Local Monotonicity Reconstruction

We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] = {1, . . . , n} (where d is viewed as a constant). We would like to closely approximate f by a monotone function g. This should be done by a procedure (a filter) that given as input a point x ∈ [n] outputs the value of g(x), and runs in time that is polylogarithmic in n. The procedure can (indeed must) be randomized, but we require that all of the randomness be specified in advance by a single short random seed. We construct such an implementation where the time and space per query is (log n) and the size of the seed is polynomial in log n and d. Furthermore, 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 allows for a local implementation: one can initialize many copies of the filter with the same short random seed, and they can autonomously handle queries, while producing outputs that are consistent with the same approximating function g. A preliminary version of this paper was titled “Parallel Monotonicity Reconstruction” [32]. This work was supported in part by NSF under grants CCF-0515201 and CCF-0832787.