Find three numbers appeared only once

Question

In a sequence of length n, where n=2k+3, that is there are k unique numbers appeared twice and three numbers appeared only once.

The question is: how to fin

Answer 1

The problem gets harder and harder as you add more unique values, mainly because you can choose A,B,C such that A xor B xor C = 0. It gets harder and harder to detect if a subset of the values has the same checksum because it contains all the unique values, or because it omitted values which happened to xor to 0.

You can do 3 values in constant space and O(n*k) time, where k is the number of bits in the largest integer. (So O(n) time for your typical case: 32-bit integers.)

It would be interesting to find out if the time bound becomes non-linear in N as the number of unique values increases and you continue to require constant space.

//Special check for 0, because otherwise we don't know A xor B xor C != A xor B
if items unique-contains 0 then
    return 0 ++ SubProblem2Unique(items - 0)
//Compute A xor B xor C
val x = fold xor items
//Try to find a split which separates A and B from C.
for i in 0..WORD_SIZE
    //see if the checksum splits
    val x1 = fold xor [e in items where e & (1<