Abstract
The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining efficient constant-factor approximation algorithms for the mobile piercing set (MPS) problem on unit-disks for standard metrics in fixed dimension vector spaces. More specifically, we provide low constant approximations for L 1 and L ∞ norms on a d-dimensional space, for any fixed d>0, and for the L 2 norm on two- and three-dimensional spaces. Our framework provides a family of fully-distributed and decentralized algorithms, which adapt (asymptotically) optimally to the mobility of disks, at the expense of a low degradation on the best known approximation factors of the respective centralized algorithms: Our algorithms take O(1) time to update the piercing set maintained, per movement of a disk. We also present a family of fully-distributed algorithms for the MPS problem which either match or improve the best known approximation bounds of centralized algorithms for the respective norms and space dimensions.
Second, we show how the proposed algorithms can be directly applied to provide theoretical performance analyses for two popular 1-hop clustering algorithms in ad-hoc networks: the lowest-id algorithm and the Least Cluster Change (LCC) algorithm. More specifically, we formally prove that the LCC algorithm adapts in constant time to the mobility of the network nodes, and minimizes (up to low constant factors) the number of 1-hop clusters maintained. While there is a vast literature on simulation results for the LCC and the lowest-id algorithms, these had not been formally analyzed prior to this work.
We also present an O(log n)-approximation algorithm for the mobile piercing set problem for nonuniform disks (i.e., disks that may have different radii), with constant update time.
Similar content being viewed by others
References
P.K. Agarwal and C.M. Procopiuc, Exact and approximation algorithms for clustering, in: Proc. of 9th ACM-SIAM Sympos. on Discrete Algorithms (1998) pp. 658–667.
P.K. Agarwal and M. Sharir, Efficient algorithms for geometric optimization, ACM Comput. Surv. 30 (1998) 412–458.
S. Basagni, Distributed and mobility-adaptive clustering for multimedia support in multi-hop wireless networks, in: Proc. of IEEE Vehicular Tech. Conf. (1999) pp. 19–22.
S. Basagni, Distributed clustering for ad-hoc networks, in: Proc. of 1999 Int. Sympos. on Parallel Architectures (1999) pp. 310–315.
J. Basch, L.J. Guibas and J. Hershberger, Data structures for mobile data, in: Proc. of 8th ACM-SIAM Sympos. on Discrete Algorithms (1997) pp. 747–756.
S. Bepamyatnikh, B. Bhattacharya, D. Kirkpatrick and M. Segal, Mobile facility location, in: Proc. of ACM Int. Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (2000) pp. 46–53.
C.-C. Chiang, H.-K. Wu, W. Liu and M. Gerla, Routing in clustered multihop, mobile wireless networks with fading channel, in: Proc. of IEEE Singapore Int. Conf. on Networks (1997) pp. 197–211.
J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices, and Groups (Springer, 1999).
Z. Drezner, The p-center problem: heuristic and optimal algorithms, J. Oper. Res. Soc. 35 (1984) 741–748.
A. Efrat, M.J. Katz, F. Nielsen and M. Sharir, Dynamic data structures for fat objects and their applications, in: Proc. of Workshop on Algorithms and Data Structures (1997) pp. 297–306.
T. Feder and D.H. Greene, Optimal algorithms for approximate clustering, in: Proc. of 20th Annu. ACM Sympos. on Theory of Computing (1988) pp. 434–444.
R.J. Fowler, M.S. Paterson and S.L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett. 12(3) (1981) 133–137.
G.N. Frederickson and D.B. Johnson, Generalized selection and ranking: sorted matrices, SIAM J. Comput. 13 (1984) 14–30.
J. Gao, L.J. Guibas, J. Hershburger, L. Zhang and A. Zhu, Discrete mobile centers, in: Proc. of 17th ACM Sympos. on Computational Geometry (2001) pp. 188–196.
M. Gerla and J.T.C. Tsai, Multicluster mobile multimedia radio networks, ACM-Baltzer J. Wireless Networks 1(3) (1995) 255–256.
M. Golumbic, Algorithmic Graph Theory (Academic Press, New York, 1980).
T. Gonzalez, Covering a set of points in multidimensional space, Inform. Process. Lett. 40 (1991) 181–188.
S. Har-Peled, Clustering motion, in: Proc. of 42nd Annu. IEEE Sympos. on Foundations of Computer Science (2001) pp. 84–93.
D.S. Hochbaum and W. Maas, Approximation schemes for covering and packing problems in image processing and vlsi, J. of the ACM 32 (1985) 130–136.
D.S. Hochbaum and D. Shmoys, A best possible heuristic for the k-center problem, Math. Oper. Res. 10 (1985) 180–184.
D.S. Hochbaum and D. Shmoys, A unified approach to approximation algorithms for bottleneck problems, J. of the ACM 33 (1986) 533–550.
H. Huang, A.W. Richa and M. Segal, Approximation algorithms for the mobile piercing set problem with applications to clustering, Technical Report TR-01-007, Department of Computer Science and Engineering, Arizona State University, Tempe, AZ (2001).
R.Z. Hwang, R.C. Chang and R.C.T. Lee, The generalized searching over separators strategy to solve some NP-hard problems in subexponential time, Algorithmica 9 (1993) 398–423.
R.Z. Hwang, R.C.T. Lee and R.C. Chang, The slab dividing approach to solve the Euclidean p-center problem, Algorithmica 9 (1993) 1–22.
M.J. Katz, F. Nielsen and M. Segal, Maintenance of a piercing set for intervals with applications, in: Proc. of 11th Int. Symp. on Algorithms and Computation (2000) pp. 552–563.
M.T. Ko, R.C.T. Lee and J.S. Chang, An optimal approximation algorithm for the rectilinear m-center problem, Algorithmica 5 (1990) 341–352.
C.R. Lin and M. Gerla, Adaptive clustering for mobile wireless networks, IEEE J. Selected Areas Commun. 15(7) (1997) 1265–1275.
A.B. McDonald and T. Znati, A mobility-based framework for adaptive clustering in wireless ad-hoc networks, IEEE J. Selected Areas Commun. 17(8) (1999).
N. Megiddo and K.J. Supowit, On the complexity of some common geometric location problems, SIAM J. Comput. 13(1) (1984) 182–196.
F. Nielsen, Fast stabbing of boxes in high dimensions, in: Proc. of 8th Canad. Conf. in Computational Geometry (1996) pp. 87–92.
R. Ramanathan and M. Steenstrup, Hierarchically-organized, multihop mobile wireless for quality-of-service support, Mobile Networks and Applications 3 (1998) 101–119.
M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, in: Proc. of 12th Annu. ACM Sympos. on Computational Geometry (1996) pp. 122–132.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Huang, H., Richa, A.W. & Segal, M. Approximation Algorithms for the Mobile Piercing Set Problem with Applications to Clustering in Ad-Hoc Networks. Mobile Networks and Applications 9, 151–161 (2004). https://doi.org/10.1023/B:MONE.0000013626.53247.1c
Issue Date:
DOI: https://doi.org/10.1023/B:MONE.0000013626.53247.1c