On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs

  • Andrea E. F. Clementi
  • Pilu Crescenzi
  • Paolo Penna
  • Gianluca Rossi
  • Paola Vocca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2010)

Abstract

We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broad- cast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a sub-logarithmic factor (unless P=NP).We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

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References

  1. 1.
    G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation-Combinatorial optimization problems and their approximability properties. Springer Verlag, 1999.Google Scholar
  2. 2.
    R. Bar-Yehuda, O. Goldreich, and A. Itai. On the time complexity of broadcast operations in multi-hop radio networks: an exponential gap between determinism and randomization. J. Computer and Systems Science, 45:104–126, 1992.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Bar-Yehuda, A. Israeli, and A. Itai. Multiple communication in multi-hop radio networks. SIAM J. on Computing, 22:875–887, 1993.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. P. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. Prentice Hall, 1994.Google Scholar
  5. 5.
    A.E.F. Clementi, A. Ferreira, P. Penna, S. Perennes, and R. Silvestri. The minimum range assignment problem on linear radio networks. In Proc. 8th Annual European Symposium on Algorithms, volume 1879 of LNCS, pages 143–154, 2000.Google Scholar
  6. 6.
    A.E.F. Clementi, P. Penna, and R. Silvestri. Hardness results for the power range assignment problem in packet radio networks. In Proc. of Randomization, Approximation and Combinatorial Optimization, volume 1671 of LNCS, pages 197–208, 1999. Full version available as ECCC Report TR00-54.Google Scholar
  7. 7.
    A.E.F. Clementi, P. Penna, and R. Silvestri. The power range assignment problem in radio networks on the plane. In Proc. 17th Annual Symposium on Theoretical Aspects of Computer Science, volume 1770 of LNCS, pages 651–660, 2000. Full version available as ECCC Report TR00-54.Google Scholar
  8. 8.
    J.H. Conway and N.J.A. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, 1988.Google Scholar
  9. 9.
    B. Dixon, M. Rauch, and R.E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21:1184–1192, 1992.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Ephremides. Complicating factors for the use of distributed algorithms in wireless networks. In 1st Int. Workshop on Approximation and Randomized Algorithms in Communication Networks, invited talk, 2000.Google Scholar
  11. 11.
    D. Eppstein. Offline algorithms for dynamic minimum spanning tree problem. J. of Algorithms, 17:237–250, 1994.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Z. Haas and S. Tabrizi. On some challenges and design choices in ad hoc communications. In Proc. IEEE MILCOM’98, 1998.Google Scholar
  13. 13.
    G.A. Kabatiansky and V.I. Levenshtein. Bounds for packings on a sphere and in space (in russian). Problemy Peredachi Informatsii, 14(1):3–25, 1978. English translation: Problems of Information Theory, 14(1):1-17, 1978.Google Scholar
  14. 14.
    L.M. Kirousis, E. Kranakis, D. Krizanc, and A. Pelc. Power consumption in packet radio networks. Theoretical Computer Science, 243:289–306, 2000.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    G.S. Lauer. Packet radio routing, chapter 11. Prentice-Hall, 1995.Google Scholar
  16. 16.
    R. Mathar and J. Mattfeldt. Optimal transmission ranges for mobile communication in linear multi-hop packet radio networks. Wireless Networks, 2:329–342, 1996.CrossRefGoogle Scholar
  17. 17.
    C. Monma and S. Suri. Transitions in geometric minimum spanning tree. Discrete and Computational Geometry, 8:265–293, 1992.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Nardelli, G. Proietti, and P. Widmayer. Maintainig a minimum spanning tree under transient node failures. In Proc. 8th Annual European Symposium on Algorithms, to appear, 2000.Google Scholar
  19. 19.
    K. Pahlavan and A. Levesque. Wireless information networks. Wiley-Interscience, 1995.Google Scholar
  20. 20.
    C. H. Papadimitriou. Computational Complexity. Addison Wesley, 1994.Google Scholar
  21. 21.
    P. Piret. On the connectivity of radio networks. IEEE Trans. on Inform. Theory, 37:1490–1492, 1991.CrossRefGoogle Scholar
  22. 22.
    D. Raychaudhuri and N.D. Wilson. ATM-based transport architecture for multiservices wireless personal communication networks. IEEE J. Selected Areas in Communications, 12:1401–1414, 1994.CrossRefGoogle Scholar
  23. 23.
    R. Raz and S. Safra. A sub-constant error-probability low-degree test, and sub-constant error-probability pcp characterization of np. In Proc. 29th Ann. ACM Symp. on Theory of Comp., pages 784–798, 1997.Google Scholar
  24. 24.
    S. Ulukus and R.D. Yates. Stocastic power control for cellular radio systems. IEEE Trans. Comm., 46:784–798, 1996.CrossRefGoogle Scholar
  25. 25.
    A.D. Wyner. Capabilities of bounded discrepancy decoding. BSTJ, 44:1061–1122, 1965.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Andrea E. F. Clementi
    • 1
  • Pilu Crescenzi
    • 2
  • Paolo Penna
    • 1
  • Gianluca Rossi
    • 2
  • Paola Vocca
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Dipartimento di Sistemi e InformaticaUniversità di FirenzeFirenzeItaly

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