Advertisement

Improved Bounds for the Randomized Decision Tree Complexity of Recursive Majority

  • Frédéric Magniez
  • Ashwin Nayak
  • Miklos Santha
  • David Xiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We consider the randomized decision tree complexity of the recursive 3-majority function. For evaluating height h formulae, we prove a lower bound for the δ-two-sided-error randomized decision tree complexity of (1 − 2δ)(5/2) h , improving the lower bound of (1 − 2δ)(7/3) h given by Jayram, Kumar, and Sivakumar (STOC ’03). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most (1.007) ·2.64946 h . The previous best known algorithm achieved complexity (1.004) ·2.65622 h . The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel “interleaving” of two recursive algorithms.

Keywords

Decision Tree Boolean Function Recursive Algorithm Hard Distribution Decision Tree Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blum, M., Impagliazzo, R.: General oracle and oracle classes. In: Proc. FOCS 1987, pp. 118–126 (1987)Google Scholar
  2. 2.
    Hartmanis, J., Hemachandra, L.: One-way functions, robustness, and non-isomorphism of NP-complete sets. In: Proc. Struc. in Complexity Th. 1987, pp. 160–173 (1987)Google Scholar
  3. 3.
    Heiman, R., Newman, I., Wigderson, A.: On read-once threshold formulae and their randomized decision tree complexity. In: Proc. Struc. in Complexity Th. 1990, pp. 78–87 (1990)Google Scholar
  4. 4.
    Heiman, R., Wigderson, A.: Randomized versus deterministic decision tree complexity for read-once boolean functions. In: Proc. Struc. in Complexity Th. 1991, pp. 172–179 (1991)Google Scholar
  5. 5.
    Jayram, T., Kumar, R., Sivakumar, D.: Two applications of information complexity. In: Proc. STOC 2003, pp. 673–682 (2003)Google Scholar
  6. 6.
    Landau, I., Nachmias, A., Peres, Y., Vanniasegaram, S.: The lower bound for evaluating a recursive ternary majority function: an entropy-free proof. Tech. rep., Dep. of Stat., UC Berkeley (2006) (undergraduate Research Report), http://www.stat.berkeley.edu/110
  7. 7.
    Nisan, N.: CREW PRAMs and decision trees. In: Proc. STOC 1989, pp. 327–335. ACM, New York (1989)Google Scholar
  8. 8.
    Reichardt, B.W., Špalek, R.: Span-program-based quantum algorithm for evaluating formulas. In: Proc. 40th STOC, pp. 103–112. ACM, New York (2008)Google Scholar
  9. 9.
    Saks, M., Wigderson, A.: Probabilistic boolean decision trees and the complexity of evaluating game trees. In: Proc. FOCS 1986, pp. 29–38 (1986)Google Scholar
  10. 10.
    Santha, M.: On the Monte Carlo boolean decision tree complexity of read-once formulae. Random Structures and Algorithms 6(1), 75–87 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Snir, M.: Lower bounds for probabilistic linear decision trees. Combinatorica 9, 385–392 (1990)Google Scholar
  12. 12.
    Tardos, G.: Query complexity or why is it difficult to separate NP A ∩ coNP A from P A by a random oracle. Combinatorica 9, 385–392 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Frédéric Magniez
    • 1
  • Ashwin Nayak
    • 2
  • Miklos Santha
    • 1
    • 3
  • David Xiao
    • 1
    • 4
  1. 1.LIAFA, Univ. Paris 7, CNRSParisFrance
  2. 2.C&O and IQCU. Waterloo; and Perimeter InstituteWaterlooCanada
  3. 3.Centre for Quantum TechnologiesNational U. of SingaporeSingapore
  4. 4.Univ. Paris-SudOrsayFrance

Personalised recommendations