Local Lipschitz Filters for Bounded-Range Functions

We study local filters for the Lipschitz property of real-valued functions $f: V \to [0,r]$, where the Lipschitz property is defined with respect to an arbitrary undirected graph $G=(V,E)$. We give nearly optimal local Lipschitz filters both with respect to $\ell_1$ distance and $\ell_0$ distance. Previous work only considered unbounded-range functions over $[n]^d$. Jha and Raskhodnikova (SICOMP `13) gave an algorithm for such functions with lookup complexity exponential in $d$, which Awasthi et al.\ (ACM Trans. Comput. Theory) showed was necessary in this setting. By considering the natural class of functions whose range is bounded in $[0,r]$, we circumvent this lower bound and achieve running time $(d^r\log n)^{O(\log r)}$ for the $\ell_1$-respecting filter and $d^{O(r)}\text{polylog }n$ for the $\ell_0$-respecting filter for functions over $[n]^d$. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on $r$ for the domain $\{0,1\}^d$, an important special case of the domain $[n]^d$. In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing. Finally, we provide two applications of our local filters. First, they can be used in conjunction with the Laplace mechanism for differential privacy to provide filter mechanisms for privately releasing outputs of black box functions even in the presence of malicious clients. Second, we use them to obtain the first tolerant testers for the Lipschitz property.