Minimum Array End

Problem Description

You are given two integers n and x. You have to construct an array of positive integers nums of size n where for every 0 <= i < n - 1, nums[i + 1] is greater than nums[i], and the result of the bitwise AND operation between all elements of nums is x.

Return the minimum possible value of nums[n - 1].

Key Insights

• The last element in the array must be greater than all previous elements.
• The bitwise AND operation will yield x only if all elements in the array have the bits set that are present in x.
• To ensure the elements are strictly increasing and maintain the required AND condition, a systematic approach can be used to find the minimum last element.

Space and Time Complexity

Time Complexity: O(1) - The solution involves constant time calculations based on the input values. Space Complexity: O(1) - No additional space is used that scales with input size.

Solution

To solve the problem, we need to ensure that the array nums constructed meets two conditions:

1. Each element must be greater than the previous one.
2. The bitwise AND of all elements must equal x.

We can start by initializing the first element as x itself, and for the remaining n-1 elements, we need to find the smallest integers that maintain strict growth and still satisfy the AND condition.

1. The first element can be set to x.
2. Each subsequent element can be chosen as x with additional bits set to ensure it is greater than the previous element, usually by setting bits to 1 that are 0 in x.
3. The final element will be x ORed with the smallest integer that ensures the array grows strictly.

Code Solutions