I'm designing a tree data structure to store strings in. One classic solution is prefix tree, but I am looking for a solution that the time to check if the string is in the storage is O(logm*logn) where m is the length of the string and n is the total number of character in Alphabet(if the string can be lower case aliphatic a-z only, n is 26).
I checked this post Data structure for storing strings, but the solution of prefix tree will take O(m) to insert.
Can I get some idea about the data structure with O(logm * logn) traverse and O(n * logm) insert time? Any general design ideas will be appreciated!