Minimum Stability Factor of Array - Solution & Explanation

Problem Overview: You are given an array and must compute the minimum possible stability factor such that the array satisfies a stability condition defined over its elements. The goal is to minimize this factor while ensuring the array remains valid according to the constraints. The challenge comes from large value ranges and the need to efficiently validate candidate stability values.

Approach 1: Brute Force Stability Check (O(n * V), Space O(1))

The most direct approach is to try every possible stability factor from the smallest feasible value up to the maximum element range. For each candidate factor, iterate through the array and verify whether the stability condition holds for all elements or segments. This involves repeated scans of the array and recomputation of constraints such as pair differences or divisibility relationships. While straightforward, this approach becomes infeasible when the value range V is large.

Approach 2: Binary Search + Greedy Validation (O(n log V), Space O(1))

A key observation is that if a stability factor S works, any larger factor will also work. This monotonic property allows you to apply binary search on the answer. For each candidate value, perform a greedy validation pass through the array. The validation checks whether the array can satisfy the stability constraint using local adjustments or allowed operations while keeping differences within the candidate bound. Each check runs in linear time, so the total complexity becomes O(n log V), which is efficient for large inputs.

Approach 3: Binary Search with Segment Tree Optimization (O(n log n log V), Space O(n))

When validation requires frequent range queries such as computing minimums, maximums, or gcd values across subarrays, a segment tree can speed up checks. Preprocess the array into a segment tree so each query runs in O(log n). During the binary search feasibility test, query ranges to verify whether the stability constraint holds across segments. This approach trades extra memory for faster constraint checks when the validation logic depends on aggregated range information.

Approach 4: Greedy + Number Theory Observations (O(n log V), Space O(1))

Some variations of the stability constraint rely on divisibility or gcd relationships between elements. Using insights from number theory, you can simplify validation by computing gcd transitions or allowable reductions directly while scanning the array. The greedy step keeps track of the minimal feasible value for each position while ensuring the candidate stability bound is not violated.

Recommended for interviews: Binary search on the answer combined with a greedy feasibility check is the approach most interviewers expect. The brute force approach demonstrates understanding of the constraint being tested, but the binary search solution shows you recognize the monotonic structure of the problem and can reduce the complexity from linear-in-range to logarithmic-in-range.