What is the optimal time complexity of solving 0-1 Knapsack?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for 0-1 Knapsack?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in 0-1 Knapsack?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for 0-1 Knapsack?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingEasy

0-1 Knapsack

Learn how to solve the '0-1 Knapsack' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function knapsack(W, wt, val, n) that finds the maximum value that can be put in a knapsack of capacity W using items with weights wt and values val. You cannot split items; you either take an item or leave it.

Constraints

•1 <= n <= 100
•1 <= W <= 1000
•1 <= wt[i], val[i] <= 1000

Examples

Example 1

Input

knapsack(50, [10, 20, 30], [60, 100, 120], 3)

Output

Explanation

By taking items with weight 20 and 30, the total weight is 50, and the value is 100 + 120 = 220.

Example 2

Input

knapsack(10, [5, 4, 6, 3], [10, 40, 30, 50], 4)

Output

Explanation

By taking items of weight 4 and 3, the total weight is 7, and the value is 40 + 50 = 90.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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