My first instinct was BFS on a state space where each state is the remaining need vector, then I pivoted to DP with memoization.
Model the problem as a dynamic programming problem where the state is the vector of remaining quantities for each item. Use memoization to compute the minimum cost by trying all possible combos and individual items that do not exceed the remaining quantities. Discuss the trade-offs between this exact DP approach and potential optimizations like integer linear programming or greedy heuristics.
Pro tip: Emphasize that the problem is NP-hard in general (it's a multidimensional knapsack variant), so for large inputs you'd need to consider approximation algorithms or ILP solvers. Showing awareness of practical constraints and trade-offs will impress interviewers.
Restate the problem to ensure understanding: we have a set of items with individual prices, a set of combos each with a price and quantities of items, and a target order with exact quantities. We must minimize cost without exceeding any item quantity. Ask about input size limits, whether prices are integers, and if combos can be used multiple times.
Let dp[q1][q2]...[qn] be the minimum cost to fulfill an order with remaining quantities q1..qn. The recurrence is dp[q] = min( min over items i with qi>0 of price_i + dp[q - e_i], min over combos c that fit in q of price_c + dp[q - quantities_c] ). Base case: dp[0] = 0.
Use memoization (top-down) or iterate over all possible quantity vectors in increasing order (bottom-up). Since the state space is the product of (required quantity + 1) for each item, it can be large. For small n and quantities, this is feasible.
Prove correctness by induction on the total remaining quantity: the optimal solution either uses an individual item or a combo first, and the recurrence considers all possibilities. Time complexity: O((Q1+1)*...*(Qn+1) * (n + m)) where m is number of combos. Space: O((Q1+1)*...*(Qn+1)).
Mention that the problem is NP-hard (reduction from multidimensional knapsack), so for large inputs, consider ILP, branch-and-bound, or approximation algorithms. Also, note that if combos are limited or quantities are small, DP is efficient. Discuss potential pruning: remove dominated combos (higher price for same or fewer items).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the original problem context (likely a DP for order fulfillment with multiple item types). Then, explain that with only three item types, the state can be reduced to a 3D vector of remaining needs, and transitions can be optimized using techniques like BFS with memoization or Dijkstra on a smaller graph. Finally, analyze the complexity improvement from O(N^K) to O(N^3) or better, emphasizing the practical impact.
Pro tip: Mention that the optimization not only reduces time complexity but also simplifies the code and makes it more maintainable, which is crucial for production systems at scale like Airbnb's.
Restate the problem to ensure understanding: likely a dynamic programming problem for order fulfillment where each order requires certain quantities of various item types, and we want to minimize cost or maximize efficiency.
With only three item types, represent the state as a tuple (a, b, c) where a, b, c are the remaining needs for each type. This reduces the state space from potentially high-dimensional to 3D.
Transitions correspond to actions that reduce the needs, such as fulfilling part of the order with a particular combination of items. Use BFS or Dijkstra if costs are involved, and memoize results to avoid recomputation.
The state space is O(N^3) where N is the maximum need per item type. Transitions per state are constant (e.g., a few possible actions). Thus, time complexity improves from O(N^K) to O(N^3), and space similarly.
Mention that this optimization is specific to three types; for more types, other techniques like meet-in-the-middle or approximation may be needed. Also, note that if N is large, further optimizations like using A* or pruning may help.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the problem context and the definition of 'dominated' or 'irrelevant' combos/items. Then describe specific preprocessing techniques (e.g., dominance pruning, bounding, symmetry breaking) and explain how each preserves optimality using exchange arguments or feasibility arguments. Finally, discuss the trade-offs and potential pitfalls.
Pro tip: Emphasize that pruning must be safe: always prove that for any pruned solution, there exists an unpruned solution at least as good. Mention that in practice, you'd validate pruning empirically by comparing results on small instances.
Restate the problem to ensure you understand what 'dominated' and 'irrelevant' mean in this context. Ask clarifying questions if needed.
List specific techniques such as dominance pruning, bounding, symmetry breaking, or removing items that can never be part of an optimal solution.
For each pruning method, provide a proof that removing those elements does not eliminate all optimal solutions. Use exchange arguments or feasibility arguments.
Explain how to implement the pruning efficiently and discuss the trade-offs between pruning overhead and algorithm speedup.
Mention the importance of testing the pruning on small cases and comparing with brute force to ensure correctness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.