Sensitivity of string compressors and repetitiveness measures

07/19/2021 ∙ by Tooru Akagi, et al. ∙ 0

The sensitivity of a string compression algorithm C asks how much the output size C(T) for an input string T can increase when a single character edit operation is performed on T. This notion enables one to measure the robustness of compression algorithms in terms of errors and/or dynamic changes occurring in the input string. In this paper, we analyze the worst-case multiplicative sensitivity of string compression algorithms, defined by max_T ∈Σ^n{C(T')/C(T) : ed(T, T') = 1}, where ed(T, T') denotes the edit distance between T and T'. For the most common versions of the Lempel-Ziv 77 compressors, we prove that the worst-case multiplicative sensitivity is only a small constant (2 or 3, depending on the version of the Lempel-Ziv 77 and the edit operation type). We strengthen our upper bound results by presenting matching lower bounds on the worst-case sensitivity for all these major versions of the Lempel-Ziv 77 factorizations. This contrasts with the previously known related results such that the size z_ 78 of the Lempel-Ziv 78 factorization can increase by a factor of Ω(n^3/4) [Lagarde and Perifel, 2018], and the number r of runs in the Burrows-Wheeler transform can increase by a factor of Ω(log n) [Giuliani et al., 2021] when a character is prepended to an input string of length n. We also study the worst-case sensitivity of several grammar compression algorithms including Bisection, AVL-grammar, GCIS, and CDAWG. Further, we extend the notion of the worst-case sensitivity to string repetitiveness measures such as the smallest string attractor size γ and the substring complexity δ, and present matching upper and lower bounds of the worst-case multiplicative sensitivity for γ and δ.

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