What is the optimal time complexity of solving Fractional 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 Fractional Knapsack?

The space complexity is O(n) 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 Fractional 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 Fractional 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

Fractional Knapsack

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

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Problem Statement

Easy

Write a function fractional_knapsack(W, items) that takes the knapsack capacity W and a list of tuples items where each tuple is (value, weight). Find the maximum total value obtainable in the knapsack, allowing items to be broken/fractioned. Round the output to 2 decimal places.

Constraints

•1 <= len(items) <= 1000
•1 <= W <= 10^5
•1 <= value, weight <= 10^4

Examples

Example 1

Input

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

Output

240.0

Explanation

Take the first and second item fully, and 2/3 of the third item. Total value: 60 + 100 + 120 * (20/30) = 240.0.

Example 2

Input

fractional_knapsack(15, [(24, 10), (18, 10), (15, 10)])

Output

33.0

Explanation

Take the first item fully, and 1/2 of the second item. Total value: 24 + 18 * 0.5 = 33.0.

Need a Hint?

Analyze the input constraints. Try sorting first (O(n log n)) or using a hash map/set to track seen elements in O(n) time.

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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