Make the XOR of All Segments Equal to Zero - Video Solutions

Problem Overview: You are given an array nums and an integer k. Every contiguous segment of length k must have XOR equal to 0. You can change any element. The goal is to minimize the number of modifications.

Approach 1: Greedy Frequency Selection (Approximate Greedy) (Time: O(n + k * 1024), Space: O(1024))

Observe that elements contributing to the same segment positions repeat every k indices. Group elements by i % k. Each group affects the XOR of every segment exactly once. A greedy idea is to replace values in each group with the most frequent value in that group, minimizing local changes. Track frequencies using hash maps or fixed arrays since values are within a small bit range.

The limitation is that independent greedy choices might not produce a global XOR of zero across groups. You still compute the XOR of chosen representatives and may need extra adjustments. This method is useful for building intuition about the repeating structure of the problem and why group-based reasoning works.

Approach 2: Dynamic Programming over XOR States (Optimal) (Time: O(n + k * 1024), Space: O(1024))

Split the array into k groups by index modulo k. For each group, count the frequency of every value. Since numbers are < 2^10, there are only 1024 possible XOR states. Define dp[x] as the minimum changes needed so processed groups produce XOR x.

For each group, try assigning a value v. The cost equals group_size - freq[v] (number of elements you must change). Update the next DP state with new_xor = prev_xor ^ v. To avoid iterating all possibilities repeatedly, track the minimum value in the previous DP layer and use it as a fallback when assigning unseen values. This reduces the transition cost dramatically.

This technique combines array grouping, frequency counting, and state transitions typical in dynamic programming. XOR transitions rely on properties from bit manipulation. After processing all k groups, the answer is dp[0] because the final XOR must be zero.

Recommended for interviews: The dynamic programming approach is the expected solution. Interviewers want to see recognition of the i % k grouping pattern and a DP state representing XOR possibilities. The greedy reasoning helps explain the structure, but the DP solution demonstrates strong problem‑solving skills with bitwise state transitions.