Minimum Operations to Transform Array into Alternating Prime - Solution & Explanation

Problem Overview: You are given an array of integers and must transform it into an alternating pattern involving prime numbers. Each operation modifies an element (typically by incrementing or decrementing its value) so that the final array alternates between prime and non‑prime values. The goal is to minimize the total number of operations.

Approach 1: Brute Force Prime Adjustment (O(n * sqrt(V)) time, O(1) space)

Check both possible alternating patterns: prime at even indices and non‑prime at odd indices, or the reverse. For each element, determine whether it already satisfies the required state (prime or non‑prime). If not, compute the minimum number of increments or decrements required to reach the nearest valid number. Primality is checked using trial division up to sqrt(num). This approach is straightforward and useful for smaller value ranges but becomes slow when many prime checks are required.

Approach 2: Precompute Primes with Sieve (O(V log log V + n) time, O(V) space)

Instead of checking primality repeatedly, precompute all primes up to the maximum value range using the Sieve of Eratosthenes. Store them in a boolean array so prime checks become O(1). Then evaluate both alternating patterns by iterating through the array once and computing the minimal distance to a valid prime or non‑prime value when needed. Precomputation removes the repeated sqrt(V) checks and significantly speeds up the evaluation when the array is large.

Approach 3: Greedy Pattern Evaluation (O(n) time after preprocessing, O(V) space)

After building the prime lookup table, iterate through the array while maintaining the expected state (prime or non‑prime) for the current index. If the element already matches the requirement, move forward. Otherwise compute the nearest valid value and add the adjustment cost. Run this process twice—once assuming index 0 should be prime and once assuming it should be non‑prime—and take the minimum result. The greedy insight is that each position can be optimized independently once the pattern is fixed.

Recommended for interviews: The sieve + greedy evaluation approach. Interviewers expect you to reduce repeated prime checks and reason about evaluating two possible patterns. Mention the brute force method first to show baseline reasoning, then optimize with precomputation. This combines concepts from arrays, math, and number theory.