A Note on the Distance to Monotonicity of Boolean Functions

Given a function f : {0, 1}n {0, 1}, let M (f) denote the smallest distance between f and a monotone function on {0, 1}n . Let M (f) denote the fraction of hypercube edges where f violates monotonicity. We give an alternative proof of the tight bound: M (f) 2 n M (f) for any boolean function f. This was already shown by Raskhodnikova in [Ras99]. Let U be a set of objects and let P U be a property of the elements of U. For many natural definitions of U and P, an object in U that is "globally" far from being in P also exhibits many "local" discrepancies. Thus, to test whether an object is globally far from being in P, one often only needs to make a few local checks for discrepancies. In this note, we characterize the relationship between global and local farness with respect to the property of monotonicity of boolean functions. First, we fix some notation. For two elements x, y {0, 1}n, x is said to be less than y, or x y, if x = y and for all i [n], xi yi. We ...

• Type: Journal
• Year: 2008
• Where: ECCC
• Authors: Arnab Bhattacharyya
• Keywords: Boolean Functions, ECCC 2008, ECCC 2010, Monotone Function, Smallest Distance