Unbounded Knapsack Problem in DSA Typescript - Time & Space Complexity

Time Complexity: Unbounded Knapsack Problem

O(n * m)

Understanding Time Complexity

We want to understand how the time needed to solve the Unbounded Knapsack problem grows as the input size increases.

Specifically, how the number of items and the knapsack capacity affect the work done.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function unboundedKnapsack(capacity: number, weights: number[], values: number[]): number {
  const dp = new Array(capacity + 1).fill(0);
  for (let c = 1; c <= capacity; c++) {
    for (let i = 0; i < weights.length; i++) {
      if (weights[i] <= c) {
        dp[c] = Math.max(dp[c], dp[c - weights[i]] + values[i]);
      }
    }
  }
  return dp[capacity];
}
    

This code finds the maximum value achievable with unlimited copies of given items within the capacity.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Nested loops - outer loop over capacity, inner loop over items.
• How many times: Outer loop runs capacity times, inner loop runs number of items times for each capacity.

How Execution Grows With Input

As capacity or number of items grows, the total operations increase by multiplying these two factors.

Input Size (capacity n, items m)	Approx. Operations
n=10, m=5	About 50 operations
n=100, m=5	About 500 operations
n=1000, m=5	About 5000 operations

Pattern observation: Operations grow roughly by multiplying capacity and number of items.

Final Time Complexity

Time Complexity: O(n * m)

This means the time needed grows proportionally with both the knapsack capacity and the number of items.

Common Mistake

[X] Wrong: "The time depends only on the number of items, not the capacity."

[OK] Correct: The outer loop runs for each capacity value, so capacity directly affects total work.

Interview Connect

Understanding this time complexity helps you explain how dynamic programming solves problems efficiently by breaking them into smaller parts.

Self-Check

"What if we used recursion with memoization instead of this bottom-up approach? How would the time complexity change?"

Practice

1. What is the main difference between the 0/1 Knapsack Problem and the Unbounded Knapsack Problem?

easy

A. In Unbounded Knapsack, you can use unlimited copies of each item.

B. In Unbounded Knapsack, items cannot be repeated.

C. Unbounded Knapsack only works with fractional items.

D. Unbounded Knapsack has a fixed number of items to choose from.

Solution

1. Step 1: Understand the item usage rules

   In 0/1 Knapsack, each item can be chosen at most once. In Unbounded Knapsack, items can be chosen unlimited times.

2. Step 2: Compare the problem constraints

   Unbounded Knapsack allows repeated use of the same item, unlike 0/1 Knapsack.

3. Final Answer:

   In Unbounded Knapsack, you can use unlimited copies of each item. -> Option A

4. Quick Check:

   Unlimited item usage = In Unbounded Knapsack, you can use unlimited copies of each item. [OK]

Hint: Unbounded means unlimited items allowed [OK]

Common Mistakes:

• Confusing unbounded with fractional knapsack
• Thinking items cannot repeat
• Assuming fixed item count

2. Which of the following TypeScript function signatures correctly represents the Unbounded Knapsack function?

easy

A. function unboundedKnapsack(W: number, wt: string[], val: number[], n: number): number

B. function unboundedKnapsack(W: string, wt: number[], val: number[], n: number): number

C. function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number

D. function unboundedKnapsack(W: number, wt: number[], val: number[], n: string): number

Solution

1. Step 1: Check parameter types for correctness

   Weight capacity W should be a number, weights wt and values val arrays of numbers, and n the number of items as a number.

2. Step 2: Identify the correct signature

   function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number matches all types correctly. Others have wrong types like string instead of number.

3. Final Answer:

   function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number -> Option C

4. Quick Check:

   Correct types for all parameters = function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number [OK]

Hint: Check parameter types carefully [OK]

Common Mistakes:

• Using string instead of number for weights or capacity
• Mixing array types
• Incorrect return type

3. What will be the output of the following TypeScript code snippet for Unbounded Knapsack?

const val = [10, 40, 50];
const wt = [1, 3, 4];
const W = 6;

function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number {
  const dp = Array(W + 1).fill(0);
  for (let i = 1; i <= W; i++) {
    for (let j = 0; j < n; j++) {
      if (wt[j] <= i) {
        dp[i] = Math.max(dp[i], dp[i - wt[j]] + val[j]);
      }
    }
  }
  return dp[W];
}

console.log(unboundedKnapsack(W, wt, val, val.length));

medium

A. 90

B. 100

C. 120

D. 80

Solution

1. Step 1: Understand the DP array construction

   We build dp array from 0 to W=6. For each capacity, we check all items and update dp[i] with max value including unlimited items.

2. Step 2: Calculate dp values step-by-step

   dp[1]=max(0, dp[0]+10)=10
   dp[2]=max(0, dp[1]+10)=20
   dp[3]=max(dp[3], dp[0]+40)=40, also dp[3]=max(40, dp[2]+10)=40
   dp[4]=max(dp[4], dp[1]+40)=50, also dp[4]=max(50, dp[0]+50)=50
   dp[5]=max(dp[5], dp[2]+40)=60, also dp[5]=max(60, dp[1]+50)=60
   dp[6]=max(dp[6], dp[3]+40)=80, also dp[6]=max(80, dp[2]+50)=90

3. Final Answer:

   90 -> Option A

4. Quick Check:

   Max value for W=6 = 90 [OK]

Hint: Build dp array incrementally for each capacity [OK]

Common Mistakes:

• Not considering multiple copies of items
• Incorrect dp array updates
• Mixing 0/1 and unbounded logic

4. Identify the bug in this Unbounded Knapsack TypeScript code snippet:

function unboundedKnapsack(W: number, wt: number[], val: number[], n: number): number {
  const dp = Array(W + 1).fill(0);
  for (let j = 0; j < n; j++) {
    for (let i = 1; i <= W; i++) {
      if (wt[j] <= i) {
        dp[i] = Math.max(dp[i], dp[i - wt[j]] + val[j]);
      }
    }
  }
  return dp[W];
}

medium

A. The condition wt[j] <= i should be wt[j] < i

B. Initialization of dp array is incorrect

C. Return statement should be dp[0] instead of dp[W]

D. Loops are reversed; inner loop should be over items, outer over capacity

Solution

1. Step 1: Analyze loop order for unbounded knapsack

   For unbounded knapsack, outer loop should iterate over capacities (i), inner loop over items (j) to allow multiple uses of items.

2. Step 2: Identify the bug in loop order

   Here, outer loop is over items and inner over capacity, which is incorrect and leads to wrong results.

3. Final Answer:

   Loops are reversed; inner loop should be over items, outer over capacity -> Option D

4. Quick Check:

   Correct loop order = capacity outer, items inner [OK]

Hint: Outer loop on capacity for unbounded knapsack [OK]

Common Mistakes:

• Swapping loop order
• Wrong dp initialization
• Incorrect return index

5. Given items with values val = [5, 10, 15] and weights wt = [1, 3, 4], and a knapsack capacity W = 7, what is the maximum value achievable using the Unbounded Knapsack approach?

hard

A. 25

B. 30

C. 35

D. 40

Solution

1. Step 1: Calculate dp array values for capacities 1 to 7

   dp[0]=0
   dp[1]=max(0, dp[0]+5)=5
   dp[2]=max(0, dp[1]+5)=10
   dp[3]=max(dp[3], dp[0]+10)=10, also dp[3]=max(10, dp[2]+5)=10
   dp[4]=max(dp[4], dp[1]+10)=15, also dp[4]=max(15, dp[0]+15)=15
   dp[5]=max(dp[5], dp[2]+10)=20, also dp[5]=max(20, dp[1]+15)=20
   dp[6]=max(dp[6], dp[3]+10)=20, also dp[6]=max(20, dp[2]+15)=25
   dp[7]=max(dp[7], dp[4]+10)=25, also dp[7]=max(25, dp[3]+15)=25

2. Step 2: Find maximum value at capacity 7

   The maximum value achievable is 25, but check if better combinations exist:
   Try dp[7] = dp[3] + val[2] = 10 + 15 = 25
   Try dp[7] = dp[4] + val[1] = 15 + 10 = 25
   Try dp[7] = dp[6] + val[0] = 25 + 5 = 30 (better)
   Update dp[7] = 30
   Try dp[7] = dp[3] + val[2] = 10 + 15 = 25 (less than 30)
   So final dp[7] = 30

3. Final Answer:

   30 -> Option B

4. Quick Check:

   Maximum value for W=7 = 30 [OK]

Hint: Check all item combinations for each capacity [OK]

Common Mistakes:

• Ignoring multiple item usage
• Stopping at first max value found
• Not updating dp array correctly