k+ Decision Trees

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.