Problem 14: Two missing numbers

def find_two_missing(numbers: list[int], n: int) -> set[int]: """ Finds two missing numbers from a list of integers from 0 to n-1 using Bit Manipulation. 1. XOR all numbers from 0 to n-1. 2. XOR all numbers present in the input list. 3. The result is x ^ y. 4. Find a set bit in x ^ y. 5. Partition the numbers into two groups based on this bit. 6. XOR the groups to find x and y. """ xor_all = 0 # XOR all numbers from 0 to n-1 for i in range(n): xor_all ^= i # XOR with numbers present in the list for num in numbers: xor_all ^= num # At this point, xor_all = x ^ y # Find the rightmost set bit (a differing bit between x and y) # Using x & -x to isolate the lowest set bit set_bit = xor_all & -xor_all missing_a = 0 missing_b = 0 # Iterate through the range and list to partition for i in range(n): if i & set_bit: missing_a ^= i else: missing_b ^= i for num in numbers: if num & set_bit: missing_a ^= num else: missing_b ^= num return {missing_a, missing_b}