发表机构
Nanyang Technological University; University of Technology Sydney（南洋理工大学; 悉尼科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究解决同步字符串的三元情况，证明3是最小常数字母表大小，通过局部熵转移定理等技术构造三元同步字符串，还得到指数级多有效单词等延伸成果。
AI 中文摘要
同步字符串是为流附加确定性位置标签，且可在插入和删除操作中保持有效。Haeupler与Shahrasbi引入了这类对象，后续研究证明，对于固定参数ε<1，4个符号足够，而2个符号无法支持任意长度的同步字符串。我们解决了剩余的三元情况：任意长度都存在三元2001/2002-同步字符串，因此3是最小的常数字母表大小。基于更大的54-均匀族，计算机辅助的改进方案可对任意ε>226/227生成三元ε-同步字符串。此前的四符号构造使用三元无平方主干排除短重复，第四个符号用于承载长程同步标记。我们的主要技术贡献是局部熵转移定理：每个具有正条件最小熵的无平方块局部源，都支持具有固定间隙的同步字符串。我们通过Brinkhuis族中按出现位置的分支实例化该定理，所有结果仍为三元且无平方，且在暴露某长区间外的所有选择后，该长区间仍保留线性条件最小熵。删除球估计将此熵转化为高最长公共子序列的反集中性，非对称Lovasz局部引理同时强制执行所有区间约束。该框架还产生指数级多的有效单词、拉斯维加斯多项式时间构造、同步圆，以及一类极端无平方单词内的同步。
英文摘要
Synchronization strings provide deterministic position labels for recovering coordinates after insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 215/216. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with a positive interval conditional min-entropy rate supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into an exponentially small probability of a near-complete common subsequence between adjacent intervals, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, synchronization circles, and synchronization within a class of extremal square-free words. Adding constraints on distant intervals gives a Las Vegas construction in expected O(n^2 log^3(n+2)) time.
Comments46 pages
