Fractional Knapsack Problem in DSA Typescript - 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, we ask: how does the work increase when we have more items to choose from?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function fractionalKnapsack(weights: number[], values: number[], capacity: number): number {
  const n = weights.length;
  const items = [];
  for (let i = 0; i < n; i++) {
    items.push({
      weight: weights[i],
      value: values[i],
      ratio: values[i] / weights[i]
    });
  }
  items.sort((a, b) => b.ratio - a.ratio);
  let totalValue = 0;
  let remaining = capacity;
  for (const item of items) {
    if (item.weight <= remaining) {
      totalValue += item.value;
      remaining -= item.weight;
    } else {
      totalValue += item.ratio * remaining;
      break;
    }
  }
  return totalValue;
}
    

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

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Sorting the items by their value-to-weight ratio.
• How many times: Sorting runs once over all items, which takes time depending on the number of items.
• Two loops run over all items: one to create the list and one to pick items.

How Execution Grows With Input

As the number of items increases, sorting takes more time, growing faster than just looking at items one by one.

Input Size (n)	Approx. Operations
10	About 10 log 10 (sorting) + 2*10 (loops) = ~40 operations
100	About 100 log 100 + 200 = ~900 operations
1000	About 1000 log 1000 + 2000 = ~13000 operations

Pattern observation: The sorting step dominates and grows faster than the loops, roughly like n times log n.

Final Time Complexity

Time Complexity: O(n log n)

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

Common Mistake

[X] Wrong: "The main time cost is picking items, so the complexity is O(n)."

[OK] Correct: Sorting the items by ratio takes more time than just picking, so sorting dominates the total time.

Interview Connect

Understanding this time complexity helps you explain why sorting is important before choosing items, a key skill in optimization problems.

Self-Check

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

Practice

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

easy

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

B. Minimize total weight of items selected

C. Select only whole items to maximize value

D. Sort items by weight only

Solution

1. Step 1: Understand problem goal

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

2. Step 2: Compare options

   Maximize total value by taking fractions of items within weight limit correctly states maximizing total value by taking fractions. Others focus on minimizing weight or whole items only, which is incorrect.

3. Final Answer:

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

4. Quick Check:

   Fractional knapsack = maximize value with fractions [OK]

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

Common Mistakes:

• Thinking only whole items can be taken
• Confusing minimization with maximization
• Ignoring fractional parts of items

2. Which TypeScript syntax correctly declares an array of objects representing items with weight and value for the fractional knapsack?

easy

A. var items = weight: 10, value: 60; weight: 20, value: 100;

B. let items = (weight: 10, value: 60), (weight: 20, value: 100);

C. items = [{10, 60}, {20, 100}];

D. const items = [{weight: 10, value: 60}, {weight: 20, value: 100}];

Solution

1. Step 1: Check array and object syntax

   const items = [{weight: 10, value: 60}, {weight: 20, value: 100}]; uses correct array of objects syntax with curly braces and key-value pairs.

2. Step 2: Identify syntax errors in other options

   Options A, B, and D have invalid TypeScript syntax for arrays or objects.

3. Final Answer:

   const items = [{weight: 10, value: 60}, {weight: 20, value: 100}]; -> Option D

4. Quick Check:

   Array of objects syntax = const items = [{weight: 10, value: 60}, {weight: 20, value: 100}]; [OK]

Hint: Use square brackets for arrays and curly braces for objects [OK]

Common Mistakes:

• Using parentheses instead of square brackets for arrays
• Missing key names in objects
• Incorrect object literal syntax

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

const items = [
  {weight: 10, value: 60},
  {weight: 20, value: 100},
  {weight: 30, value: 120}
];
const capacity = 50;

function fractionalKnapsack(items, capacity) {
  items.sort((a, b) => (b.value / b.weight) - (a.value / a.weight));
  let totalValue = 0;
  let remaining = capacity;
  for (const item of items) {
    if (item.weight <= remaining) {
      totalValue += item.value;
      remaining -= item.weight;
    } else {
      totalValue += item.value * (remaining / item.weight);
      break;
    }
  }
  return totalValue;
}

console.log(fractionalKnapsack(items, capacity));

medium

A. 200

B. 220

C. 240

D. 180

Solution

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

   Ratios: 60/10=6, 100/20=5, 120/30=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.
   Total value = 60 + 100 + 80 = 240.

3. Step 3: Check options

   Matches 240.

4. Final Answer:

   240 -> Option C

5. Quick Check:

   Sum values with fractions = 240 [OK]

Hint: Sort by value/weight ratio, take full items first [OK]

Common Mistakes:

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

4. Identify the bug in this TypeScript function for fractional knapsack:

function fractionalKnapsack(items, capacity) {
  items.sort((a, b) => (a.value / a.weight) - (b.value / b.weight));
  let totalValue = 0;
  let remaining = capacity;
  for (const item of items) {
    if (item.weight <= remaining) {
      totalValue += item.value;
      remaining -= item.weight;
    } else {
      totalValue += item.value * (remaining / item.weight);
      break;
    }
  }
  return totalValue;
}

medium

A. Using 'let' instead of 'const' for totalValue

B. Sorting order is ascending instead of descending by value/weight

C. Missing return statement

D. Incorrect fraction calculation in else block

Solution

1. Step 1: Analyze sorting function

   The sort compares (a.value/a.weight) - (b.value/b.weight), which sorts ascending by ratio.

2. Step 2: Understand impact on knapsack

   Fractional knapsack requires descending order to pick highest value/weight first. Ascending order causes wrong selection.

3. Final Answer:

   Sorting order is ascending instead of descending by value/weight -> Option B

4. Quick Check:

   Sort descending by ratio [OK]

Hint: Sort descending by value/weight ratio, not ascending [OK]

Common Mistakes:

• Sorting ascending instead of descending
• Ignoring sorting impact on result
• Confusing variable declarations

5. Given items with weights and values: [{weight: 5, value: 30}, {weight: 10, value: 40}, {weight: 20, value: 140}], and a knapsack capacity of 15, what is the maximum value achievable using the fractional knapsack approach?

hard

A. 105

B. 90

C. 120

D. 100

Solution

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

   Ratios: 30/5=6, 40/10=4, 140/20=7.
   Sorted order: item3 (7), item1 (6), item2 (4).

2. Step 2: Fill knapsack capacity 15

   Take item3 partially: since 20 > 15, value = 140 * (15/20) = 105.
   Total value = 105.
   Remaining capacity 0, stop.

3. Final Answer:

   105 -> Option A

4. Quick Check:

   Fractional knapsack max value = 105 [OK]

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

Common Mistakes:

• Not sorting by value/weight ratio
• Trying to take full items exceeding capacity
• Miscomputing fractional values