Local Property Reconstruction and Monotonicity

We propose a general model of local property reconstruction. Suppose we have a function f on domain Γ, which is supposed to have a particular property P, but may not have the property. We would like a procedure which 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 Γ and outputs g(x). The procedure has oracle access to the function f and uses a single short random string ρ, 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 thie property. The input f is a real valued function defined on the domain {1, . . . , n} (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) time and uses a random seed ρ 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). ∗This is an extended abstract of work that will appear as “Local Monotonicity Reconstruction” in SIAM Journal of Computing. A preliminary version of this work appeared as “Parallel Monotonicity Reconstruction” [29]. †This work was supported in part by NSF under grants CCF-0515201 and CCF-0832787. ‡This paper is partly based on material that appeared in this author’s Ph.D. dissertation for the Department of Computer Science, Princeton University.