Approximating the distance to monotonicity in high dimensions

In this article we study the problem of approximating the distance of a function Ÿ: [n]d → R to monotonicity where [n] = {1, ⋯, n} and R is some fully ordered range. Namely, we are interested in randomized sublinear algorithms that approximate the Hamming distance between a given function and the closest monotone function. We allow both an additive error, parameterized by δ, and a multiplicative error. Previous work on distance approximation to monotonicity focused on the one-dimensional case and the only explicit extension to higher dimensions was with a multiplicative approximation factor exponential in the dimension d. Building on Goldreich et al. [2000] and Dodis et al. [1999], in which there are better implicit results for the case n = 2, we describe a reduction from the case of functions over the d-dimensional hypercube [n]d to the case of functions over the κ-dimensional hypercube [n]κ, where 1 ≤ κ ≤ d. The quality of estimation that this reduction provides is linear in [d/κ] and logarithmic in the size of the range [R] (if the range is infinite or just very large, then log [R] can be replaced by d log n). Using this reduction and a known distance approximation algorithm for the one-dimensional case, we obtain a distance approximation algorithm for functions over the d-dimensional hypercube, with any range R, which has a multiplicative approximation factor of 0(d log [R]). For the case of a binary range, we present algorithms for distance approximation to monotonicity of functions over one dimension, two dimensions, and the κ-dimensional hypercube (for any κ ≥ 1). Applying these algorithms and the reduction described before, we obtain a variety of distance approximation algorithms for Boolean functions over the d-dimensional hypercube which suggest a trade-off between quality of estimation and efficiency of computation. In particular, the multiplicative error ranges between O(d) and 0(1).