#3022

Hard

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Minimize OR of Remaining Elements Using Operations

Minimize the bitwise OR of the remaining elements of an array after applying at most k operations.

Problem Statement

You are given a 0-indexed integer array nums and an integer k. In each operation, you can pick an index i (0 <= i < nums.length - 1) and replace nums[i] and nums[i + 1] with the result of nums[i] & nums[i + 1], where '&' represents the bitwise AND operator.

Your task is to return the minimum possible value of the bitwise OR of the remaining elements after performing at most k operations.

Examples

Example 1

Input: nums = [3,5,3,2,7], k = 2

Output: 3

Let's do the following operations:

Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [1,3,2,7].
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

Let's do the following operations:

Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,15,14,2,8].
Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [3,14,2,8].
Replace nums[0] and nums[1] with (nums[0] & nums[1]) so that nums becomes equal to [2,2,8].
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

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

Solution Approach

Greedy Approach with Bitwise Manipulation

Start by analyzing the array from the most significant bit to the least significant bit. Maintain a variable to track bits that will not contribute to the final result. Apply operations to reduce the number of non-contributing bits, ensuring that the OR of the remaining elements is minimized after applying the allowed k operations.

Minimize OR by Grouping Elements

Use the bitwise AND operation on adjacent elements to group them together. By performing the operation on pairs of elements that result in smaller OR values, you can strategically reduce the bits that would increase the final OR value.

Track State of Bits

Keep track of which bits are still active in the array after each operation. By selectively choosing which pairs of elements to combine based on their bitwise AND results, you can ensure that the final OR value is minimized after k operations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the approach chosen to track and apply the bitwise AND operations. The greedy approach typically requires iterating through the array, and the space complexity depends on the number of bits being tracked, which is constant in most cases (as the maximum value is under 2^30). Thus, the time complexity is O(n) in the best case, with space complexity being O(1).

What Interviewers Usually Probe

Candidate demonstrates strong understanding of bitwise operations.
Candidate utilizes a greedy approach to minimize OR effectively.
Candidate maintains a correct invariant throughout the problem.

Common Pitfalls or Variants

Common pitfalls

Forgetting to prioritize the most significant bits first.
Making unnecessary operations that don't reduce the OR value.
Confusing the bitwise OR operation with AND when deciding the next pair to merge.

Follow-up variants

What if the operations were allowed on non-adjacent elements?
How would this problem change if the goal was to maximize the OR instead?
What if there was a constraint on which elements could be combined (e.g., only adjacent even or odd elements)?

FAQ

What is the optimal way to minimize OR in this problem?

The optimal approach involves using a greedy strategy, focusing on reducing the number of non-contributing bits and applying bitwise AND operations on adjacent pairs to minimize the final OR value.

How does the greedy approach apply to the bitwise OR problem?

The greedy approach works by choosing the most beneficial operations that reduce the active bits, starting with the most significant bits, to minimize the OR of the remaining elements.

What role do bitwise operations play in this problem?

Bitwise operations, specifically AND and OR, are used to manipulate the array's elements by combining adjacent elements and reducing the number of bits that contribute to the final OR value.

How does GhostInterview help with this bitwise OR problem?

GhostInterview provides a detailed step-by-step approach to solving bitwise OR problems using greedy strategies, including helpful hints to track critical bits and ensure optimal operations.

What are the most common mistakes to avoid in this problem?

Common mistakes include failing to prioritize the most significant bits, making unnecessary operations, or incorrectly applying bitwise AND instead of OR when choosing operations.

terminal

Solution

Solution 1

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class Solution:
    def minOrAfterOperations(self, nums: List[int], k: int) -> int:
        ans = 0
        rans = 0
        for i in range(29, -1, -1):
            test = ans + (1 << i)
            cnt = 0
            val = 0
            for num in nums:
                if val == 0:
                    val = test & num
                else:
                    val &= test & num
                if val:
                    cnt += 1
            if cnt > k:
                rans += 1 << i
            else:
                ans += 1 << i
        return rans

Continue Practicing

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