Minimum Operations to Make XOR of All Elements Zero solution and explanation
The problem
You are given an integer array nums consisting of positive integers.
You may perform the following operation any number of times:
- Choose two distinct indices
iandjsuch thatnums[i] != nums[j], and replace eithernums[i]ornums[j]withnums[i] ^ nums[j], where^denotes the bitwise XOR.
Return the minimum number of operations required to make the bitwise XOR of all elements in nums equal to 0. If it is impossible, return -1.
Example 1:
Input: nums = [8,1,4,8,2]
Output: 3
Explanation:
One optimal sequence of operations is:
- Choose indices 0 and 1, and replace
nums[0]with8 ^ 1 = 9. The array becomes[9, 1, 4, 8, 2]. - Choose indices 2 and 3, and replace
nums[3]with4 ^ 8 = 12. The array becomes[9, 1, 4, 12, 2]. - Choose indices 0 and 4, and replace
nums[0]with9 ^ 2 = 11. The array becomes[11, 1, 4, 12, 2].
The XOR of all elements of nums is 11 ^ 1 ^ 4 ^ 12 ^ 2 = 0, so the answer is 3.
Example 2:
Input: nums = [1,2,3]
Output: 0
Explanation:
The XOR of all elements of nums is 1 ^ 2 ^ 3 = 0, so no operations are required.
Example 3:
Input: nums = [1,2,4]
Output: -1
Explanation:
It is impossible to make the XOR of all elements of nums equal to 0, so the answer is -1.
Constraints:
2 <= nums.length <= 1051 <= nums[i] <= 2000
Approach overview
Problem Overview: Given an array of integers, you can change any element to any value. Find the minimum number of operations required to make the XOR of all elements zero.
Approach 1: XOR and Check (O(n) Time, O(1) Space)
Compute the XOR of every element in the array. If the result is already zero, no operations are needed. If it's non-zero, you can always change exactly one element to the XOR of the remaining elements, which forces the total XOR to zero. This works because XOR is associative and commutative; setting any element to the XOR of the others cancels out the original contribution and leaves the overall XOR at zero.
The key insight is that you never need more than one operation. The problem is a classic bitwise trick — it tests whether you recognize that a single change suffices, rather than attempting to simulate or backtrack. Use a simple loop to accumulate the XOR, then compare to zero. This approach is optimal and works for all cases.
Recommended for interviews: Interviewers expect the O(n) XOR method. It shows you understand bitwise properties and can identify the minimal change. Mentioning the proof (changing one element to the XOR of the rest) demonstrates clarity. The brute force (trying all possible changes) is unnecessary and would be rejected.
For more bit manipulation patterns, see our bit manipulation guide.
The written solution for this problem is still being prepared. Try it yourself in the meantime.
Try solving it yourselfComplexity analysis
| Approach | Time | Space | When to Use |
|---|---|---|---|
| XOR and Check | O(n) | O(1) | Always — optimal and simplest |
Frequently asked questions
Is Minimum Operations to Make XOR of All Elements Zero easy or hard?
The problem is marked Hard, but the solution is surprisingly simple. The difficulty lies in recognizing that at most one operation is ever needed. Once you see the XOR trick, it becomes an Easy-level problem.
Minimum Operations to Make XOR of All Elements Zero Python/Java solution
Python: def min_operations(nums): return 0 if reduce(xor, nums) == 0 else 1. Java: int xor=0; for(int n: nums) xor^=n; return xor==0?0:1. Both run in O(n) time and O(1) space.
How to solve Minimum Operations to Make XOR of All Elements Zero in O(n)?
Iterate through the array and maintain a running XOR. After the loop, check if the result equals zero. If yes, output 0; otherwise, output 1. This is a single pass, O(n) time and O(1) space.
What is the best approach for Minimum Operations to Make XOR of All Elements Zero?
The best approach is to compute the XOR of all elements in O(n) time. If the XOR is zero, return 0; otherwise, return 1. This works because changing any one element to the XOR of the remaining elements makes the total XOR zero.
Is Minimum Operations to Make XOR of All Elements Zero asked at Google/Amazon/Meta?
While not explicitly tagged with company names, this problem is a common interview trick question at top tech companies. It tests fundamental bitwise XOR properties and problem simplification skills, so it may appear in phone screens or onsite interviews.
What data structure is used in Minimum Operations to Make XOR of All Elements Zero?
No special data structure is needed. The solution only requires a single integer variable to accumulate the XOR. The algorithm is purely arithmetic and bitwise.
What is the time complexity of Minimum Operations to Make XOR of All Elements Zero?
The optimal solution runs in O(n) time, where n is the number of elements. Space complexity is O(1) because only a single variable is needed to store the cumulative XOR.