Finding an element in an array where every element is repeated odd number of times (but more than single occurrence) and only one appears once

Question

You have an array in which every number is repeated odd number of times (but more than single occurrence). Exactly one number appears once. How do you find the number that a

Answer 1

For the problem as stated it is most likely that the most efficient answer is the O(n) space answer. On the other hand, if we narrow the problem to be "All numbers appear n times except for one which only appears once" or even "All numbers appear a multiple of n times except for one which only appears once" then there's a fairly straightforward solution for any n (greater than 1, obviously) which takes only O(1) space, which is to break each number into bits and then count how many times each bit is turned on and take that modulo n. If the result is 1, then it should be turned on in the answer. If it is 0, then it should be turned off. (Any other answer shows that the parameters of the problem did not hold). If we examine the situation where n is 2, we can see that using XOR does exactly this (bitwise addition modulo 2). We're just generalizing things to do bitwise addition modulo n for other values of n.

This, by the way, is what the other answer for n=3 does, it's actually just a complex way of doing bit-wise addition where it stores a 2-bit count for each bit. The int called "ones" contains the ones bit of the count and the int called "twos" contains the twos bit of the count. The int not_threes is used to set both bits back to zero when the count reaches 3, thus counting the bits modulo 3 rather than normally (which would be modulo 4 since the bits would wrap around). The easiest way to understand his code is as a 2-bit accumulator with an extra part to make it work modulo 3.

So, for the case of all numbers appearing a multiple of 3 times except the one unique number, we can write the following code for 32 bit integers:

int findUnique(int A[], int size) {
  // First we set up a bit vector and initialize it to 0.
  int count[32];
  for (int j=0;j<32;j++) {
    count[j] = 0;
  }

  // Then we go through each number in the list.
  for (int i=0;i>= 1;
    }
  }

  // Then we just have to reassemble the answer by putting together any
  // bits which didn't appear a multiple of 3 times.
  int answer = 0;
  for (int j=31;j>=0;j--) {
    answer <<= 1;
    if (count[j] == 1) {
      answer |= 1;
    }
  }

  return answer;
}

This code is slightly longer than the other answer (and superficially looks more complex due to the additional loops, but they're each constant time), but is hopefully easier to understand. Obviously, we could decrease the memory space by packing the bits more densely since we never use more than two of them for any number in count. But I haven't bothered to do that since it has no effect on the asymptotic complexity.

If we wish to change the parameters of the problem so that instead the numbers are repeated 5 times, we just change the 3s to 5s. Or we can do likewise for 7, 11, 137, 727, or any other number (including even numbers). But instead of using the actual number, we can use any factor of it, so for 9, we could just leave it as 3, and for even numbers we can just use 2 (and hence just use xor).

However, there is no general bit-counting based solution for the original problem where a number can be repeated any odd number of times. This is because even if we count the bits exactly without using modulo, when we look at a particular bit, we simply can't know whether the 9 times it appears represents 3 + 3 + 3 or 1 + 3 + 5. If it was turned on in three different numbers which each appeared three times, then it should be turned off in our answer. If it was turned on in a number which appeared once, a number which appeared three times, and a number which appeared five times, then it should be turned on in our answer. But with just the count of the bits, it's impossible for us to know this.

This is why the other answer doesn't generalize and the clever idea to handle the special cases is not going to materialize: any scheme based on looking at things bit by bit to figure out which bits should be turned on or off does not generalize. Given this, I don't think that any scheme which takes space O(1) works for the general case. It is possible that there are clever schemes which use O(lg n) space or so forth, but I would doubt it. I think that the O(n) space approach is probably the best which can be done in the problem as proposed. I can't prove it, but at this point, it's what my gut tells me and I hope that I've at least convinced you that small tweaks to the "even number" technique are not going to cut it.