Spent the first few minutes just drawing out the [2,2,1,2,1] example trying to convince myself I understood what a horizontal stroke even meant physically.
Recognize this as a classic dynamic programming problem (similar to 'Strange Printer' or 'Fence Painting'). Define a DP state over intervals and consider the optimal strategy: either paint the first plank vertically and recurse on the rest, or use a horizontal stroke to cover multiple planks of the same height, which can be modeled by splitting at matching heights. Derive the recurrence and then optimize to O(n^2) or O(n^3) depending on the approach.
Pro tip: Start by discussing the brute-force recursive solution and then optimize with memoization; this shows you understand the problem deeply and can improve efficiency. Also, clarify that horizontal strokes can only be applied at the same height across consecutive planks, which is a key constraint.
Restate the problem: each operation is either a vertical stroke on one plank or a horizontal stroke across consecutive planks at the same height. The goal is to minimize the total number of strokes. Clarify that horizontal strokes can only be applied if all planks in the range have at least that height (i.e., the stroke is at a height ≤ the minimum height of the range).
Let dp[i][j] be the minimum number of strokes to paint the subarray of planks from index i to j. Base case: dp[i][i] = 1 (one vertical stroke). For a range, the initial upper bound is the length of the range (all vertical strokes).
Consider the first plank i. We can paint it vertically, which costs 1 + dp[i+1][j]. Alternatively, if there is an index k > i such that height[k] == height[i], we can paint a horizontal stroke covering from i to k (and possibly beyond) at that height, which effectively merges the painting of these planks. The recurrence: dp[i][j] = min(1 + dp[i+1][j], min_{k: height[k]==height[i]} dp[i+1][k-1] + dp[k][j]). This accounts for the horizontal stroke covering i and k, and then solving the subproblems.
Use memoization or bottom-up DP to compute dp[0][n-1]. The time complexity is O(n^3) in the naive implementation, but can be optimized to O(n^2) by precomputing next occurrence of each height or using a different DP formulation. Discuss potential optimizations and trade-offs.
Walk through a small example, e.g., heights = [1,2,1], to verify the recurrence. Consider edge cases: all planks same height (answer 1), strictly increasing heights (answer n), and empty input (answer 0).
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