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, which is defined by $\max_{T \in Σ^n}\{C(T')/C(T) : ed(T, T') = 1\}$, 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 upper bounded by a small constant, and give matching lower bounds. We generalize these results to the smallest bidirectional scheme $b$. In addition, we show that the sensitivity of a grammar-based compressor called GCIS is also a small constant. 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 show that the worst-case sensitivity of $δ$ is also a small constant. These results contrast with the previously known related results such that the size $z_{\rm 78}$ 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 $n$. By applying our sensitivity bounds of $δ$ or the smallest grammar to known results (c.f. [Navarro, 2021]), some non-trivial upper bounds for the sensitivities of important string compressors and repetitiveness measures including $γ$, $r$, LZ-End, RePair, LongestMatch, and AVL-grammar are derived.
LZBE: an LZ-style compressor supporting $O(\log n)$-time random access
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；Department of Informatics, Kyushu University（九州大学信息学系）；Department of Information and Physical Sciences, Osaka University（大阪大学信息与物理科学系）
String Representation Based on Substring Equation Systems
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；M&D Data Science Center, Institute of Integrated Research, Institute of Science Tokyo（东京科学大学综合研究所M&D数据中心）
Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation
Kyushu University（九州大学）；University of Chile（智利大学）；Center for Biotechnology and Bioengineering (CeBiB)（生物技术与生物工程中心）；University of Palermo（巴勒莫大学）；University of Warsaw（华沙大学）
