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That's not what O(1) means. If your strings are short, complexity doesn't matter either way.

if all your strings are of size < c, you can do any string operation on a single string in constant time.

There are certain (randomized) operations that you can’t…

That's still misleading.

If all my input is small then O(n^2) is basically O(1) as well.

Everything is O(1) when you have finite bounded input.