James Aspnes, Eric Blais, Murat Demirbas, Ryan O'Donnell, Atri Rudra, and Steve Uurtamo. k+ decision trees. Submitted to STACS 2010.
Consider a wireless sensor network in which each node possesses a bit of information. Suppose all sensors with the bit 1 broadcast this fact to a central processor. If zero or one sensors broadcast, the central processor can detect this fact. If two or more sensors broadcast, the central processor can only detect that there is a ``collision.'' Although collisions may seem to be a nuisance, they can in some cases help the central processor 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, demonstrating that ``collisions can help.'' 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 Ω~(t⋅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.
@unpublished{AspnesBDORU2009,
author = {James Aspnes and Eric Blais and Murat Demirbas and Ryan O'Donnell and Atri Rudra and Steve Uurtamo},
title = {$k^+$ decision trees},
month=sep,
year = 2009,
note = "Submitted to STACS 2010"
}