Minimum Operations to Make Binary Palindrome - Video Solutions

Problem Overview: You are given a binary array and need the minimum number of operations required to transform it into a palindrome. A palindrome reads the same from both ends, so elements at index i and n-1-i must match. The challenge is determining the smallest number of operations needed while efficiently handling mismatched pairs.

Approach 1: Two Pointers Pair Analysis (O(n) time, O(1) space)

Use the classic two pointers pattern. Start one pointer at the beginning and another at the end. For every pair (i, n-1-i), check if the bits differ. Each mismatch represents a required correction because a palindrome requires symmetric values. Iterate toward the center and count mismatches. This approach works well when each operation can directly fix a mismatched pair and is often the fastest baseline solution.

Approach 2: Preprocessing + Binary Search (O(n log n) time, O(n) space)

When operations affect ranges or require validation under constraints, combine preprocessing with binary search. First preprocess mismatched symmetric pairs and store them in a prefix structure so you can quickly determine how many conflicts exist in any region. Then binary search on the number of operations k. For each candidate k, run a feasibility check that uses the precomputed mismatch data to determine whether those operations can resolve all symmetric conflicts.

The preprocessing step compresses the mismatch information so validation becomes a quick scan rather than recomputing comparisons each time. During the check phase, you effectively simulate whether k operations can eliminate every pair conflict. The combination of prefix preprocessing and binary search dramatically reduces repeated work.

Because the array contains only binary values, operations and comparisons can also leverage bit manipulation tricks such as XOR to quickly detect mismatches (a ^ b == 1 means the pair differs). This keeps pair validation constant time.

Recommended for interviews: Start with the two‑pointer mismatch counting approach to demonstrate understanding of palindrome symmetry. Then discuss the preprocessing + binary search strategy for scenarios where operations affect ranges or require feasibility checks. Interviewers typically expect you to recognize the symmetric pair structure first, then optimize validation with prefix preprocessing.