First, clarify the problem and constraints, then derive the minimum total increment by analyzing the difference array. The key insight is that each operation on a contiguous subarray corresponds to adding a value to a range in the difference array, and the minimum total sum is the sum of all negative differences (or equivalently, the sum of positive differences).
Pro tip: Always discuss edge cases like already non-decreasing arrays or arrays with all equal elements, and mention that the solution runs in O(n) time and O(1) extra space, which is optimal.
Restate the problem in your own words and ask clarifying questions about constraints, input size, and whether the added values must be integers.
Recognize that adding a value to a contiguous subarray is equivalent to adding that value to a range in the difference array, which simplifies the problem.
Show that to make the array non-decreasing, the total added sum must be at least the sum of all negative differences, and this bound is achievable by incrementing appropriate subarrays.
Explain why the sum of negative differences is a lower bound and how to construct operations that achieve it, ensuring the array becomes non-decreasing.
State that the algorithm runs in O(n) time and O(1) space, and discuss edge cases such as already non-decreasing arrays or arrays with all equal elements.
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