Monotonicity Testing and Directed Isoperimetric Inequalities

Abstract

2.1.1 Boolean Isoperimetric Type Theorems n Given a function f : {0, 1} ↦ {0, 1}, define the variance of the function as var(f ) = p(1 − p), where p = Prx[f (x) = 1]. Let Sf denote the set of sensitive edges, that is, the set of pairs (x, y) such that x, y ∈ {0, 1}n differ in exactly one coordinate, f (x) = 1 and f (y) = 0. Let If = |Sf | denote the “total influence” of the function. A 2n folklore theorem states:1