You don't write O(N) class-specific implementations. You write only one function implementation.
You only have to write N class specific implementations if each class shows a different behaviour and thus requires a different implementation. If one function code can handle N different classes that share 1 behaviour, you encode this behavioural information in the class hierarchies: by inheritance/delegation, you inform the type system that the N classes share a common behaviour. You write the code in one class, and have the N classes either inherit from that class or delegate onto that class (inheritance vs delegation is orthogonal to this discussion).
You only have to write N class specific implementations if each class shows a different behaviour and thus requires a different implementation. If one function code can handle N different classes that share 1 behaviour, you encode this behavioural information in the class hierarchies: by inheritance/delegation, you inform the type system that the N classes share a common behaviour. You write the code in one class, and have the N classes either inherit from that class or delegate onto that class (inheritance vs delegation is orthogonal to this discussion).