The Maximized Threshold Divergence problem asks for the largest possible gap that can be created by placing a threshold between two values of a numeric sequence. By sorting the sequence in non‑decreasing order, every potential threshold can be represented as a cut between two consecutive elements. The divergence of a particular cut is simply the difference between the right‑hand element and the left‑hand element. A naïve solution would examine every possible pair of indices (i, j) and compute |a[j]‑a[i]|, leading to O(N^2) time, which quickly becomes infeasible for N up to 10^5 or higher. The optimal paradigm leverages the monotonic property of a sorted array: the maximum gap must occur between two adjacent elements after sorting, because any larger distance can be decomposed into a sum of smaller adjacent gaps, none of which exceed the maximum adjacent gap. Therefore, a single pass over the sorted array yields the answer in linear time after the O(N log N) sort.

The underlying algorithmic pattern is a classic "sort‑and‑scan" greedy technique. Sorting imposes order, collapsing the combinatorial explosion of O(N^2) pairwise comparisons into a deterministic sequence where the optimal decision is locally observable. This reduction is a cornerstone of many hard‑level sorting problems, such as maximizing the minimum distance between placed items or minimizing the maximum subarray sum after partitioning. The key insight is that the global optimum is encoded in a simple local property once the data is ordered.

Why this matters for large inputs is twofold: first, the O(N log N) bound is asymptotically optimal for comparison‑based sorting, and second, the subsequent linear scan adds negligible overhead. Any attempt to avoid sorting—e.g., using bucket sort or counting sort—must respect the value range constraints, otherwise the algorithm degrades to the naïve quadratic approach. Hence, the sort‑then‑scan strategy is both theoretically sound and practically efficient for the hard difficulty tier.