k +  Decision Trees

(Extended Abstract)
  • James Aspnes
  • Eric Blais
  • Murat Demirbas
  • Ryan O’Donnell
  • Atri Rudra
  • Steve Uurtamo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6451)

Abstract

Consider a wireless sensor network in which each sensor has a bit of information. Suppose all sensors with the bit 1 broadcast this fact to a basestation. If zero or one sensors broadcast, the basestation can detect this fact. If two or more sensors broadcast, the basestation can only detect that there is a ”collision.” Although collisions may seem to be a nuisance, they can in some cases help the basestation compute an aggregate function of the sensors’ data.

Motivated by this scenario, we study a new model of computation for boolean functions: the 2 +  decision tree. This model is an augmentation of the standard decision tree model: now each internal node queries an arbitrary set of literals and branches on whether 0, 1, or at least 2 of the literals are true. This model was suggested in a work of Ben-Asher and Newman but does not seem to have been studied previously.

Our main result shows that 2 +  decision trees can ”count” rather effectively. Specifically, we show that zero-error 2 +  decision trees can compute the threshold-of-t symmetric function with O(t) expected queries (and that Ω(t) is a lower bound even for two-sided error 2 +  decision trees). Interestingly, this feature is not shared by 1 +  decision trees. Our result implies that the natural generalization to k +  decision trees does not give much more power than 2 +  decision trees. We also prove a lower bound of \(\tilde{\Omega}(t) \cdot \log(n/t)\) for the deterministic 2 +  complexity of the threshold-of-t function, demonstrating that the randomized 2 +  complexity can in some cases be unboundedly better than deterministic 2 +  complexity.

Finally, we generalize the above results to arbitrary symmetric functions, and we discuss the relationship between k +  decision trees and other complexity notions such as decision tree rank and communication complexity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aigner, M.: Combinatorial Search. Wiley-Teubner Series in Computer Science (1988)Google Scholar
  2. 2.
    Arora, A., Dutta, P., Bapat, S., Kulathumani, V., Zhang, H., Naik, V., Mittal, V., Cao, H., Demirbas, M., Gouda, M., Choi, Y.R., Herman, T., Kulkarni, S.S., Arumugam, U., Nesterenko, M., Vora, A., Miyashita, M.: A line in the sand: A wireless sensor network for target detection, classification, and tracking. Computer Networks (Elsevier) 46(5), 605–634 (2004)CrossRefGoogle Scholar
  3. 3.
    Ben-Asher, Y., Newman, I.: Decision trees with boolean threshold queries. J. Comput. Syst. Sci. 51(3), 495–502 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: STOC 1983, pp. 80–86 (1983)Google Scholar
  5. 5.
    Bshouty, N.H.: A subexponential exact learning algorithm for DNF using equivalence queries. Information Processing Letters” 59(3), 37–39 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: A survey. Theoretical Computer Science 288(1), 21–43 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chockler, G., Demirbas, M., Gilbert, S., Lynch, N.A., Newport, C.C., Nolte, T.: Consensus and collision detectors in radio networks. Distributed Computing 21(1), 55–84 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Demirbas, M., Soysal, O., Hussain, M.: Singlehop collaborative feedback primitives for wireless sensor networks. In: INFOCOM, pp. 2047–2055 (2008)Google Scholar
  9. 9.
    Dobkin, D., Lipton, R.J.: Multidimensional searching problems. SIAM Journal on Computing 5(2), 181–186 (1976)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Du, D.Z., Hwang, F.K.: Combinatorial Group Testing and its Applications. World Scientific, Singapore (2000)MATHGoogle Scholar
  11. 11.
    Dutta, P., Musaloiu-e, R., Stoica, I., Terzis, A.: Wireless ack collisions not considered harmful. In: HotNets-VII: The Seventh Workshop on Hot Topics in Networks (2008)Google Scholar
  12. 12.
    Ehrenfeucht, A., Haussler, D.: Learning decision trees from random examples. Information and Computation 82(3), 231–246 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Goodrich, M.T., Hirschberg, D.S.: Efficient parallel algorithms for dead sensor diagnosis and multiple access channels. In: SPAA 2006: Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures, pp. 118–127. ACM, New York (2006)CrossRefGoogle Scholar
  14. 14.
    Hamza, K.: The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions. Statistics and Probability Letters 23(1), 21–25 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jurdzinski, T., Kutylowski, M., Zatopianski, J.: Energy-efficient size approximation of radio networks with no collision detection. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 279–289. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Kushilevitz, E., Mansour, Y.: Learning decision trees using the fourier spectrum. SIAM Journal on Computing 22(6), 1331–1348 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  18. 18.
    Moran, S., Snir, M., Manber, U.: Applications of ramsey’s theorem to decision tree complexity. J. ACM 32(4), 938–949 (1985)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nisan, N.: CREW PRAMs and decision trees. SIAM Journal on Computing 20(6), 999–1007 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Paturi, R.: On the degree of polynomials that approximate symmetric boolean functions (preliminary version). In: STOC 1992, pp. 468–474 (1992)Google Scholar
  21. 21.
    Rosenberg, A.L.: On the time required to recognize properties of graphs: a problem. SIGACT News 5(4), 15–16 (1973)CrossRefGoogle Scholar
  22. 22.
    Yao, A.C.C.: Monotone bipartite graph properties are evasive. SIAM Journal on Computing 17(3), 517–520 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • James Aspnes
    • 1
  • Eric Blais
    • 2
  • Murat Demirbas
    • 3
  • Ryan O’Donnell
    • 2
  • Atri Rudra
    • 3
  • Steve Uurtamo
    • 3
  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA
  2. 2.Department of Computer ScienceCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of Computer Science and EngineeringBuffalo State University of New YorkBuffaloUSA

Personalised recommendations