We study the fully dynamic pattern matching problem where the pattern may contain up to kwildcard symbols, each matching any symbol of the alphabet. Both the text and the pattern are subject to updates (insert, delete, change). We design an algorithm with O(nlog^2 n) preprocessing and update/query time O(knk/k+1 + k2 log n). The bound is truly sublinear for a constant k, and sublinear when k= o(log n). We further complement our results with a conditional lower bound: assuming subquadratic preprocessing time, achieving truly sublinear update time for the case k = Ω(log n) would contradict the Strong Exponential Time Hypothesis (SETH). Finally, we develop sublinear algorithms for two special cases: - If the pattern contains w non-wildcard symbols, we give an algorithm with preprocessing time O(nw) and update time O(w + log n), which is truly sublinear whenever wis truly sublinear. - Using FFT technique combined with block decomposition, we design a deterministic truly sublinear algorithm with preprocessing time O(n^1.8) and update time O(n^0.8 log n) for the case that there are at most two non-wildcards.

翻译：我们研究了完全动态模式匹配问题，其中模式可能包含最多k个通配符，每个通配符可匹配字母表中的任意符号。文本和模式均可进行更新操作（插入、删除、修改）。我们设计了一种算法，其预处理时间复杂度为O(nlog² n)，更新/查询时间复杂度为O(kn^{k/(k+1)} + k² log n)。当k为常数时该边界是严格亚线性的，当k=o(log n)时仍保持亚线性。我们进一步通过条件性下界补充了研究结果：在假设预处理时间为亚二次的前提下，若对k=Ω(log n)的情况实现严格亚线性更新时间，将与强指数时间假设（SETH）相矛盾。最后，我们针对两种特殊情况开发了亚线性算法：- 若模式包含w个非通配符符号，我们提出预处理时间为O(nw)、更新时间为O(w + log n)的算法，当w为严格亚线性时该算法亦为严格亚线性。- 结合FFT技术与分块分解方法，我们针对最多包含两个非通配符的情况，设计了确定性严格亚线性算法，其预处理时间为O(n^{1.8})，更新时间为O(n^{0.8} log n)。