k ⁺ Decision Trees

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.

James Aspnes is supported in part by NSF grant CCF-0916389. Murat Demirbas’ work was partially supported by NSF Career award #0747209. Ryan O’Donnell is supported in part by NSF grants CCF-0747250 and CCF-0915893, a Sloan fellowship, and an Okawa fellowship. Atri Rudra and Steve Uurtamo are supported in part by NSF CAREER grant CCF-0844796.