Minimize the Maximum Adjacent Element Difference - Video Solutions

Problem Overview: You are given an array where some elements may be missing and must be replaced. The goal is to choose replacement values so that the maximum absolute difference between adjacent elements is minimized. The challenge is deciding which values to insert while keeping the largest neighbor gap as small as possible.

Approach 1: Greedy with Adjacent Difference Calculation (O(n) time, O(1) space)

Scan the array and focus on positions where a missing value sits between known neighbors. Track the minimum and maximum values around these gaps and estimate the best replacement that minimizes the largest adjacent difference. This method computes candidate values directly while iterating once through the array. It works well when the replacement pattern is straightforward and most constraints come from immediate neighbors.

Approach 2: Greedy Replacement with Median Strategy (O(n) time, O(1) space)

Instead of testing many replacements, compute the range of values surrounding missing segments and place a value near the median of that range. The intuition is that choosing a midpoint balances the difference on both sides of the gap. While iterating through the array, maintain boundary values for each missing block and greedily assign replacements that minimize the worst adjacent difference. This technique reduces variance across neighbors and often produces the optimal answer in a single pass.

Approach 3: Binary Search on Possible Values (O(n log M) time, O(1) space)

Binary search the answer for the smallest possible maximum difference d. For each candidate d, iterate through the array and check whether it is possible to fill missing values so every adjacent pair differs by at most d. The check function greedily determines valid ranges for replacements and verifies that constraints are satisfiable. Binary search works because the feasibility of a given d is monotonic.

Approach 4: Binary Search on Minimum Difference (O(n log M) time, O(1) space)

This variation also uses binary search but frames the feasibility check more explicitly around valid intervals for replacements. As you iterate through the array, compute allowable ranges that keep differences within d. Intersect these ranges for consecutive constraints and confirm whether at least one valid value exists for each missing position. This approach is reliable for complex configurations of gaps and constraints.

Recommended for interviews: The Binary Search feasibility approach combined with greedy validation is the most expected solution. It clearly demonstrates how to convert an optimization problem into a monotonic decision problem. Understanding the simpler greedy observations over arrays and constraint balancing also helps justify why the greedy check works.