This feels really weird to consider in terms of "O(..)" notation as presented to be honest. Once you get into the trie and other lookups the "n" vs "w" is not something you can ignore. The real effects of cache usage and implementation details render it true-but-meaningless as written.

I don't think the function "inDict" is even defined as either local or remote query (unless I missed it), which really influences the answers. Querying a short string against in-memory structure will be based on "n", but querying a long string against any remote database will be based on "w".