Testing Lipschitz Functions on Hypergrid Domains

A function (Formula presented.) , where each input is an integer from 1 to (Formula presented.) and output is a real number, is Lipschitz if changing one of the inputs by 1 changes the output by at most 1. In other words, Lipschitz functions are not very sensitive to small changes in the input. Our main result is an efficient tester for the Lipschitz property of functions (Formula presented.) , where (Formula presented.) and (Formula presented.) is the set of integer multiples of (Formula presented.). A property tester is given an oracle access to a function (Formula presented.) and a proximity parameter (Formula presented.) , and it has to distinguish, with high probability, functions that have the property from functions that differ on at least an (Formula presented.) fraction of values from every function with the property. The Lipschitz property was first studied by Jha and Raskhodnikova (FOCS’11) who motivated it by applications to data privacy and program verification. They presented efficient testers for the Lipschitz property of functions on the domains (Formula presented.) and (Formula presented.). Our tester for functions on the more general domain (Formula presented.) runs in time (Formula presented.) for constant (Formula presented.) and (Formula presented.). The main tool in the analysis of our tester is a smoothing procedure that makes a function Lipschitz by modifying it at a few points. Its analysis is already nontrivial for the 1-dimensional version, which we call Bubble Smooth, in analogy to Bubble Sort. In one step, Bubble Smooth modifies two values that violate the Lipschitz property, namely, differ by more than 1, by transferring (Formula presented.) units from the larger to the smaller. We define a transfer graph to keep track of the transfers, and use it to show that the (Formula presented.) distance between (Formula presented.) and BubbleSmooth (Formula presented.) is at most twice the (Formula presented.) distance from (Formula presented.) to the nearest Lipschitz function. Bubble Smooth has several other important properties that allow us to obtain a dimension reduction, i.e., a reduction from testing functions on multidimensional domains to testing functions on one-dimensional domains. Our dimension reduction incurs only a small multiplicative overhead in the running time and thus avoids the exponential dependence on the dimension.