Model the problem as a sequential decision process where each warehouse's state is the number of items remaining modulo (m+n), and skips allow you to take extra turns. Use dynamic programming to compute the maximum score for each warehouse given a certain number of skips, then combine results across warehouses with a knapsack-style DP. Analyze complexity in terms of number of warehouses, items, and skips.
Pro tip: Clarify with the interviewer whether skips can be used mid-warehouse or only between warehouses, as this affects the DP state. Also, consider if the optimal strategy might involve saving skips for later warehouses based on their sizes.
Restate the problem: you and coworker alternate turns, you remove m, coworker removes n, you score when you empty a warehouse. Skips let you take an extra turn immediately. Determine if skips can be used at any time or only at warehouse boundaries.
For a warehouse with W items, compute the maximum score (0 or 1) achievable with s skips. Since you only score if you empty it, determine the minimum skips needed to make your removal exactly equal to the remaining items at some turn. This depends on W mod (m+n) and the sequence of removals.
Let dp[i][j] be the max score using j skips on the first i warehouses. For each warehouse i, precompute cost[s] = max score (0 or 1) with s skips. Then dp[i][j] = max_{s=0..j} (dp[i-1][j-s] + cost_i[s]).
For each warehouse, simulate the turn sequence with skips. Since skips only affect your turns, the number of your turns before emptying is determined by W and skips. Derive a formula: you need to remove m on your turn when remaining items ≡ 0 mod m? Actually, you score if you empty it, so you need to arrange that your removal exactly equals remaining. This happens if W ≡ m (mod m+n) after some number of full cycles, or with skips you can adjust. Precompute for each possible s up to k.
Time: O(N * k^2) if naive, but can be optimized to O(N * k) if cost_i[s] is monotonic or if we use prefix maxima. Space: O(N * k) for DP table, can be reduced to O(k) with rolling array. Discuss trade-offs.
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