Find two integers A and B such that A ^ N = A + N and B ^ N = B + N
Given a non-negative integer N, the task is to find two integers A (greatest integer smaller than N) and (smallest integer greater than N) such that A + N = A ^ N and B + N = B ^ N
Input: N = 5
Output: A = 2 and B = 8
2 + 8 = 2 ^ 8 = 10
Input: N = 10
Output: A = 5 and B = 16
5 + 16 = 5 ^ 16 = 21
Approach: Lets find A and B independently. To solve this problem we have to use the property, x + y = x^y + 2 * (x & y)
Since the problem states that xor sum is equal to the given sum which implies that their AND must be 0.
- Finding A: N can be represented as a series of bits of 0 and 1. To find A we will first have to find the most significant bit of N which is set. Since A & N = 0, The places where N has set bit, for that places we will make bits of A as unset and for the places where N has unset bit, we will make that bit set for A as we want to maximize A. This we will do for all the bits from most significant to the least significant. Hence we will get our A.
- Finding B: Finding B is easy. Let i be the position of the leftmost set bit in 1. We want B to be greater than N, also we want B & N =0. Hence using these two facts B will be always (1<< (i+1)).
Below is the implementation of the above approach:
A = 2 and B = 8
Time Complexity: O(MAX)
Auxiliary Space: O(N)