Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability
Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability
Lower bounds on the complexity of graph properties
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On the degree of Boolean functions as real polynomials
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Information Processing Letters
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Improved Lower Bounds on the Randomized Complexity of Graph Properties
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Computing Graph Properties by Randomized Subcube Partitions
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Balanced boolean functions that can be evaluated so that every input bit is unlikely to be read
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Noise stability of functions with low in.uences invariance and optimality
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Collective coin flipping, robust voting schemes and minima of Banzhaf values
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Probabilistic Boolean decision trees and the complexity of evaluating game trees
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Lower bounds to randomized algorithms for graph properties
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Generic oracles and oracle classes
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Decision trees and influences of variables over product probability spaces
Combinatorics, Probability and Computing
Variable Influences in Conjunctive Normal Forms
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
The Fourier entropy-influence conjecture for certain classes of Boolean functions
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Approximating the Influence of Monotone Boolean Functions in O(√n) Query Complexity
ACM Transactions on Computation Theory (TOCT)
Evasiveness through a circuit lens
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Gems in decision tree complexity revisited
ACM SIGACT News
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We prove that for any decision tree calculating a boolean function f : {-1,1}^n\to : {-1,1} Var [f] {\le \sum\limits_{i = 1}^n {_{} } } \delta {\rm{i Inf(f),}} where di is the probability that the ith input variable is read and Infi(f) is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {-1,1}^n. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth d has a variable with influence at least \frac{1}{d}. The only previous nontrivial lower bound known was \Omega(b^2) . Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least \Omega(v^4/3/p^1/3), where v is the number of vertices and p \le \frac{1}{2} is the critical threshold probability. This supersedes the milestone \Omega(v^4/3) bound of Hajnal [13] and is sometimes superior to the best known lower bounds of Chakrabarti- Khot [9] and Friedgut-Kah