Sensitivity of string compressors and repetitiveness measures [article]

Tooru Akagi, Mitsuru Funakoshi, Shunsuke Inenaga
2022 arXiv   pre-print
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. We analyze the worst-case multiplicative sensitivity of string compression algorithms, which is 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'. In particular, for the most common versions of the Lempel-Ziv 77 compressors, we prove that the worst-case
more » ... licative sensitivity is only a small constant. We strengthen our upper bound results by presenting matching lower bounds. We also generalize these results to the smallest bidirectional scheme b. These results contrast 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^1/4) [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. In addition, we show that the worst-case multiplicative sensitivity of r is upper bounded by O(log r log n). We also study the worst-case sensitivity of several grammar-based compressors including RePair, LongestMatch, 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. We present some non-trivial upper and lower bounds of the worst-case multiplicative sensitivity for γ and matching upper and lower bounds of the worst-case multiplicative sensitivity for δ. We also exhibit the worst-case additive sensitivity max_T ∈Σ^n{C(T') - C(T) : ed(T, T') = 1}.
arXiv:2107.08615v4 fatcat:qndn2atlzvbghoqjvjgpt77euq