High-frequency problem so I'd seen it before, which helped.
Use dynamic programming where dp[i] represents the maximum sum for the first i elements. For each i, consider all possible last partitions of length 1 to k, compute the sum as dp[i-j] + j * max(arr[i-j..i-1]), and take the maximum. This yields an O(n*k) solution.
Pro tip: After presenting the DP solution, mention that the time complexity is O(n*k) and space is O(n), and note that if k is large, a monotonic queue could optimize to O(n), but for typical constraints O(n*k) is acceptable. This shows you consider scalability and trade-offs.
Restate the problem: partition array into contiguous subarrays of length at most k, replace each element in a subarray with the subarray's maximum, and maximize the total sum. Ask about constraints (n, k, element ranges) to determine optimal complexity.
Let dp[i] be the maximum sum for the first i elements. For each i, iterate j from 1 to min(k, i), compute the maximum in arr[i-j..i-1], and update dp[i] = max(dp[i], dp[i-j] + j * max_val).
While iterating j, maintain the maximum of the last j elements incrementally to avoid O(k) max computation per j. This keeps the inner loop O(k) and overall O(n*k).
State time O(n*k) and space O(n). Discuss edge cases: k=1 (no change), k >= n (whole array becomes max), and arrays with negative numbers (though problem typically assumes positive).
Walk through a small example to verify correctness. Optionally mention that a monotonic queue can reduce time to O(n) if k is large, but the DP is sufficient for typical constraints.
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