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)


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.


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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

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