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Mobile Networks and Applications

, Volume 11, Issue 2, pp 129–142 | Cite as

Approximating the Minimum Number of Maximum Power Users in Ad hoc Networks

  • Errol L. Lloyd
  • Rui Liu
  • S. S. Ravi
Article

Abstract

Topology control is the problem of assigning transmission power values to the nodes of an ad hoc network so that the induced graph satisfies some specified property. The most fundamental such property is that the network/graph be connected. For connectivity, prior work on topology control gave a polynomial time algorithm for minimizing the maximum power assigned to any node (such that the induced graph is connected). In this paper we study the problem of minimizing the number of maximum power nodes. After establishing that this minimization problem is NP-complete, we focus on approximation algorithms for graphs with symmetric power thresholds. We first show that the problem is reducible in an approximation preserving manner to the problem of assigning power values so that the sum of the powers is minimized. Using known results for that problem, this provides a family of approximation algorithms for the problem of minimizing the number of maximum power nodes with approximation ratios of 5/3 + ε for every ε > 0. Unfortunately, these algorithms, based on solving large linear programming problems, are not practical. The main results of this paper are practical algorithms with approximation ratios of 7/4 and 5/3 (exactly). In addition, we present experimental results, both on randomly generated networks, and on two networks derived from proximity data associated with the TRANSIMS project of Los Alamos National Labs. Finally, based on the reduction to the problem of minimizing the total power, we describe some additional results for minimizing the number of maximum power users, both for graph properties other than connectivity and for graphs with asymmetric power thresholds.

Keywords

adhoc network topology control maximum power users approximation algorithm 

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References

  1. [1]
    E. Althaus, G. Calinescu, I. Mandoiu, S. Prasad, N. Tchervenski and A. Zelikovksy, Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks, Private communication. (The paper is available from www.cs.iit.edu/~calinescu.)
  2. [2]
    S. Arora and C. Lund, Hardness of approximations, in: Approximation Algorithms for NP-Hard Problems, D. Hochbaum (eds.), (PWS Publishing Company, Boston, MA, 1997).Google Scholar
  3. [3]
    W. Chen and N. Huang, The strongly connecting problem on multihop packet radio networks, IEEE Trans. Communication, 37(3) (1989) 293–295.CrossRefMathSciNetGoogle Scholar
  4. [4]
    T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction to Algorithms (MIT Press and McGraw-Hill, Cambridge, MA, 2000).Google Scholar
  5. [5]
    A.E.F. Clementi, P. Penna and R. Silvestri, Hardness results for the power range assignment problem in packet radio networks, in: Proc. Third International Workshop on Randomization and Approximation in Computer Science (APPROX 1999), Lecture Notes in Computer Science, vol. 1671 (Springer-Verlag, July 1999) pp. 195–208.Google Scholar
  6. [6]
    A.E.F. Clementi, P. Penna and R. Silvestri, The power range assignment problem in packet radio networks in the plane, in: Proc. 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2000) (Feb. 2000) pp. 651–660.Google Scholar
  7. [7]
    G. Calinescu and P.-J. Wan, Symmetric high connectivity with minimum total power consumption in multihop packet radio networks, in: Proc. International Conference on Ad hoc and Wireless Networks (ADHOC-NOW’03), Lecture Notes in CS, vol. 2865, S. Pierre, M. Barbeau and E. Kranakis (eds.) (Montreal, Canada, Oct. 2003) pp. 235–246.Google Scholar
  8. [8]
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman and Co., San Francisco, CA, 1979).MATHGoogle Scholar
  9. [9]
    L.M. Kirousis, E. Kranakis, D. Krizanc and A. Pelc, in: Power consumption in packet radio networks, in: Proc. 14th Annual Symposium on Theoretical Aspects of Computer Science (STACS 97), Lecture Notes in Computer Science, vol. 1200 (Springer-Verlag, Feb. 1997) pp. 363–374. (Complete version under review for Theoretical Computer Science.)MathSciNetCrossRefGoogle Scholar
  10. [10]
    S.O. Krumke, R. Liu, E.L. Lloyd, M.V. Marathe, R. Ramanathan and S.S. Ravi, Topology control problems under symmetric and asymmetric power thresholds, in: Proc. International Conference on Ad hoc and Wireless Networks (ADHOC-NOW’03), Lecture Notes in CS, vol. 2865, S. Pierre, M. Barbeau and E. Kranakis (eds.) (Montreal, Canada, Oct. 2003) pp. 187–198.Google Scholar
  11. [11]
    E. Lloyd, R. Liu, M. Marathe, R. Ramanathan and S.S. Ravi, Algorithmic aspects of topology control problems for ad hoc networks, in: Proc. Third ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) (2002) pp. 123–134. (Complete version to appear in Mobile Networks and Applications.)Google Scholar
  12. [12]
    L. Li, J.Y. Halpern, P. Bahl, Y. Wang and R. Wattenhofer, Analysis of cone-based distributed topology control algorithm for wireless multi-hop networks, in: Proc. ACM Principles of Distributed ComputingConference (PODC’01) (Aug. 2001) pp. 264–273.Google Scholar
  13. [13]
    M.V. Marathe, R. Ravi, R. Sundaram, S.S. Ravi, D.J. Rosenkrantz and H.B. Hunt III, Bicriteria network design problems, Journal of Algorithms 28(1) (1998) 142–171.CrossRefMathSciNetMATHGoogle Scholar
  14. [14]
    T.S. Rappaport, Wireless Communications: Principles and Practice (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996).Google Scholar
  15. [15]
    V. Radoplu and T.H. Meng, Minimum energy mobile wireless networks, IEEE J. Selected Areas in Communications 17(8) (1999) 1333–1344.Google Scholar
  16. [16]
    E.M. Royer, P. Melliar-Smith and L. Moser, An analysis of the optimum node density for ad hoc mobile networks, in: Proc. IEEE Intl. Conf. on Communication (ICC’01) (Helsinki, Finland, June 2001) pp. 857–861.Google Scholar
  17. [17]
    R. Ramanathan and R. Rosales-Hain, Topology control of multihop wireless networks using transmit power adjustment, in: Proc. IEEE INFOCOM 2000 (Tel Aviv, Israel, March 2000) pp. 404–413.Google Scholar
  18. [18]
    H. Takagi, and L. Kleinrock, Optimal transmission ranges for randomly distributed packet radio terminals, IEEE Transactions on Communications, COM-32(3) (1984) 246–257. Also appears in Multiple Access Communications, Foundations for Emerging Technologies, Norman Abramson (Ed.) (IEEE Press, 1992) pp. 342–353.CrossRefGoogle Scholar
  19. [19]
  20. [20]
    R. Wattenhofer, L. Li, P. Bahl and Y. Wang, Distributed topology control for power efficient operation in multihop wireless ad hoc networks, in: Proc. IEEE INFOCOM 2001 (Anchorage, Alaska, April 2001) pp. 1388–1397.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Computer and Information SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of Computer ScienceUniversity at Albany – SUNYAlbanyUSA

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