Introduction

The Fractional Knapsack Problem is a classic optimization problem in Data Structures and Algorithms (DSA) and a perfect example of a greedy algorithm. If you are enrolled in a DSA course in Jaipur, mastering this problem will help you understand how to maximize profit using greedy strategies.

Unlike the 0/1 Knapsack problem, here you can take fractions of items.

What is Fractional Knapsack Problem?

You are given:

• Items with weight and value
• A knapsack with limited capacity

Goal:

• Maximize total value

Condition:

• You can take fractions of items

Example

Items:
Weight = [10, 20, 30]
Value = [60, 100, 120]
Capacity = 50

Value/Weight ratio:

• Item 1 → 6
• Item 2 → 5
• Item 3 → 4

Solution:

• Take full item 1 → value = 60
• Take full item 2 → value = 100
• Take 20 weight of item 3 → value = 80

Total value = 240

Greedy Strategy

• Select items based on highest value/weight ratio
• Take full item if possible
• Otherwise take fraction

Algorithm Steps

1. Calculate value/weight ratio
2. Sort items in descending order of ratio
3. Pick items greedily
4. Take fraction if capacity is not enough

Time Complexity

O(n log⁡ n)

• Sorting → O(n log n)
• Selection → O(n)

Why Greedy Works Here

• Taking highest ratio item gives maximum profit
• Fractional choice allows optimal solution

Fractional vs 0/1 Knapsack

• Fractional allows partial items
• 0/1 does not allow splitting
• Fractional uses greedy
• 0/1 uses dynamic programming

Advantages

• Optimal solution guaranteed
• Simple and efficient
• Easy implementation

Disadvantages

• Not applicable when items cannot be divided
• Limited real-world applicability in some cases

Real-World Applications

• Resource allocation
• Investment optimization
• Cargo loading
• Budget planning

Common Interview Questions

• Implement fractional knapsack
• Compare with 0/1 knapsack
• Maximize profit problems

Best Practices

• Always sort by value/weight ratio
• Handle fractional calculations carefully
• Understand greedy choice logic

Summary

• Fractional knapsack uses greedy approach
• Select items based on value/weight ratio
• Allows fractional selection
• Time complexity is O(n log n)
• Important for coding interviews

FAQs

Q1. What is fractional knapsack?
It is a problem where items can be taken partially.

Q2. Why does greedy work here?
Because selecting highest ratio maximizes value.

Q3. What is time complexity?
O(n log n).

Q4. What is difference from 0/1 knapsack?
Fractional allows splitting, 0/1 does not.

Q5. Is it important for interviews?
Yes, it is a classic greedy problem.

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