Ramesh Krishnan Pallavoor 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. 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 \Omega(log n) lower bound by Fischer (Inf. Comput., 2004) for this problem. Joint work with Sofya Raskhodnikova and Nithin Varma (TOCT, 2018).