Maximize the Bitwise OR Within a Doubling Budget

# Maximize the Bitwise OR Within a Doubling Budget You are given a zero-indexed array `nums` of positive integers and an integer `k`. One operation chooses one array element and multiplies it by `2`. You may choose the same element more than once. Using at most `k` operations in total across the array, maximize the bitwise OR of all resulting elements. Return that maximum value. The symbol `|` denotes bitwise OR. ### Examples **Example 1** ```text Input: nums = [5,3], k = 2 Output: 23 ``` Doubling `5` twice gives `[20,3]`, whose OR is `23`. For comparison, doubling each element once gives `[10,6]`, whose OR is `14`. **Example 2** ```text Input: nums = [6,5,1], k = 1 Output: 15 ``` Doubling the middle element gives `[6,10,1]` and an OR of `15`. Doubling the first element instead gives an OR of `13`; doubling the last gives `7`. ### Constraints - `1 <= nums.size() <= 100000`. - `1 <= nums[i] <= 1000000000`. - `1 <= k <= 15`. ### Clarifying Questions - Is `k` a separate allowance for each element? No; it is one global operation budget. - Must different operations choose different positions? No; one position may be chosen repeatedly. - Can the result exceed a 32-bit signed integer? Yes; preserve the `long long` return type. - Is the modified array returned? No; only its maximum attainable OR is required. ### C++ Interface ```cpp class Solution { public: long long maximumOr(vector<int>& nums, int k) { } }; ``` ### Reasoning Checkpoints - Respect the global budget and the phrase “at most.” - Compare alternatives by the numerical value of their resulting OR. - Account for arithmetic width before carrying out any doubling or shift. ```hint Compare significant bits How does the highest set bit constrain the result of an operation allocation? ```