发表机构
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了LZ分解对四类结构改变型操作的乘法敏感性问题，构造了敏感性为Ω(log n)的字符串族，还确定了LZ分解与拼贴系统、词法解析的规模关系，给出了渐近紧的下界与上界。
AI 中文摘要
莱姆尔-齐夫（Lempel-Ziv, LZ）分解是压缩高度重复字符串的最基础方法之一，其分解得到的短语数量被视为重复度的度量。对编辑操作的敏感性是指，当对一个字符串应用该操作时，重复度度量的最大增量。虽然已有渐近紧界适用于LZ分解对单字符编辑的敏感性，但对于改变字符串结构大部分的操作（如前缀删除、子串删除、循环旋转和字符串反转），其乘法敏感性是否受常数约束仍是未解决的问题。我们解决了该问题：针对上述四种操作，我们构造了长度为n的字符串族，其对对应操作的敏感性为Ω(log n)。我们还确定了LZ分解、拼贴系统（collage systems）和词法解析（lex-parse）之间的规模关系：构造了一类字符串，其LZ分解的规模是最小拼贴系统的Ω(log n)倍；还构造了另一类字符串，其词法解析的规模是LZ分解的Ω(log n)倍。所有这些下界均为渐近紧界，与O(log n)的上界匹配。
英文摘要
The Lempel-Ziv (LZ) factorization is a fundamental method for compressing repetitive strings. Its multiplicative sensitivity to an operation measures how much that operation can increase the number of phrases, as the maximum ratio of the phrase counts after and before the operation. While its sensitivity to single-character edits is known, its sensitivity to more global operations such as prefix deletion and string reversal remains unknown. Further open questions concern its size relationships with other compressed representations and whether it can be converted into an LZ-like encoding of polylogarithmic height without asymptotically increasing its size. In this paper, We determine the multiplicative sensitivity of the LZ factorization to prefix deletion, cyclic rotation, and string reversal by establishing an $Ω(\log n)$ lower bound for each operation that matches the known upper bound, where $n$ is the input length. Using the result for prefix deletion, we show that the combined size of the relative LZ factorization and the LZ encoding of its reference string can be asymptotically smaller than that of the LZ factorization. We also obtain asymptotically tight worst-case ratios of $Θ(\log n)$ for the size of the LZ factorization divided by the minimum size of a collage system and for the size of the lex-parse divided by that of the LZ factorization. We give lower and upper bounds on the worst-case ratio of the minimum size of an LZ-like encoding of height at most $h$ to the size of the LZ factorization, with matching bounds for $h \geq \lg n$. In particular, we show that the minimum size of an LZ-like encoding with height at most $h \in O({\rm poly}\,\log n)$ can be larger than the size of the LZ factorization by a factor of $Ω(\log n / \log\log n)$. All our lower bounds follow from a common string construction based on the bit-reversal permutation.
