Sample-Based Distance-Approximation for Subsequence-Freeness

In this work, we study the problem of approximating the distance to subsequence-freeness in the sample-based distribution-free model. For a given subsequence (word) w = w1 . . . w_k, a sequence (text) T = t1 . . . tn is said to contain w if there exist indices 1 ≤ i1 < · · · < i_k ≤ n such that ti_j = wj for every 1 ≤ j ≤ k. Otherwise, T is w-free. Ron and Rosin (ACM TOCT 2022) showed that the number of samples both necessary and sufficient for one-sided error testing of subsequence-freeness in the sample-based distribution-free model is Θ(k/ϵ). Denoting by ∆(T, w, p) the distance of T to w-freeness under a distribution p : [n] → [0, 1], we are interested in obtaining an estimate ∆^b , such that |∆^b − ∆(T, w, p)| ≤ δ with probability at least 2/3, for a given distance parameter δ. Our main result is an algorithm whose sample complexity is Õ(k²/δ²). We first present an algorithm that works when the underlying distribution p is uniform, and then show how it can be modified to work for any (unknown) distribution p. We also show that a quadratic dependence on 1/δ is necessary.