Falsehoods Programmers Believe
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Those Chinese customers all had 2 letters last name (mostly 'Xi' I think) but when I developed the software I thought any last name would be at least 3 letters long and to avoid taxing the database, I didn't start autocompletion before 3 letters where entered in the textbox.
Now the search starts at 2 letters, but I fully expect to find a one letter last name customer before retirement ;)
Personally, I wouldn't risk it and would just make single character lookups reasonably fast. I doubt the number of relevant records exceed 100k, so the savings shouldn't be significantly enough to bother with. However, I've seen worse things done with poorer justifications.
The system is already quite slow as is (the DB is distant and on an underpowered server) so I don't want to tax it even more. What I may do next change is to cache locally all names at the start of the client and hit the DB only when it didn't find the name in the cache.