huang, Y., Gao, X., Zhang, Z., Wu, W.: A better constant-factor approximation for weighted dominating set in unit disk graph. Journal of Combinatorial Optimization (2008)
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. J. ACM 48(2), 274–296 (2001)
Frank, C.: Algorithms for Sensor and Ad Hoc Networks. Springer, Heidelberg (2007)
Kuehn, A.A., Hamburger, M.J.: A heuristic program for locating warehouses. Management Science 9(4), 643–666 (1963)
Stollsteimer, J.F.: A working model for plant numbers and locations. Management Science 45(3), 631–645 (1963)
Balinski, M.L.: On finding integer solutions to linear programs. In: Proceedings of IBM Scientific Computing Symposium on Combinatorial Problems, pp. 225–248 (1966)
Kaufman, L., Eede, M.V., Hansen, P.: A plant and warehouse location problem. Operational Research Quarterly 28(3), 547–554 (1977)
Cornuejols, G., Nemhouser, G., Wolsey, L.: Discrete Location Theory. Wiley, Chichester (1990)
Hochbaum, D.S.: Heuristics for the fixed cost median problem. Mathematical Programming 22(1), 148–162 (1982)
Lin, J.H., Vitter, J.S.: e-approximations with minimum packing constraint violation (extended abstract). In: STOC 1992: Proceedings of the twenty-fourth annual ACM symposium on Theory of computing, pp. 771–782 (1992)
Shmoys, D.B., Tardos, É., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 265–274 (1997)
Moscibroda, T., Wattenhofer, R.: Facility location: distributed approximation. In: PODC 2005: Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing, pp. 108–117 (2005)
Peleg, D.: Distributed computing: a locality-sensitive approach. Society for Industrial and Applied Mathematics (2000)
Frank, C., Römer, K.: Distributed facility location algorithms for flexible configuration of wireless sensor networks. In: Aspnes, J., Scheideler, C., Arora, A., Madden, S. (eds.) DCOSS 2007. LNCS, vol. 4549, pp. 124–141. Springer, Heidelberg (2007)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp. J. ACM 50(6), 795–824 (2003)
Gehweiler, J., Lammersen, C., Sohler, C.: A distributed O(1)-approximation algorithm for the uniform facility location problem. In: SPAA 2006: Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures, pp. 237–243. ACM, New York (2006)
Schneider, J., Wattenhofer, R.: A Log-Star Distributed Maximal Independent Set Algorithm for growth-Bounded Graphs. In: 27th ACM Symposium on Principles of Distributed Computing (PODC), Toronto, Canada (2008)
Pemmaraju, S., Pirwani, I.: Good quality virtual realization of unit ball graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 311–322. Springer, Heidelberg (2007)
Pandit, S., Pemmaraju, S.: Finding facilities fast. Full Paper (2009), http://cs.uiowa.edu/~spandit/research/icdcn2009.pdf
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Pandit, S., Pemmaraju, S.V. (2008). Finding Facilities Fast. In: Garg, V., Wattenhofer, R., Kothapalli, K. (eds) Distributed Computing and Networking. ICDCN 2009. Lecture Notes in Computer Science, vol 5408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92295-7_5
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DOI: https://doi.org/10.1007/978-3-540-92295-7_5
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