Wireless Networks

, Volume 21, Issue 1, pp 281–295 | Cite as

An exact algorithm for maximum lifetime data gathering tree without aggregation in wireless sensor networks

  • Xiaojun Zhu
  • Xiaobing Wu
  • Guihai Chen


In wireless sensor networks, maximizing the lifetime of a data gathering tree without aggregation has been proved to be NP-complete. In this paper, we prove that, unless P = NP, no polynomial-time algorithm can approximate the problem with a factor strictly greater than 2/3. The result even holds in the special case where all sensors have the same initial energy. Existing works for the problem focus on approximation algorithms, but these algorithms only find sub-optimal spanning trees and none of them can guarantee to find an optimal tree. We propose the first non-trivial exact algorithm to find an optimal spanning tree. Due to the NP-hardness nature of the problem, this proposed algorithm runs in exponential time in the worst case, but the consumed time is much less than enumerating all spanning trees. This is done by several techniques for speeding up the search. Featured techniques include how to grow the initial spanning tree and how to divide the problem into subproblems. The algorithm can handle small networks and be used as a benchmark for evaluating approximation algorithms.


Wireless sensor networks Data gathering tree Maximum lifetime Exact algorithm Inapproximability 



This work was partly supported by the New-Faculty Research Grant of Nanjing University of Aeronautics and Astronautics, China NSF Grants (61321491, 61373130, 61133006), NSF of Jiangsu Province Grant (BK20141319), and China 973 projects (2014CB340303, 2012CB316201).


  1. 1.
    Altinkemer, K., & Gavish, B. (1988). Heuristics with constant error guarantees for the design of tree networks. Management Science, 34(3), 331–341.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Arkin, E. M., Guttmann-Beck, N., & Hassin, R. (2012). The (k, k)-capacitated spanning tree problem. Discrete Optimization, 9(4), 258–266.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berkelaar, M., Eikland, K., & Notebaert, P. (2010). lp_solve Accessed 2014-3-2.
  4. 4.
    Cormen, T. H., Stein, C., Rivest, R. L., & Leiserson, C. E. (2001). Introduction to algorithms (2nd ed.). New York: McGraw-Hill Higher Education.zbMATHGoogle Scholar
  5. 5.
    Diestel, R. (2006). Graph theory (3rd ed.). Berlin: Springer.Google Scholar
  6. 6.
    Gabow, H. N., & Myers, E. W. (1978). Finding all spanning trees of directed and undirected graphs. SIAM Journal on Computing, 7(3), 280–287.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W. H. Freeman & Co.zbMATHGoogle Scholar
  8. 8.
    Garey, M. R., & Johnson, D. S. (1983). Formulations and algorithms for the capacitated minimal directed tree problem. Journal of the ACM, 30(1), 118–132. doi: 10.1145/322358.322367.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopcroft, J., & Tarjan, R. (1973). Algorithm 447: Efficient algorithms for graph manipulation. Communications of ACM, 16(6), 372–378.CrossRefGoogle Scholar
  10. 10.
    Intanagonwiwat, C., Govindan, R., & Estrin, D. (2000). Directed diffusion: A scalable and robust communication paradigm for sensor networks. In Proceedings of ACM MobiCom.Google Scholar
  11. 11.
    Jothi, R., & Raghavachari, B. (2005). Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design. ACM Transations on Algorithms, 1(2), 265–282.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kuo, T. W., & Tsai, M. J. (2012). On the construction of data aggregation tree with minimum energy cost in wireless sensor networks: Np-completeness and approximation algorithms. In Proceedings of IEEE INFOCOM.Google Scholar
  13. 13.
    Lenstra, J. K., Shmoys, D. B., & Tardos, E. (1990). Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46(3), 259–271.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Liang, J., Wang, J., Cao, J., Chen, J., & Lu, M. (2010). An efficient algorithm for constructing maximum lifetime tree for data gathering without aggregation in wireless sensor networks. In Proceedings of IEEE INFOCOM.Google Scholar
  15. 15.
    Luo, D., Zhu, X., Wu, X., & Chen, G. (2011). Maximizing lifetime for the shortest path aggregation tree in wireless sensor networks. In Proceedings of IEEE INFOCOM.Google Scholar
  16. 16.
    Peng, Y., Li, Z., Qiao, D., & Zhang, W. (2013). I2c: A holistic approach to prolong the sensor network lifetime. In Proceedings of IEEE INFOCOM.Google Scholar
  17. 17.
    Read, R. C., & Tarjan, R. (1975). Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5, 237–252.zbMATHMathSciNetGoogle Scholar
  18. 18.
    Shan, M., Chen, G., Luo, D., Zhu, X., & Wu, X. (2014). Building optimal shortest path data aggregation trees in wireless sensor networks. ACM Transactions on Sensor Networks, 11(1). Article No. 11.Google Scholar
  19. 19.
    Shioura, A., Tamura, A., & Uno, T. (1997). An optimal algorithm for scanning all spanning trees of undirected graphs. SIAM Journal on Computing, 26(3), 678–692.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wikipedia Contributors. (2014). Linear programming. Accessed 2014-3-2.
  21. 21.
    Williamson, D. P., & Shmoys, D. B. (2011). The design of approximation algorithms (1st ed.). New York, NY: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  22. 22.
    Wu, Y., Mao, Z., Fahmy, S., & Shroff, N. (2010). Constructing maximum-lifetime data-gathering forests in sensor networks. IEEE/ACM Transactions on Networking, 18(5), 1571–1584.CrossRefGoogle Scholar
  23. 23.
    Xiang, L., Luo, J., & Rosenberg, C. (2013). Compressed data aggregation: Energy efficient and high fidelity data collection. IEEE/ACM Transactions on Networking, 21(6), 1722–1735.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Computer Science and TechnologyNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.State Key Laboratory for Novel Software TechnologyNanjing UniversityNanjingChina
  3. 3.Shanghai Key Laboratory of Scalable Computing and SystemsShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations