Fractional Knapsack Problem in DSA C# - Time & Space Complexity

Time Complexity: Fractional Knapsack Problem

O(n log n)

Understanding Time Complexity

We want to understand how the time needed to solve the Fractional Knapsack Problem changes as the number of items grows.

Specifically, how the sorting and selection steps affect the total work done.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Fractional Knapsack Problem
// items: array of (value, weight) pairs
// capacity: max weight allowed

Array.Sort(items, (a, b) => ((float)b.value / b.weight).CompareTo((float)a.value / a.weight));

float totalValue = 0;
int remainingCapacity = capacity;

foreach (var item in items) {
    if (item.weight <= remainingCapacity) {
        totalValue += item.value;
        remainingCapacity -= item.weight;
    } else {
        totalValue += item.value * ((float)remainingCapacity / item.weight);
        break;
    }
}
    

This code sorts items by value per weight and then picks as much as possible from the highest ratio items until capacity is full.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Sorting the array of items by value-to-weight ratio.
• How many times: Sorting typically takes time proportional to n log n, where n is the number of items.
• Secondary operation: One loop through all items to pick fractions or whole items.
• How many times: This loop runs once over all n items.

How Execution Grows With Input

As the number of items increases, sorting takes more time, growing faster than just going through the list.

Input Size (n)	Approx. Operations
10	About 10 * log 10 ≈ 33 operations for sorting + 10 for picking
100	About 100 * log 100 ≈ 664 operations for sorting + 100 for picking
1000	About 1000 * log 1000 ≈ 9966 operations for sorting + 1000 for picking

Pattern observation: Sorting dominates and grows faster than the simple loop, so total time grows roughly like n log n.

Final Time Complexity

Time Complexity: O(n log n)

This means the time needed grows a bit faster than just the number of items, mainly because of sorting.

Common Mistake

[X] Wrong: "The fractional knapsack can be solved in linear time because we just pick items until full."

[OK] Correct: We must first sort items by value per weight, which takes more than linear time, so sorting dominates the total time.

Interview Connect

Understanding this time complexity helps you explain why sorting is necessary before selection, showing you grasp the problem's core steps clearly.

Self-Check

"What if the items were already sorted by value per weight? How would the time complexity change?"

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