Fractional Knapsack Problem
Given items with values and weights, and a knapsack with capacity, select items (or fractions thereof) to maximize total value. Unlike the 0/1 knapsack where items must be taken whole, the fractional knapsack allows taking fractions of items. The greedy approach of taking items in order of value-to-weight ratio yields the optimal solution in O(n log n) time. This demonstrates when greedy algorithms work vs. when dynamic programming is needed.
Visualization
Interactive visualization for Fractional Knapsack Problem
Fractional Knapsack (Greedy)
• Time: O(n log n) - sorting
• Greedy: take highest value/weight ratio first
• Can take fractions unlike 0/1 knapsack
Interactive visualization with step-by-step execution
Implementation
1interface Item {
2 value: number;
3 weight: number;
4}
5
6function fractionalKnapsack(items: Item[], capacity: number): number {
7 items.sort((a, b) => (b.value / b.weight) - (a.value / a.weight));
8
9 let totalValue = 0;
10 let remaining = capacity;
11
12 for (const item of items) {
13 if (remaining >= item.weight) {
14 totalValue += item.value;
15 remaining -= item.weight;
16 } else {
17 totalValue += (item.value / item.weight) * remaining;
18 break;
19 }
20 }
21
22 return totalValue;
23}Deep Dive
Theoretical Foundation
Fractional Knapsack has optimal substructure and greedy choice property. Greedy strategy: sort items by value/weight ratio (value density) in descending order, take items greedily. Take as much as possible of highest ratio item, then next, until capacity filled. Last item may be taken partially. Why greedy works: taking item with highest ratio always contributes most value per unit weight. Proof: exchange argument shows swapping lower ratio for higher ratio always improves solution. Time: O(n log n) for sorting + O(n) for selection. Unlike 0/1 knapsack (requires DP), fractions make greedy optimal.
Complexity
Time
O(n log n)
O(n log n)
O(n log n)
Space
O(1)
Applications
Industry Use
Resource allocation (divisible resources)
Investment portfolios (fractional investments)
Network bandwidth allocation
Fuel loading optimization
Continuous material selection
Load balancing in distributed systems
Use Cases
Related Algorithms
Huffman Coding
Huffman Coding is a lossless data compression algorithm that creates optimal prefix-free variable-length codes based on character frequencies. Developed by David A. Huffman in 1952 as a student at MIT, it uses a greedy approach to build a binary tree where frequent characters get shorter codes. This algorithm is fundamental in ZIP, JPEG, MP3, and many compression formats.
Activity Selection Problem
Select the maximum number of non-overlapping activities from a set, where each activity has a start and end time. This classic greedy algorithm demonstrates the greedy choice property: always selecting the activity that finishes earliest leaves the most room for remaining activities. Used in scheduling problems, resource allocation, and interval management. Achieves optimal solution with O(n log n) time complexity.
Job Sequencing with Deadlines
Schedule jobs with deadlines and profits to maximize total profit. Each job takes 1 unit time and has a deadline and profit. The greedy strategy is to sort jobs by profit (descending) and greedily schedule each job as late as possible before its deadline. This maximizes profit while respecting constraints. Used in task scheduling, CPU job management, and project planning.
Minimum Platforms Problem
Find the minimum number of platforms required for a railway station given arrival and departure times of trains. No train should wait. The elegant solution uses the sorted merge technique: sort arrivals and departures separately, then use two pointers to track platforms needed at each moment. This greedy approach simulates the timeline efficiently in O(n log n) time, commonly asked in interviews.