Parameterized Property Testing of Functions

Abstract

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.

Testing local properties of arrays

“…Small alphabets The results in this work are alphabet independent, and in particular, they work for alphabets over any size. An intriguing direction of research is to understand whether one can obtain more efficient general testability results for local properties of multi-dimensional arrays over smaller alphabets; this line of research has been conducted for specific properties of interest, like monotonicity and convexity [4,28]. Note that the one-sided non-adaptive lower bound we prove here can be adapted to yield a |Σ| Ω(1) lower bound for testing local properties over alphabets Σ of size smaller than n d .…”

Isoperimetric inequalities for real‐valued functions with applications to monotonicity testing

“…Monotonicity of functions, first studied in the context of property testing by Goldreich et al [31], is one of the most widely investigated properties in this model [1, 2, 4, 6–9, 13, 14, 16–22, 24, 26, 27, 29, 30, 32–34, 36, 38, 39]. A function is $$$ \varepsilon $$$‐far from monotone if its distance to monotonicity is at least $$$ \varepsilon $$$; otherwise, it is $$$ \varepsilon $$$‐close to monotone.…”

“…Its query complexity has been completely pinned down in terms of $$$ n $$$ and $$$ \varepsilon $$$ by [5, 19, 27, 29]: it is $$$ \Theta \left(\frac{\log \left(\varepsilon n\right)}{\varepsilon}\right) $$$. Pallavoor et al [36, 40] considered the setting when the tester is given an additional parameter $$$ r $$$, the number of distinct elements in the array, and obtained an $$$ O\left(\left(\log r\right)/\varepsilon \right) $$$‐query algorithm. There are also lower bounds for this setting: $$$ \Omega \left(\log r\right) $$$ for nonadaptive algorithms by [14] and $$$ \Omega \left(\frac{\log r}{\mathrm{loglog}r}\right) $$$ for all testers for the case when $$$ r={n}^{1/3} $$$ by [5].…”

Approximating the distance to monotonicity of Boolean functions

“…Testing monotonicity and unateness (first studied in [41]), as well as ‐juntas (first studied in [38]), are among the most widely investigated problems in property testing ([1, 2, 4, 5, 8–10, 13, 15, 16, 19–25, 28, 33, 34, 36, 39, 44, 45, 48, 53, 55, 56] study monotonicity testing, [3, 27–29, 46] study unateness testing, and [11, 12, 17, 26, 30, 60, 61] study ‐junta testing). Nearly all the previous work on these properties is in the standard testing model.…”