Applications of Ramsey's theorem to decision tree complexity
Journal of the ACM (JACM)
Monotone bipartite graph properties are evasive
SIAM Journal on Computing
Combinatorial search
Learning decision trees from random examples needed for learning
Information and Computation
SIAM Journal on Computing
On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
Decision trees with Boolean threshold queries
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
A subexponential exact learning algorithm for DNF using equivalence queries
Information Processing Letters
Communication complexity
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Energy-Efficient Size Approximation of Radio Networks with No Collision Detection
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
A line in the sand: a wireless sensor network for target detection, classification, and tracking
Computer Networks: The International Journal of Computer and Telecommunications Networking - Special issue: Military communications systems and technologies
Efficient parallel algorithms for dead sensor diagnosis and multiple access channels
Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures
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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 Ω(t) ċ 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.