Start by clarifying the problem constraints (e.g., item weights/values, capacity, whether items can be taken fractionally). Then explain that this is the classic 0/1 knapsack problem, which can be solved optimally using dynamic programming in O(n*W) time and space. Walk through the DP formulation, recurrence, and how to reconstruct the selected items.
Pro tip: Mention that while the DP solution is optimal for small capacities, for large capacities you might consider approximation algorithms or meet-in-the-middle for small n. Also, discuss how Uber might use this in real-world scenarios like resource allocation or delivery packing.
Ask if items can be taken fractionally (0/1 vs fractional knapsack) and confirm the goal is to maximize value without exceeding capacity. Also check constraints on n and capacity.
For 0/1 knapsack, dynamic programming is optimal. For fractional, a greedy approach by value/weight ratio works. Explain why DP is needed for 0/1.
Let dp[i][w] be the max value using first i items with capacity w. Recurrence: dp[i][w] = max(dp[i-1][w], dp[i-1][w-w_i] + v_i) if w_i <= w.
Use a 1D array to reduce space to O(W). To reconstruct items, either keep a 2D table or store decisions. Discuss trade-offs.
Time O(nW), space O(W) with 1D DP. Handle edge cases: zero capacity, items heavier than capacity, negative values (if allowed).
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