Parameterized Convexity Testing
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In this work, we develop new insights into the fundamental problem of convexity testing of real-valued functions over the domain [n]. Specifically, we present a nonadaptive algorithm that, given inputs ∈ (0,1), s ∈ℕ, and oracle access to a function, -tests convexity in O(log (s)/), where s is an upper bound on the number of distinct discrete derivatives of the function. We also show that this bound is tight. Since s ≤ n, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity O(log n/); our bound is strictly better when s = o(n). The main contribution of our work is to appropriately parameterize the complexity of convexity testing to circumvent the worst-case lower bound (Belovs et al.; SODA 2020) of Ω(log ( n)/) expressed in terms of the input size and obtain a more efficient algorithm.
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