Fractional Knapsack Problem

This problem can be solved using the Greedy Approach.

Goal: Given a knapsack that has a maximum capacity W, and a set of items with corresponding values and weights, we must choose items in such a way that the total value is maximum, and their total weight doesn't exceed W. Since this is the fractional knapsack problem, we can choose parts (fractions) of the items instead of the entire items.

Algorithm:

  1. Calculate the value/weight ratio for all the items
  2. Sort items according to this ratio in descending order
  3. Take the item with the highest ratio and keep adding items till we can't add the next item as a whole
  4. Add a part (fraction) of the next item so as to completely fill the knapsack
  5. Calculate total value of items in knapsack accordingly

Example:

  • W = 50
  • Items: A, B, C
  • Weights: 10, 20, 30
  • Values: 60, 100, 120

Calculating v/w ratios: 6, 5, 4

Sort items according to this ratio in descending order: A, B, C

Items added to the knapsack: A (w=10), B (w=20), C (w=50-(10+20)=20 i.e. 2/3rds)

Total value of added items: 60 + 100 + (2/3)*120 = 240

Time Complexity: O(nlgn)

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