Minimum Impossible OR

Problem Description

You are given a 0-indexed integer array nums. We say that an integer x is expressible from nums if there exist some integers 0 <= index1 < index2 < ... < indexk < nums.length for which nums[index1] | nums[index2] | ... | nums[indexk] = x. In other words, an integer is expressible if it can be written as the bitwise OR of some subsequence of nums. Return the minimum positive non-zero integer that is not expressible from nums.

Key Insights

• The bitwise OR operation accumulates bits, meaning that any number formed by a combination of elements in nums will have bits set that are also present in nums.
• The minimum positive integer that is not expressible will typically be found by checking each integer starting from 1 and seeing if it can be represented by the OR of any subset of nums.
• The algorithm must efficiently determine the smallest non-expressible integer without generating all possible subsets of nums.

Space and Time Complexity

Time Complexity: O(n log(max(nums))), where n is the number of elements in nums and max(nums) is the maximum element in nums.
Space Complexity: O(1), as we only use a few additional variables.

Solution

The solution involves using a set to track all expressible integers. We iteratively check each integer starting from 1 to see if it can be formed using the bitwise OR of elements in nums. To efficiently determine expressibility, we can use a greedy approach that accumulates the OR results and checks for gaps in the expressible integers.

Code Solutions