#2997

Medium

LeetCode Problem Workspace

Minimum Number of Operations to Make Array XOR Equal to K

Find the minimum number of operations to make the XOR of an array equal to a given value k by modifying array elements.

Problem Statement

You are given a 0-indexed integer array nums and a positive integer k. Your goal is to determine the minimum number of operations needed to make the bitwise XOR of all elements of the array equal to k. In each operation, you can flip any bit of any element in the array.

The key observation is that we need to compare the current XOR of the array with k and apply bit-flipping operations to achieve the desired result. The challenge lies in determining the minimum operations, especially when dealing with large arrays and numbers.

Examples

Example 1

Input: nums = [2,1,3,4], k = 1

Output: 2

We can do the following operations:

Choose element 2 which is 3 == (011)2, we flip the first bit and we obtain (010)2 == 2. nums becomes [2,1,2,4].
Choose element 0 which is 2 == (010)2, we flip the third bit and we obtain (110)2 = 6. nums becomes [6,1,2,4]. The XOR of elements of the final array is (6 XOR 1 XOR 2 XOR 4) == 1 == k. It can be shown that we cannot make the XOR equal to k in less than 2 operations.

Example 2

Input: nums = [2,0,2,0], k = 0

Output: 0

The XOR of elements of the array is (2 XOR 0 XOR 2 XOR 0) == 0 == k. So no operation is needed.

Constraints

1 <= nums.length <= 105
0 <= nums[i] <= 106
0 <= k <= 106

Solution Approach

Initial XOR Calculation

Start by calculating the XOR of all elements in the array. This gives the initial XOR value which needs to be adjusted to match k.

Identify Bit Differences

Next, identify which bits of the current XOR result differ from the target k. These bit positions will guide the necessary flips on array elements.

Apply Minimum Flips

Finally, flip the required bits on the array elements and count the number of operations. The goal is to make the XOR of the final array equal to k in the minimum number of moves.

Complexity Analysis

Metric	Value
Time	O(N)
Space	O(1)

The time complexity is O(N) due to the need to process each element of the array, where N is the number of elements. The space complexity is O(1), as we only need a few variables to track the current XOR and target k, without using extra space proportional to input size.

What Interviewers Usually Probe

Ensure the candidate recognizes the importance of bitwise operations in solving the problem efficiently.
Watch for understanding of XOR properties and how they relate to the problem's requirements.
Look for a clear plan to minimize the number of operations by flipping only necessary bits.

Common Pitfalls or Variants

Common pitfalls

Failing to properly calculate the initial XOR of the array, leading to incorrect results.
Misunderstanding the bit manipulation, leading to unnecessary or excessive operations.
Not considering the constraints on the size of the array and the values of the elements, potentially missing optimizations.

Follow-up variants

A variant could involve applying bit flips to certain subarrays, not the entire array.
Another variant could restrict the number of times each element can be modified.
A harder variant could involve multiple target XOR values instead of just one k.

FAQ

What is the minimum number of operations to make array XOR equal to k?

The minimum number of operations is determined by comparing the initial XOR of the array to the target value k and flipping the necessary bits of array elements.

How does XOR help in solving this problem?

XOR helps by providing a way to compare the current state of the array with the target value k, allowing for efficient bit manipulation to reach the desired result.

What is the time complexity of solving this problem?

The time complexity is O(N), where N is the number of elements in the array, since each element needs to be processed once.

Can this problem be solved without using bit manipulation?

No, bit manipulation is essential to efficiently solve this problem, as the core operation involves flipping specific bits in the array elements.

What are the constraints on the problem?

The array length must be between 1 and 100,000 elements, and each element and the target k are non-negative integers up to 1,000,000.

terminal

Solution

Solution 1: Bit Manipulation

We can perform a bitwise XOR operation on all elements in the array $nums$. The number of bits that differ from the binary representation of $k$ in the result is the minimum number of operations.

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class Solution:
    def minOperations(self, nums: List[int], k: int) -> int:
        return reduce(xor, nums, k).bit_count()

Continue Practicing

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