We study the sensitivity of the Lempel-Ziv 77 compression algorithm to edits, showing how modifying a string $w$ can deteriorate or improve its compression. Our first result is a tight upper bound for $k$ edits: $\forall w' \in B(w,k)$, we have $C_{\mathrm{LZ77}}(w') \leq 3 \cdot C_{\mathrm{LZ77}}(w) + 4k$. This result contrasts with Lempel-Ziv 78, where a single edit can significantly deteriorate compressibility, a phenomenon known as a *one-bit catastrophe*. We further refine this bound, focusing on the coefficient $3$ in front of $C_{\mathrm{LZ77}}(w)$, and establish a surprising trichotomy based on the compressibility of $w$. More precisely we prove the following bounds: - if $C_{\mathrm{LZ77}}(w) \lesssim k^{3/2}\sqrt{n}$, the compression may increase by up to a factor of $\approx 3$, - if $k^{3/2}\sqrt{n} \lesssim C_{\mathrm{LZ77}}(w) \lesssim k^{1/3}n^{2/3}$, this factor is at most $\approx 2$, - if $C_{\mathrm{LZ77}}(w) \gtrsim k^{1/3}n^{2/3}$, the factor is at most $\approx 1$. Finally, we present an $\varepsilon$-approximation algorithm to pre-edit a word $w$ with a budget of $k$ modifications to improve its compression. In favorable scenarios, this approach yields a total compressed size reduction by up to a factor of~$3$, accounting for both the LZ77 compression of the modified word and the cost of storing the edits, $C_{\mathrm{LZ77}}(w') + k \log |w|$.

翻译：本研究探讨了Lempel-Ziv 77压缩算法对编辑操作的敏感性，揭示了字符串$w$的修改如何影响其压缩效率。我们的首要成果是针对$k$次编辑的紧致上界：对于任意$w' \in B(w,k)$，均有$C_{\mathrm{LZ77}}(w') \leq 3 \cdot C_{\mathrm{LZ77}}(w) + 4k$。该结论与Lempel-Ziv 78算法形成鲜明对比——后者在单次编辑下可能引发压缩性能的显著退化，即所谓的*单比特灾难*。我们进一步细化该上界，聚焦于$C_{\mathrm{LZ77}}(w)$前的系数$3$，并基于$w$的可压缩性建立了令人惊异的三分定理。具体而言我们证明了以下界限：- 若$C_{\mathrm{LZ77}}(w) \lesssim k^{3/2}\sqrt{n}$，压缩量可能增加至多约$3$倍；- 若$k^{3/2}\sqrt{n} \lesssim C_{\mathrm{LZ77}}(w) \lesssim k^{1/3}n^{2/3}$，该系数至多约为$2$；- 若$C_{\mathrm{LZ77}}(w) \gtrsim k^{1/3}n^{2/3}$，则系数至多约为$1$。最后，我们提出一种$\varepsilon$近似算法，可在$k$次修改的预算内对单词$w$进行预编辑以优化其压缩效果。在理想情况下，该方法能使总压缩规模（包含修改后单词的LZ77压缩量$C_{\mathrm{LZ77}}(w')$与编辑操作存储开销$k \log |w|$）降低至多约$3$倍。