Testing juntas nearly optimally

A function on n variables is called a k-junta if it depends on at most k of its variables. In this article, we show that it is possible to test whether a function is a k-junta or is "far" from being a k-junta with O(k/ + k log k) queries, where is the approximation parameter. This result improves on the previous best upper bound of ~O(k3/2 )/ queries and is asymptotically optimal, up to a logarithmic factor. We obtain the improved upper bound by introducing a new algorithm with one-sided error for testing juntas. Notably, the algorithm is a valid junta tester under very general conditions: it holds for functions with arbitrary finite domains and ranges, and it holds under any product distribution over the domain. A key component of the analysis of the new algorithm is a new structural result on juntas: roughly, we show that if a function f is "far" from being a k-junta, then f is "far" from being determined by k parts in a random partition of the variable...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Eric Blais
• Keywords: Algorithms, General Terms Algorithms, Recherche Sur La, STOC 2009, Valid Junta Tester