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• W117090335 abstract "Given a function f : {0, 1} → {0, 1}, let M (f) denote the smallest distance between f and a monotone function on {0, 1}. Let δM (f) denote the fraction of hypercube edges where f violates monotonicity. We give an alternative proof of the tight bound: δM (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}, x is said to be less than y, or x ≺ y, if x 6= y and for all i ∈ [n], xi ≤ yi. We view the set {0, 1} n as vertices of the n-dimensional hypercube graph. An edge (x, y) in this graph denotes a pair of strings x and y such that x ≺ y and the Hamming distance between x and y is exactly 1. Note that the number of edges in {0, 1} is exactly 1 2n2 . For a function f : {0, 1} → {0, 1}, we say that an edge (x, y) is violated by f if x ≺ y but f(x) > f(y). The function f is monotone if and only if no edge in the hypercube is violated by f . Now, let us define the following two quantities: Definition 1. For a function f : {0, 1} → {0, 1}, • M (f) def = min g Pr x∈{0,1} [f(x) 6= g(x)], where g : {0, 1} → {0, 1} ranges over all monotone functions • δM (f) def = Pr e edge in {0,1} [e violated by f ] M (f) represents the global distance of f from the monotonicity property. δM (f) is a local distance measure corresponding to the following natural test of monotonicity (analyzed, e.g., in [DGL+99, GGL+00]): choose some random edges in the hypercube and check whether they are violated by f . The combinatorial question that now arises is the characterization of the relationship between the two distance measures. Goldreich et al. in [GGL+00] observed that this relationship is not simply determined; that is, M (·) is not just a function of δM (·) or vice versa. In fact, they proved the following: Theorem 2 (Proposition 4 of [GGL+00]). For every c < 1, for any sufficiently large n, and for any α such that 2−c·n ≤ α ≤ 1 2 : 1. There exists a function f : {0, 1} → {0, 1} such that α ≤ M (f) ≤ 2α and δM (f) = 2 n M (f) 2. There exists a function f : {0, 1} → {0, 1} such that (1− o(1))α ≤ M (f) ≤ 2α and δM (f) = (1± o(1)) · (1− c) · M (f) ∗Massachusetts Institute of Technology. Email:abhatt@mit.edu." @default.
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• W117090335 date "2008-01-01" @default.
• W117090335 modified "2023-09-23" @default.
• W117090335 title "A Note on the Distance to Monotonicity of Boolean Functions." @default.
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