Simple routine to find elements of an input array that add up to zero. So, 5 and -5
would be found as duplicate elements.
The algorithm goes through each element in the input array and breaks it up into two separate arrays, one to hold the positive elements, and one to hold the negative elements. When inserting new elements into each array, these two arrays are kept sorted by using a binary search to find the correct position. Since there are n elements in the input array and each insert operation uses a binary search that will take log(n) time. Creating these two subarrays will take O(nlogn) time. If we are lucky and there are approximately equal numbers of positive and negative elements in the input array, we could get that this part of the algorithm only takes O(n log(n/2)) time.
Once the two subarrays are built, we just march through them seeing if the elements in each subarray at the current index add up to zero. This part of the algorithm adds another O(n) time to the algorithm.
Note that we find pairs of zeroes in the first pass of the input array. Every time that we find a zero we just toggle a flag between 0 and 1, and if the flag is 1 when we find a zero, we announce that we have found a pair.