Probabilistic Analysis of Power Assignments

  • Maurits de Graaf
  • Bodo Manthey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)


A fundamental problem for wireless ad hoc networks is the assignment of suitable transmission powers to the wireless devices such that the resulting communication graph is connected. The goal is to minimize the total transmit power in order to maximize the life-time of the network. Our aim is a probabilistic analysis of this power assignment problem. We prove complete convergence for arbitrary combinations of the dimension d and the distance-power gradient p. Furthermore, we prove that the expected approximation ratio of the simple spanning tree heuristic is strictly less than its worst-case ratio of 2.

Our main technical novelties are two-fold: First, we find a way to deal with the unbounded degree that the communication network induced by the optimal power assignment can have. Minimum spanning trees and traveling salesman tours, for which strong concentration results are known in Euclidean space, have bounded degree, which is heavily exploited in their analysis. Second, we apply a recent generalization of Azuma-Hoeffding’s inequality to prove complete convergence for the case p ≥ d for both power assignments and minimum spanning trees (MSTs). As far as we are aware, complete convergence for p > d has not been proved yet for any Euclidean functional.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Althaus, E., Calinescu, G., Mandoiu, I.I., Prasad, S.K., Tchervenski, N., Zelikovsky, A.: Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks. Wireless Networks 12(3), 287–299 (2006)CrossRefGoogle Scholar
  2. 2.
    Bläser, M., Manthey, B., Rao, B.V.R.: Smoothed analysis of partitioning algorithms for Euclidean functionals. Algorithmica 66(2), 397–418 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bläser, M., Panagiotou, K., Rao, B.V.R.: A probabilistic analysis of Christofides’ algorithm. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 225–236. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Clementi, A.E.F., Penna, P., Silvestri, R.: On the power assignment problem in radio networks. Mobile Networks and Applications 9(2), 125–140 (2004)CrossRefGoogle Scholar
  5. 5.
    Funke, S., Laue, S., Lotker, Z., Naujoks, R.: Power assignment problems in wireless communication: Covering points by disks, reaching few receivers quickly, and energy-efficient travelling salesman tours. Ad Hoc Networks 9(6), 1028–1035 (2011)CrossRefGoogle Scholar
  6. 6.
    de Graaf, M., Boucherie, R.J., Hurink, J.L., van Ommeren, J.K.: An average case analysis of the minimum spanning tree heuristic for the range assignment problem. Memorandum 11259 (revised version), Department of Applied Mathematics, University of Twente (2013)Google Scholar
  7. 7.
    Halldórsson, M.M., Holzer, S., Mitra, P., Wattenhofer, R.: The power of non-uniform wireless power. In: Proc. of the 24th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1595–1606. SIAM (2013)Google Scholar
  8. 8.
    Halldórsson, M.M., Mitra, P.: Wireless connectivity and capacity. In: Rabani, Y. (ed.) Proc. of the 23rd Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 516–526. SIAM (2012)Google Scholar
  9. 9.
    Kesselheim, T.: A constant-factor approximation for wireless capacity maximization with power control in the SINR model. In: Proc. of the 22nd Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1549–1559. SIAM (2011)Google Scholar
  10. 10.
    Khan, M., Pandurangan, G., Pei, G., Vullikanti, A.K.S.: Brief announcement: A fast distributed approximation algorithm for minimum spanning trees in the SINR model. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 409–410. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Kirousis, L.M., Kranakis, E., Krizanc, D., Pelc, A.: Power consumption in packet radio networks. Theoretical Computer Science 243(1-2), 289–305 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lloyd, E.L., Liu, R., Marathe, M.V., Ramanathan, R., Ravi, S.S.: Algorithmic aspects of topology control problems for ad hoc networks. Mobile Networks and Applications 10(1-2), 19–34 (2005)CrossRefGoogle Scholar
  13. 13.
    Pahlavan, K., Levesque, A.H.: Wireless Information Networks. Wiley (1995)Google Scholar
  14. 14.
    Rappaport, T.S.: Wireless Communication. Prentice Hall (2002)Google Scholar
  15. 15.
    Warnke, L.: On the method of typical bounded differences. Computing Research Repository 1212.5796 [math.CO], arXiv (2012)Google Scholar
  16. 16.
    Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer, Heidelberg (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Maurits de Graaf
    • 1
    • 2
  • Bodo Manthey
    • 1
  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  2. 2.Thales Nederland B.V., HuizenThe Netherlands

Personalised recommendations