LeetCode #2997 — MEDIUM

Minimum Number of Operations to Make Array XOR Equal to K

Move from brute-force thinking to an efficient approach using array strategy.

Solve on LeetCode
The Problem

Problem Statement

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

You can apply the following operation on the array any number of times:

  • Choose any element of the array and flip a bit in its binary representation. Flipping a bit means changing a 0 to 1 or vice versa.

Return the minimum number of operations required to make the bitwise XOR of all elements of the final array equal to k.

Note that you can flip leading zero bits in the binary representation of elements. For example, for the number (101)2 you can flip the fourth bit and obtain (1101)2.

Example 1:

Input: nums = [2,1,3,4], k = 1
Output: 2
Explanation: 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
Explanation: 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
Patterns Used

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: You are given a 0-indexed integer array nums and a positive integer k. You can apply the following operation on the array any number of times: Choose any element of the array and flip a bit in its binary representation. Flipping a bit means changing a 0 to 1 or vice versa. Return the minimum number of operations required to make the bitwise XOR of all elements of the final array equal to k. Note that you can flip leading zero bits in the binary representation of elements. For example, for the number (101)2 you can flip the fourth bit and obtain (1101)2.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Bit Manipulation

Example 1

[2,1,3,4]
1

Example 2

[2,0,2,0]
0

Related Problems

  • Minimum Bit Flips to Convert Number (minimum-bit-flips-to-convert-number)
Step 02

Core Insight

What unlocks the optimal approach

  • Calculate the bitwise <code>XOR</code> of all elements of the original array and compare it to <code>k</code> in their binary representation.
  • For each different bit between the bitwise <code>XOR</code> of elements of the original array and <code>k</code> we have to flip <strong>exactly</strong> one bit of an element in <code>nums</code> to make that bit equal.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. Continue until all input is consumed.
Use the first example testcase as your mental trace to verify each transition.
Step 04

Edge Cases

Minimum Input
Single element / shortest valid input
Validate boundary behavior before entering the main loop or recursion.
Duplicates & Repeats
Repeated values / repeated states
Decide whether duplicates should be merged, skipped, or counted explicitly.
Extreme Constraints
Upper-end input sizes
Re-check complexity target against constraints to avoid time-limit issues.
Invalid / Corner Shape
Empty collections, zeros, or disconnected structures
Handle special-case structure before the core algorithm path.
Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2997: Minimum Number of Operations to Make Array XOR Equal to K
class Solution {
    public int minOperations(int[] nums, int k) {
        for (int x : nums) {
            k ^= x;
        }
        return Integer.bitCount(k);
    }
}
Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n)
Space
O(1)

Approach Breakdown

SORT + SCAN
O(n log n) time
O(n) space

Sort the array in O(n log n), then scan for the missing or unique element by comparing adjacent pairs. Sorting requires O(n) auxiliary space (or O(1) with in-place sort but O(n log n) time remains). The sort step dominates.

BIT MANIPULATION
O(n) time
O(1) space

Bitwise operations (AND, OR, XOR, shifts) are O(1) per operation on fixed-width integers. A single pass through the input with bit operations gives O(n) time. The key insight: XOR of a number with itself is 0, which eliminates duplicates without extra space.

Shortcut: Bit operations are O(1). XOR cancels duplicates. Single pass → O(n) time, O(1) space.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.