基于贪心的文法压缩的非恒定下界
A Non-constant Lower Bound for Grammar-Based Compression with Greedy
AI总结：
本文证明了贪心文法压缩算法Greedy的近似比具有非恒定下界，解决了二十余年的开放问题，并在Lean 4中形式化验证。
AI中文摘要：
我们证明了全局文法压缩算法Greedy的近似比的下界为{\Omega}(log n/ log log n)。据我们所知，此前最好的下界是常数，而非恒定下界的存在性已悬而未决二十余年。我们的下界在无限族长度为n的单词上成立，这些单词的字母表规模不断增长，且适用于使用从左到右出现替换和任意平局打破的每次执行。该下界也在Lean 4中得到了形式化验证。
英文摘要：
We prove a lower bound of Ω(log n/ log log n) on the approximation ratio of the global grammar-based compression algorithm Greedy. To our knowledge, the previously best lower bound was a constant, and the existence of a nonconstant lower bound had remained open for more than twenty years. Our bound holds on an infinite family of words of length n, over alphabets of growing size, for every execution using left-to-right occurrence replacement and arbitrary tie-breaking. The lower bound is also formally verified in Lean 4.
