Minimum Increase to Maximize Special Indices - Solution & Explanation

Problem Overview: You are given an array and can increase elements by any amount. The goal is to apply the minimum total increase so the number of special indices becomes as large as possible. A special index is defined by a condition involving prefix and suffix values, which means the solution depends on efficiently tracking partial sums and evaluating each position.

Approach 1: Brute Force Simulation (O(n²) time, O(1) space)

The most direct strategy checks every index and determines how much increase is required to satisfy the special condition. For each position, recompute prefix sums and suffix sums and calculate the minimal increment needed to make the index valid. This leads to repeated scans of the array, producing quadratic time complexity. The approach is easy to reason about and useful for validating logic during development, but it becomes too slow for large inputs.

Approach 2: Prefix Sum Optimization (O(n) time, O(n) space)

Instead of recomputing sums repeatedly, precompute prefix sums and suffix sums once using a prefix sum array. With these values available, you can evaluate each index in constant time. The key insight is that the cost to make an index special can be expressed as a simple difference between prefix and suffix contributions. By iterating once through the array and computing the required increase for each position, you determine which indices can become special with minimal modification. This reduces the overall complexity to linear time.

Approach 3: Greedy with Running Prefix Tracking (O(n) time, O(1) space)

A further optimization avoids storing full auxiliary arrays. Maintain a running prefix sum while tracking the remaining suffix sum as you move through the array. For each index, compute the required increase using the current prefix state and the remaining suffix contribution. This greedy evaluation works because the cost for each index is independent once the prefix and suffix totals are known. The approach combines ideas from array traversal and prefix calculations to produce an optimal linear scan.

Recommended for interviews: Start by describing the brute force approach to show you understand the definition of a special index. Then move quickly to the prefix-sum based optimization. Interviewers typically expect the O(n) scan using prefix tracking because it demonstrates familiarity with prefix sum techniques and greedy reasoning. The optimal implementation is concise and scales well for large arrays.