Fractional Knapsack Problem in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Fractional Knapsack Problem?

To maximize the total value of items placed in a knapsack with a limited weight capacity, allowing fractions of items to be taken.

beginner

Why can we take fractions of items in the Fractional Knapsack Problem but not in the 0/1 Knapsack Problem?

Because the problem assumes items are divisible, like liquids or materials, so partial amounts can be taken to fill the knapsack optimally.

intermediate

What is the greedy strategy used in the Fractional Knapsack Problem?

Sort items by their value-to-weight ratio in descending order and take as much as possible from the highest ratio items first.

beginner

How do you calculate the value-to-weight ratio of an item?

Divide the item's value by its weight: ratio = value / weight.

intermediate

What is the time complexity of the Fractional Knapsack Problem solution using sorting?

O(n log n), where n is the number of items, due to sorting the items by value-to-weight ratio.

In the Fractional Knapsack Problem, what do you do after sorting items by value-to-weight ratio?

A Take only the first item fully

B Take items fully until the knapsack is full, then take fraction of the next item

C Take items randomly

D Take items with the lowest value first

The greedy approach takes full items from highest ratio first, then a fraction of the next item if needed.

Which of these is true about the Fractional Knapsack Problem?

A Items can be broken into smaller parts

B Items must be taken whole or not at all

C It uses dynamic programming

D It cannot be solved greedily

Fractional Knapsack allows breaking items into parts, unlike 0/1 Knapsack.

What is the key difference between Fractional and 0/1 Knapsack Problems?

A Fractional allows partial items, 0/1 does not

B 0/1 allows partial items, Fractional does not

C Both allow partial items

D Neither allow partial items

Fractional Knapsack allows fractions; 0/1 Knapsack requires whole items.

What data structure is commonly used to sort items by value-to-weight ratio?

A Linked list without sorting

B Queue

C Stack

D Array or list

Arrays or lists are sorted by ratio to apply the greedy approach.

If the knapsack capacity is 50 and the next item weight is 30, but only 20 capacity remains, what do you do?

A Take the full item anyway

B Skip the item

C Take 20/30 fraction of the item

D Remove some items already taken

Take the fraction of the item that fits in the remaining capacity.

Explain the greedy approach to solving the Fractional Knapsack Problem step-by-step.

Think about how to fill the knapsack with the most valuable parts first.

Describe the difference between Fractional Knapsack and 0/1 Knapsack problems and why the greedy method works for one but not the other.

Focus on item divisibility and algorithm choice.

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