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We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us to surpass the (worst-case) lower bounds, expressed in terms of the input size, for several problems. Our aim is to develop a similar level of understanding of the complexity of sublinear-time algorithms to the one that was enabled by research in parameterized complexity for classical algorithms. Specifically, we focus on testing properties of functions. By parameterizing the query complexity in terms of the size r of the image of the input function, we obtain testers for monotonicity and convexity of functions of the form f :[ n ]→ R with query complexity O (log r ), with no dependence on n . The result for monotonicity circumvents the Ω (log n ) lower bound by Fischer (Inf. Comput. 2004) for this problem. We present several other parameterized testers, providing compelling evidence that expressing the query complexity of property testers in terms of the input size is not always the best choice.

We give a unateness tester for functions of the form f : [n] d → R, where n, d ∈ N and R ⊆ R with query complexity O( d log(max(d,n)) ). Previously known unateness testers work only for Boolean functions over the domain {0, 1} d . We show that every unateness tester for realvalued functions over hypergrid has query complexity Ω(min{d, |R| 2 }). Consequently, our tester is nearly optimal for real-valued functions over {0, 1} d . We also prove that every nonadaptive, 1-sided error unateness tester for Boolean functions needs Ω( √ d/ ) queries. Previously, no lower bounds for testing unateness were known.

We design a nonadaptive algorithm that, given oracle access to a function f:{0,1} n→{0,1} which is α‐far from monotone, makes poly(n,1/α) queries and returns an estimate that, with high probability, is an Õ(n)‐approximation to the distance of f to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly(n,1/α)‐query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant κ>0, every nonadaptive n1/2−κ‐approximation algorithm for this problem requires 2nκ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure‐resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a k‐junta.