Approximating the influence of monotone Boolean functions in O(√n) query complexity

Dana Ron; Ronit Rubinfeld; Muli Safra; Alex Samorodnitsky; Omri Weinstein

doi:https://doi.org/10.1145/2382559.2382562

Approximating the influence of monotone Boolean functions in O(√n) query complexity

Dana Ron, Ronit Rubinfeld, Muli Safra, Alex Samorodnitsky, Omri Weinstein

Tel Aviv University, The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ∈) by performing O (equation) queries. We also prove a lower bound of Ω (equation) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I[f] = Ω(1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions, we give a lower bound of Ω ([n/I[f]]), which matches the complexity of a simple sampling algorithm.

Original language	English
Article number	11
Journal	ACM Transactions on Computation Theory
Volume	4
Issue number	4
DOIs	https://doi.org/10.1145/2382559.2382562
State	Published - Nov 2012

Keywords

Influence of a Boolean function
Sublinear query approximation algorithms
Symmetric chains

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Computational Theory and Mathematics

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https://doi.org/10.1145/2382559.2382562

Cite this

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abstract = "The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ∈) by performing O (equation) queries. We also prove a lower bound of Ω (equation) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I[f] = Ω(1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions, we give a lower bound of Ω ([n/I[f]]), which matches the complexity of a simple sampling algorithm.",

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N2 - The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ∈) by performing O (equation) queries. We also prove a lower bound of Ω (equation) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I[f] = Ω(1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions, we give a lower bound of Ω ([n/I[f]]), which matches the complexity of a simple sampling algorithm.

AB - The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ∈) by performing O (equation) queries. We also prove a lower bound of Ω (equation) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I[f] = Ω(1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions, we give a lower bound of Ω ([n/I[f]]), which matches the complexity of a simple sampling algorithm.

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Approximating the influence of monotone Boolean functions in O(√n) query complexity

Abstract

Keywords

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