Local Property Reconstruction and Monotonicity

We propose a general model of local property reconstruction. Suppose we have a function f on domain Gamma, which is supposed to have a particular property P, but may not have the property. We would like a procedure that produces a function g that has property P and is close to f (according to some suitable metric). The reconstruction procedure, called a filter, has the following form. The procedure takes as input an element x of Gamma and outputs g(x). The procedure has oracle access to the function f and uses a single short random string rho, but is otherwise deterministic. This model was inspired by a related model of online property reconstruction that was introduced by by Ailon, Chazelle, Comandur and Liu (2004). It is related to the property testing model, and extends the framework that is used in the model of locally decodable codes. A similar model, in the context of hypergraph properties, was independently proposed and studied by Austin and Tao (2008). We specifically consider the property of monotonicity and develop an efficient local filter for this property. The input f is a real valued function defined over the domain {1, ... , n}(d) (where n is viewed as large and d as a constant). The function is monotone if the following property holds: for two domain elements x and y, if x <= y (in the product order) then f(x) <= f(y). Given x, our filter outputs the value g(x) in (log n)(O(1)) time and uses a random seed rho of the same size. With high probability, the ratio of the Hamming distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n).