We all know that element distinctness in the comparison based model cannot be done in $o(n\log n)$ time. However, on a word-RAM, one can possibly achieve better.
Of course, if one assumes the existence of a perfect hash function that can be computed in linear time, we get a linear time algorithm for element distinctness: just keep hashing the numbers one by one and return 1 if there is a collision.
However, there are two issues: 1) most constructions of perfect hash functions that I could find used randomness and 2) I cannot find a discussion of the pre-processing time anywhere, i.e., the time required to decide what hash function one is going to use based on the input set of numbers.
Fredman et al.'s "Storing a sparse table with $O(1)$ worst case access time" does resolve the first issue by providing a hash function with $O(1)$ access time in the worst case, but says nothing about the second issue.
So to sum up, here's what I want:
Design an algorithm that given a set $S$ of $n$ numbers (each number being $w$ bits long) on a word-RAM with word length $w$, finds a hash function $h:S\rightarrow \{1, \ldots , m\}$ in $O(n)$ time, where $m = O(n)$. The function $h$ should have the property that for any $j \in \{1, \ldots , m\}$, the number of elements of $S$ that map to $j$ is a constant and computing $h(i)$ should take $O(1)$ time in a "reasonable" word-RAM model, i.e., the model should not allow "exotic" functions on words to be evaluated in $O(1)$ time.
I will also like to know if there are algorithms to solve element distinctness on the word-RAM that do not use hash functions at all.
