Partition Equal Subset Sum in DSA Typescript - Time & Space Complexity

Time Complexity: Partition Equal Subset Sum

O(n * s)

Understanding Time Complexity

We want to know how the time needed to check if an array can be split into two equal sum parts grows as the array gets bigger.

How does the number of steps change when the array size or sum increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function canPartition(nums: number[]): boolean {
  const total = nums.reduce((a, b) => a + b, 0);
  if (total % 2 !== 0) return false;
  const target = total / 2;
  const dp = new Array(target + 1).fill(false);
  dp[0] = true;
  for (const num of nums) {
    for (let j = target; j >= num; j--) {
      dp[j] = dp[j] || dp[j - num];
    }
  }
  return dp[target];
}
    

This code checks if the array can be split into two parts with equal sums using a dynamic programming approach.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Nested loops where the outer loop goes through each number and the inner loop goes through possible sums up to half the total.
• How many times: Outer loop runs once per number (n times), inner loop runs up to target sum (s times).

How Execution Grows With Input

As the number of elements (n) or the target sum (s) grows, the steps increase roughly by multiplying these two.

Input Size (n)	Approx. Operations (n * s)
10 (sum ~ 50)	~ 10 * 25 = 250
100 (sum ~ 500)	~ 100 * 250 = 25,000
1000 (sum ~ 5000)	~ 1000 * 2500 = 2,500,000

Pattern observation: The steps grow roughly by multiplying the number of elements and half the total sum, so it grows faster if either increases.

Final Time Complexity

Time Complexity: O(n * s)

This means the time needed grows with both the number of elements and the half of the total sum.

Common Mistake

[X] Wrong: "The time depends only on the number of elements (n)."

[OK] Correct: The sum value also affects how many steps the inner loop runs, so both n and the sum matter.

Interview Connect

Understanding this time complexity helps you explain how dynamic programming balances between input size and value range, a key skill in problem solving.

Self-Check

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

Practice

1. What does the Partition Equal Subset Sum problem ask you to find?

easy

A. If the array can be split into two subsets with equal sums

B. If the array contains any duplicate elements

C. If the array is sorted in ascending order

D. If the array has an even number of elements

Solution

1. Step 1: Understand the problem goal

   The problem asks whether the given array can be divided into two parts where the sum of elements in both parts is the same.

2. Step 2: Clarify what is being checked

   It is not about duplicates, sorting, or count of elements, but about equal sum partitioning.

3. Final Answer:

   If the array can be split into two subsets with equal sums -> Option A

4. Quick Check:

   Partition Equal Subset Sum = Split into equal sum subsets [OK]

Hint: Focus on equal sum subsets, not element order or duplicates [OK]

Common Mistakes:

• Thinking it checks for duplicates
• Confusing with sorting problems
• Assuming it checks array length parity

2. Which of the following TypeScript function signatures correctly represents a function to solve Partition Equal Subset Sum?

easy

A. function canPartition(nums: boolean[]): boolean

B. function canPartition(nums: string): number

C. function canPartition(nums: number): string

D. function canPartition(nums: number[]): boolean

Solution

1. Step 1: Identify correct parameter type

   The input is an array of numbers, so nums: number[] is correct.

2. Step 2: Identify correct return type

   The function returns true or false, so return type is boolean.

3. Final Answer:

   function canPartition(nums: number[]): boolean -> Option D

4. Quick Check:

   Input array of numbers, output boolean [OK]

Hint: Input is number array, output is boolean for yes/no [OK]

Common Mistakes:

• Using wrong input type like string or boolean array
• Returning number instead of boolean
• Incorrect function signature syntax

3. What is the output of the following TypeScript code?

function canPartition(nums: number[]): boolean {
  const total = nums.reduce((a, b) => a + b, 0);
  if (total % 2 !== 0) return false;
  const target = total / 2;
  const dp = new Array(target + 1).fill(false);
  dp[0] = true;
  for (const num of nums) {
    for (let j = target; j >= num; j--) {
      dp[j] = dp[j] || dp[j - num];
    }
  }
  return dp[target];
}

console.log(canPartition([1, 5, 11, 5]));

medium

A. Runtime error

B. true

C. undefined

D. false

Solution

1. Step 1: Calculate total sum and check parity

   The sum is 1+5+11+5 = 22, which is even, so continue.

2. Step 2: Use dynamic programming to check subset sum

   The code checks if a subset sums to 11 (half of 22). The subset [11] or [5,5,1] sums to 11, so dp[11] becomes true.

3. Final Answer:

   true -> Option B

4. Quick Check:

   Sum even and subset exists = true [OK]

Hint: Check if half sum subset exists using dp array [OK]

Common Mistakes:

• Not checking if total sum is even
• Confusing dp array update direction
• Assuming false because of large numbers

4. Find the bug in this TypeScript code for Partition Equal Subset Sum:

function canPartition(nums: number[]): boolean {
  const total = nums.reduce((a, b) => a + b);
  if (total % 2 !== 0) return false;
  const target = total / 2;
  const dp = new Array(target + 1).fill(false);
  dp[0] = true;
  for (const num of nums) {
    for (let j = num; j <= target; j++) {
      dp[j] = dp[j] || dp[j - num];
    }
  }
  return dp[target];
}

medium

A. Missing initial value in reduce causes error on empty array

B. dp array size is incorrect

C. Loop direction in inner for loop should be reversed

D. No bug, code is correct

Solution

1. Step 1: Check reduce usage

   Reduce lacks initial value, but if array is non-empty, it works. Not the main bug.

2. Step 2: Check inner loop direction

   The inner loop runs from num to target increasing, but it should run backward from target down to num to avoid using updated dp values in the same iteration.

3. Final Answer:

   Loop direction in inner for loop should be reversed -> Option C

4. Quick Check:

   Inner loop must go backward to avoid dp overwrite [OK]

Hint: Inner loop for dp must run backward to prevent reuse [OK]

Common Mistakes:

• Not reversing inner loop causes wrong dp updates
• Assuming reduce always needs initial value
• Ignoring dp array size correctness

5. Given the array [2, 2, 3, 7], which of the following subsets can be one half of the partition if the array can be split into two equal sum subsets?

hard

A. [2, 2, 3]

B. [2, 3]

C. [5]

D. [3, 5]

Solution

1. Step 1: Calculate total sum

   Sum = 2 + 2 + 3 + 7 = 14. Half sum = 7.

2. Step 2: Find subset summing to 7

   Check each option:
   A: [2,3] = 5 ≠ 7
   B: [2,2,3] = 7 == 7
   C: [5] = 5 ≠ 7 (5 not in array)
   D: [3,5] = 8 ≠ 7 (5 not in array)
   B works, with remaining [7] also 7.

3. Final Answer:

   [2, 2, 3] -> Option A

4. Quick Check:

   Sum half is 7, B sums to 7 [OK]

Hint: Sum array, find subset with half sum [OK]

Common Mistakes:

• Not calculating half sum correctly
• Choosing subsets that don't sum to half
• Ignoring that partition may not exist