Fractional Knapsack Problem in DSA C# - Execution Trace

Concept Flow - Fractional Knapsack Problem

Start with items sorted by value/weight ratio

↓

Initialize total weight = 0, total value = 0

↓

For each item in sorted list

↓

Can full item fit?

No→Take fraction to fill knapsack

|Yes↓

Add full item weight and value

↓

Update total weight and value

↓

Knapsack full?

No→Next item

|Yes↓

Done

The flow shows sorting items by value/weight, then adding full or fractional parts until knapsack is full.

Execution Sample

DSA C#

items = [(weight=10, value=60), (weight=20, value=100), (weight=30, value=120)]
capacity = 50
Sort items by value/weight ratio descending
Take items fully or fractionally until capacity filled

This code sorts items by value per weight and fills the knapsack with full or fractional items to maximize value.

Execution Table

Step	Operation	Item (W,V)	Weight Taken	Value Added	Total Weight	Total Value	Knapsack State
1	Sort items by value/weight	[(10,60), (20,100), (30,120)]	-	-	0	0	[]
2	Take full item	(10,60)	10	60	10	60	[10/10]
3	Take full item	(20,100)	20	100	30	160	[10/10, 20/20]
4	Take fractional item	(30,120)	20	80	50	240	[10/10, 20/20, 20/30]
5	Knapsack full	-	-	-	50	240	[10/10, 20/20, 20/30]

💡 Knapsack capacity reached 50, no more items can be added.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
totalWeight	0	10	30	50	50
totalValue	0	60	160	240	240
currentItemIndex	0	1	2	3	3

Key Moments - 3 Insights

Why do we take only a fraction of the last item instead of the whole?

Because adding the whole last item would exceed the knapsack capacity, so we take only the fraction that fits (see Step 4 in execution_table).

How do we decide the order of items to take?

We sort items by their value-to-weight ratio in descending order before taking them (see Step 1 in execution_table).

What happens if the knapsack capacity is exactly filled before considering all items?

We stop adding items once the total weight equals the capacity (see Step 5 in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table, what is the total value after Step 3?

A 60

B 160

C 100

D 240

💡 Hint

Check the 'Total Value' column at Step 3 in the execution_table.

At which step does the knapsack become full?

A Step 5

B Step 3

C Step 4

D Step 2

💡 Hint

Look at the 'Knapsack State' and 'Total Weight' columns to see when capacity reaches 50.

If the capacity was 60 instead of 50, what would change in the execution_table?

A We would take only fractions of all items.

B We would take only the first item fully.

C We would take full items 1, 2, and 3 completely.

D The knapsack would never be full.

💡 Hint

Sum weights of all items and compare with capacity in variable_tracker.

Concept Snapshot

Fractional Knapsack Problem:
- Sort items by value/weight ratio descending
- Take full items while possible
- Take fraction of next item if needed
- Stop when capacity is full
- Maximizes total value with fractional items allowed

Full Transcript

The Fractional Knapsack Problem involves selecting items with given weights and values to maximize total value without exceeding capacity. We first sort items by their value-to-weight ratio from highest to lowest. Then, we add items fully if they fit or fractionally if only part fits. We track total weight and value as we add items. The process stops when the knapsack is full. This approach ensures maximum value by prioritizing items with the best value density.

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