Sample-based distance-approximation for subsequence-freeness
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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 … w_k, a sequence (text) T = t_1 … t_n is said to contain w if there exist indices 1 ≤ i_1 < … < i_k ≤ n such that t_i_j = w_j 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 Δ(T,w,p) the distance of T to w-freeness under a distribution p :[n]→ [0,1], we are interested in obtaining an estimate Δ, such that |Δ - Δ(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(k^2/δ^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/δ is necessary.
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