Question

An array a[] contains all of the integers from 0 to N, except one. However, you cannot access an element with a single operation. Instead, you can call get(i, k) which returns the kth bit of a[i] or you can call swap(i, j) which swaps the ith and jth elements of a[]. Design a O(N) algorithm to find the missing integer.
(For simplicity, assume N is a power of 2.)

Answer 1

If N is a power of 2, it can be done in O(N) using divide and conquer.

Note that there are logN bits in the numbers. Now, using this information – you can use a combination of partition based selection algorithm and radix-sort.

1. Iterate the numbers for the first bit, and divide the array to two
   halves – the first half has this bit as 0, the other half has it as 1. (Use the swap() for partitioning the array).
2. Note that one half has ceil(N/2) elements, and the other has floor(N/2) elements.
3. Repeat the process for the smaller array, until you find the missing
   number.

The complexity of this approach will be N + N/2 + N/4 + ... + 1 < 2N, so it is O(n)