Alphabet-Dependent String Searching with Wexponential Search Trees

We consider finding a pattern of length \(m\) in a compacted (linear-size) trie storing strings over an alphabet of size \(\sigma \). In static tries, we achieve \(O(m+\lg \lg \sigma )\) deterministic time, whereas in dynamic tries we achieve \(O(m+\frac{\lg ^{2}\lg \sigma }{\lg \lg \lg \sigma })\) deterministic time per query or update. One particular application of the above bounds (static and dynamic) are suffix trees, where we also show how to pre- or append letters in \(O(\lg \lg n + \frac{\lg ^{2}\lg \sigma }{\lg \lg \lg \sigma })\) time. Our main technical contribution is a weighted variant of exponential search trees, which might be of independent interest.