Is Minimize OR of Remaining Elements Using Operations easy or hard?

The problem is rated Hard because it requires combining greedy reasoning with bit manipulation and a feasibility check. Understanding that AND operations only remove bits while OR aggregates them is the key insight needed to design the optimal algorithm.

Minimize OR of Remaining Elements Using Operations Python/Java solution

Implement the greedy bit-check approach. Iterate bits from high to low, assume the bit is removable, then scan the array while building AND segments. If more than k merges are required, the bit must be present in the final OR. The same logic works in Python, Java, C++, C#, and JavaScript.

How to solve Minimize OR of Remaining Elements Using Operations in O(n)?

Pure O(n) is not typical because each bit of the result must be validated. However, the practical optimal solution is O(n log M). For each candidate bit, you simulate merges with a single pass through the array and track segment AND values to determine whether the bit can be removed.

What is the best approach for Minimize OR of Remaining Elements Using Operations?

The most effective approach uses greedy bit manipulation. Iterate bits from most significant to least significant and check if that bit can be eliminated from the final OR by merging adjacent elements using AND operations. A linear scan verifies whether the required merges stay within k. This produces the minimum OR value in O(n log M) time where M is the maximum element value.

Is Minimize OR of Remaining Elements Using Operations asked at Google/Amazon/Meta?

Problems combining greedy decisions with bit manipulation are common in interviews at companies like Google, Amazon, and Meta. Variants involving minimizing OR or maximizing AND frequently appear because they test understanding of bitwise properties and greedy feasibility checks.

What data structure is used in Minimize OR of Remaining Elements Using Operations?

The problem mainly relies on arrays and bitwise operations. No complex data structures are required. The algorithm scans the array while maintaining a running AND value and counting merges needed to satisfy the bit constraints.

What is the time complexity of Minimize OR of Remaining Elements Using Operations?

The optimal solution runs in O(n log M) time and O(1) space. The algorithm checks each bit of the result (up to ~30 for typical integer constraints) and performs a linear scan of the array to verify feasibility using greedy AND merging.

Description Hints 7 Solution Video 50:08

3022. Minimize OR of Remaining Elements Using Operations

Hard30.2% acceptance1 company

You are given a 0-indexed integer array nums and an integer k.

In one operation, you can pick any index i of nums such that 0 <= i < nums.length - 1 and replace nums[i] and nums[i + 1] with a single occurrence of nums[i] & nums[i + 1], where & represents the bitwise AND operator.

Return the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.

Example 1:

Input: nums = [3,5,3,2,7], k = 2
Output: 3
Explanation: Let's do the following operations:
1. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [1,3,2,7].
2. Replace nums[2] and nums[3] with (nums[2] & nums[3]) so that nums becomes equal to [1,3,2].
The bitwise-or of the final array is 3.
It can be shown that 3 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.

Example 2:

Input: nums = [7,3,15,14,2,8], k = 4
Output: 2
Explanation: Let's do the following operations:
1. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,15,14,2,8]. 
2. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,14,2,8].
3. Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [2,2,8].
4. Replace nums[1] and nums[2] with (nums[1] & nums[2]) so that nums becomes equal to [2,0].
The bitwise-or of the final array is 2.
It can be shown that 2 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.

Example 3:

Input: nums = [10,7,10,3,9,14,9,4], k = 1
Output: 15
Explanation: Without applying any operations, the bitwise-or of nums is 15.
It can be shown that 15 is the minimum possible value of the bitwise OR of the remaining elements of nums after applying at most k operations.

Constraints:

1 <= nums.length <= 10⁵
0 <= nums[i] < 2³⁰
0 <= k < nums.length

Asked at 1 company

Aon

Topics

Hints open one at a time

Hint 1

From the most significant bit to the least significant bit, maintain the bits that will not be included in the final answer in a variable mask.

Hint 2

For a fixed bit, add it to mask then check if there exists some sequence of k operations such that mask & answer == 0 where answer is the bitwise-or of the remaining elements of nums. If there is no such sequence of operations, remove the current bit from mask. How can we perform this check?

Hint 3

Let x be the bitwise-and of all elements of nums. If x AND mask != 0, there is no sequence of operations that satisfies the condition in the previous hint. This is because even if we perform this operation n - 1 times on the array, we will end up with x as the final element.

Hint 4

Otherwise, there exists at least one such sequence. It is sufficient to check if the number of operations in such a sequence is less than k. Let’s calculate the minimum number of operations in such a sequence.

Hint 5

Iterate over the array from left to right, if nums[i] & mask != 0, apply the operation on index i.

Hint 6

After iterating over all elements, let x be the bitwise-and of all elements of nums. If x == 0, then we have found the minimum number of operations. Otherwise, It can be proven that we need exactly one more operation so that x == 0.

Hint 7

The condition in the second hint is satisfied if and only if the minimum number of operations is less than or equal to k.

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