What is the big O of a perfect hash function?

Question

Regular hash functions, in which collisions are probable, run in constant time: O(1). But what is the time complexity of a perfect hash function? Is it 1?

Answer 1

If the hash function is intended to be used to access a hash table, then there is no difference in terms of complexity between perfect and regular hash functions, since both of them may also create collisions in the table. The reason is that the index associated to an element in a hash table is the remainder of the division of the hash by the length of the table (usually a prime number). This is why two elements which hash to different values will collide if their remainder modulo the (said) prime is the same for both of them. This means that the time complexity of accessing the table is O(1) in both cases.

Note also that the computation of the hash usually depends on the size of the input. For instance, if the elements to be hashed are strings, good hashes take all their characters into account. Therefore, for the complexity to remain O(1), one has to limit the size (or length) of the inputs. Again, this applies to both perfect and regular hashes.