Block sensitivity of minterm-transitive functions - Drucker, Andrew (Theoretical computer science, 2011)
Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures. In a recent work of this type, Sun (2007) [9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy b s ( f ) = Ω ( N 1 / 3 ) . He also showed that there exists such a function for which b s ( f ) = O ( N 3 / 7 ln N ) . His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) [3]. We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy b s ( f ) = Ω ( N 3 / 7 ) . Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which b s ( f ) = O ( N 3 / 7 ln 1 / 7 N ) .