Approximating the Distance to Monotonicity and Convexity in Sublinear Time

In this thesis we study the problems of distance approximation to monotonicity and distance approximation to convexity. Namely, we are interested in (randomized) sublinear algorithms that approximate the Hamming distance between a given function and the closest monotone/convex function. For the monotonicity property, we focus on functions over the d-dimensional hyper-cube, [n]d, with any finite range. Previous work on distance approximation to monotonicity focused on the one-dimensional case and the only extension to higher dimensions was with an approximation factor exponential in the dimension d. We describe a reduction from the case of functions over the d-dimensional hyper-cube to the case of functions over the k-dimensional hyper-cube, where k < d. This reduction is efficient. That is, polynomial only in the additive error allowed, and not dependent on the size of the domain, range, or dimension. The quality of estimation that this reduction provides is linear in the size of the dimension and logarithmic in the size of the range. Using this reduction and a known distance approximation algorithm for the one dimensional case, we suggest a distance approximation algorithm for functions over the d-dimensional hyper-cube, with any finite range. For the case of the Boolean range, we present solutions for distance approximation to monotonicity of functions over one dimension, two dimensions, and the k-dimensional hypercube (for any k ≥ 1). Applying these algorithms and the reduction described above, we suggest a variety of distance approximation algorithms for Boolean range functions over the d-dimensional hyper-cube, which suggest a trade-off between quality of estimation and efficiency of computation. For the convexity property, we present an efficient distance approximation algorithm for functions over one dimension, with any range. No solution for this problem was known before.