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) \(w = w_1 \ldots w_k\), a sequence (text) \(T = t_1 \ldots t_n\) is said to contain w if there exist indices \(1 \le i_1< \cdots < i_k \le n\) such that \(t_{i_{j}} = w_j\) for every \(1 \le j \le k\). Otherwise, T is w-free. Ron and Rosin (ACM Trans Comput Theory 14(4):1–31, 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 \(\Theta (k/\epsilon )\). Denoting by \(\Delta (T,w,p)\) the distance of T to w-freeness under a distribution \(p:[n]\rightarrow [0,1]\), we are interested in obtaining an estimate \(\widehat{\Delta }\), such that \(|\widehat{\Delta }- \Delta (T,w,p)| \le \delta \) with probability at least 2/3, for a given error parameter \(\delta \). Our main result is a sample-based distribution-free algorithm whose sample complexity is \(\tilde{O}(k^2/\delta ^2)\). 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 \(1/\delta \) is necessary.