Why Fractional Knapsack Problem in DSA C#?

The Big Idea

What if you could carry the most valuable things by taking just the right parts of each item?

The Scenario

Imagine you are packing a bag for a trip. You have many items, each with a weight and value. You want to carry the most valuable items, but your bag can only hold a limited weight. If you try to pick items one by one without a plan, you might end up carrying less value.

The Problem

Trying to pick items manually is slow and confusing. You might pick heavy items with low value first, leaving no space for lighter, more valuable items. This wastes your bag space and reduces the total value you carry.

The Solution

The Fractional Knapsack Problem helps by allowing you to take parts of items, not just whole items. It sorts items by value per weight and picks the best parts first, filling your bag with the highest value possible.

Before vs After

✗ Before

double totalValue = 0;
foreach (var item in items) {
  if (bagCanHold(item.weight)) {
    totalValue += item.value;
    bagAdd(item);
  }
}

✓ After

items.Sort((a, b) => b.valuePerWeight.CompareTo(a.valuePerWeight));
double totalValue = 0;
double remainingWeight = bagCapacity;
foreach (var item in items) {
  double weightToTake = Math.Min(item.weight, remainingWeight);
  totalValue += weightToTake * item.valuePerWeight;
  remainingWeight -= weightToTake;
  if (remainingWeight == 0) break;
}

What It Enables

This method lets you maximize the total value you carry by smartly choosing whole or parts of items based on their value-to-weight ratio.

Real Life Example

Think of filling a backpack with snacks for a hike. Some snacks are heavy but less tasty, others are light and delicious. Fractional Knapsack helps you pick the tastiest mix that fits your backpack weight limit.

Key Takeaways

Manual picking wastes space and value.

Fractional Knapsack sorts items by value per weight.

It allows taking parts of items to fill the bag optimally.

Practice

1. What is the main goal of the Fractional Knapsack Problem?

easy

A. Maximize number of items regardless of weight

B. Maximize total value by taking fractions of items within weight limit

C. Minimize total weight by selecting whole items only

D. Select items randomly without considering value or weight

Solution

1. Step 1: Understand the problem goal

   The fractional knapsack problem allows taking parts of items to maximize total value within a weight limit.

2. Step 2: Compare options

   Only Maximize total value by taking fractions of items within weight limit correctly states maximizing total value by taking fractions within weight limit.

3. Final Answer:

   Maximize total value by taking fractions of items within weight limit -> Option B

4. Quick Check:

   Goal = Maximize value with fractions [OK]

Hint: Remember: fractional means partial items allowed [OK]

Common Mistakes:

• Thinking only whole items can be taken
• Confusing with minimizing weight
• Ignoring value-to-weight ratio

2. Which C# syntax correctly declares a class named Item with public fields weight and value?

easy

A. class Item { public int weight; public int value; }

B. class Item { int weight; int value; }

C. public Item { int weight; int value; }

D. class Item { public weight; public value; }

Solution

1. Step 1: Check class declaration syntax

   In C#, class fields must have type and access modifier for public fields.

2. Step 2: Validate each option

   class Item { public int weight; public int value; } correctly declares public int fields. class Item { int weight; int value; } lacks access modifier. public Item { int weight; int value; } misses class keyword. class Item { public weight; public value; } misses types.

3. Final Answer:

   class Item { public int weight; public int value; } -> Option A

4. Quick Check:

   Correct class syntax = class Item { public int weight; public int value; } [OK]

Hint: Public fields need type and access modifier [OK]

Common Mistakes:

• Omitting access modifiers
• Missing data types
• Incorrect class keyword usage

3. What is the output of this C# code snippet for fractional knapsack?

var items = new[] {
  new Item { weight = 10, value = 60 },
  new Item { weight = 20, value = 100 },
  new Item { weight = 30, value = 120 }
};
int capacity = 50;
double maxValue = 0;
Array.Sort(items, (a, b) => (b.value / (double)b.weight).CompareTo(a.value / (double)a.weight));
int remaining = capacity;
foreach (var item in items) {
  if (item.weight <= remaining) {
    maxValue += item.value;
    remaining -= item.weight;
  } else {
    maxValue += item.value * (remaining / (double)item.weight);
    break;
  }
}
Console.WriteLine(Math.Round(maxValue, 2));

medium

A. 200.00

B. 220.00

C. 180.00

D. 240.00

Solution

1. Step 1: Sort items by value/weight ratio

   Ratios: Item1=6, Item2=5, Item3=4. Sorted order: Item1, Item2, Item3.

2. Step 2: Add items until capacity 50

   Take Item1 fully (weight 10, value 60), remaining=40. Take Item2 fully (weight 20, value 100), remaining=20. Take 20/30 fraction of Item3: value = 120 * (20/30) = 80.

3. Step 3: Calculate total value

   Total = 60 + 100 + 80 = 240. Rounded to 240.00.

4. Final Answer:

   240.00 -> Option D

5. Quick Check:

   Sum values with fractions = 240.00 [OK]

Hint: Sort by value/weight, take full then fraction [OK]

Common Mistakes:

• Not sorting by value/weight ratio
• Taking items without fraction
• Incorrect fraction calculation

4. Identify the bug in this fractional knapsack code snippet:

int capacity = 50;
var items = new List<Item> {
  new Item { weight = 10, value = 60 },
  new Item { weight = 20, value = 100 },
  new Item { weight = 30, value = 120 }
};
items.Sort((a, b) => (a.value / a.weight).CompareTo(b.value / b.weight));
double maxValue = 0;
int remaining = capacity;
foreach (var item in items) {
  if (item.weight <= remaining) {
    maxValue += item.value;
    remaining -= item.weight;
  } else {
    maxValue += item.value * (remaining / item.weight);
    break;
  }
}
Console.WriteLine(maxValue);

medium

A. Sorting order is ascending instead of descending

B. Missing check for zero weight items

C. Integer division causes incorrect ratio calculation

D. Using List instead of array causes error

Solution

1. Step 1: Analyze sorting comparator

   The comparator uses integer division (a.value / a.weight) which truncates decimals, causing wrong ratios.

2. Step 2: Impact on sorting and calculation

   Integer division leads to incorrect sorting order and wrong maxValue calculation.

3. Final Answer:

   Integer division causes incorrect ratio calculation -> Option C

4. Quick Check:

   Use double division for correct ratios [OK]

Hint: Use double division for ratios, not integer division [OK]

Common Mistakes:

• Sorting ascending instead of descending
• Ignoring integer division truncation
• Assuming List causes error

5. Given items with weights and values: [(5, 30), (10, 40), (15, 45)], and knapsack capacity 20, what is the maximum value achievable using fractional knapsack?

hard

A. 85

B. 75

C. 80

D. 70

Solution

1. Step 1: Calculate value/weight ratios and sort

   Ratios: (5,30)=6, (10,40)=4, (15,45)=3. Sorted order: (5,30), (10,40), (15,45).

2. Step 2: Fill knapsack capacity 20

   Take full (5,30), remaining=15, value=30. Take full (10,40), remaining=5, value=70. Take fraction of (15,45): 5/15=1/3 fraction, value=45*(1/3)=15.

3. Step 3: Calculate total value

   Total value = 30 + 40 + 15 = 85.

4. Final Answer:

   85 -> Option A

5. Quick Check:

   Sum with fractions = 85 [OK]

Hint: Sort by ratio, take full items first, then fraction [OK]

Common Mistakes:

• Not sorting by value/weight ratio
• Taking items without fraction
• Incorrect fraction calculation