Minimum Operations to Make Array Modulo Alternating II - Solution & Explanation

Problem Overview: You are given an array and must perform the minimum number of operations so that the sequence of nums[i] % k values alternates between two different remainders. Adjacent indices cannot share the same modulo value. Each operation changes an element to any value, effectively allowing you to control its remainder.

Approach 1: Brute Force Remainder Pair Enumeration (O(n * m^2) time, O(m) space)

First compute r[i] = nums[i] % k. The final array must alternate between two remainders a and b where a != b. Try every possible ordered pair of remainders from the modulo domain. For each pair, iterate through the array and count how many positions do not match the expected alternating pattern. The minimum mismatch count is the answer. This approach works but becomes expensive if the remainder domain is large because every candidate pair must be checked.

Approach 2: Frequency Counting on Even/Odd Indices (O(n) time, O(m) space)

Observe that alternating patterns split the array into two independent groups: even indices and odd indices. Count the frequency of each remainder separately for these two groups using a hash map. The optimal pattern keeps the most frequent remainder on even positions and the most frequent different remainder on odd positions. If the top remainders for both groups differ, use them directly. If they are equal, try the second-best option for one side. The number of operations equals n - (bestEvenFreq + bestOddFreq). This converts the problem into a greedy selection over frequency counts and runs in linear time.

Approach 3: Top-Two Frequency Optimization (O(n) time, O(m) space)

Instead of scanning every remainder candidate, track only the top two frequencies for even and odd indices. This guarantees you can resolve conflicts when the most frequent remainders are identical. The algorithm iterates once to compute counts and once more to determine the best pair combination. The logic mirrors problems like minimum operations to make array alternating and relies on counting patterns in an array combined with a simple greedy decision.

Recommended for interviews: The frequency-counting approach with top-two tracking is what interviewers expect. Brute force demonstrates the correct interpretation of the alternating modulo constraint, but the optimized counting method shows you recognize the independence of even and odd positions and can reduce the search space to constant candidates.