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

Time Complexity: Partition Equal Subset Sum

O(n * sum)

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 length or sum increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

bool canPartition(int* nums, int numsSize) {
    int sum = 0;
    for (int i = 0; i < numsSize; i++) {
        sum += nums[i];
    }
    if (sum % 2 != 0) return false;
    int target = sum / 2;
    bool dp[target + 1];
    memset(dp, 0, sizeof(dp));
    dp[0] = true;
    for (int i = 0; i < numsSize; i++) {
        for (int j = target; j >= nums[i]; j--) {
            dp[j] = dp[j] || dp[j - nums[i]];
        }
    }
    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

• Primary operation: Nested loops where the outer loop runs over each number and the inner loop runs over possible sums up to half the total sum.
• How many times: Outer loop runs n times (array length), inner loop runs up to sum/2 times.

How Execution Grows With Input

As the array size n or the sum sum grows, the number of operations grows roughly by multiplying these two.

Input Size (n)	Approx. Operations (sum/2)
10	10 * (sum/2)
100	100 * (sum/2)
1000	1000 * (sum/2)

Pattern observation: The time grows linearly with the number of elements and linearly with the half sum value.

Final Time Complexity

Time Complexity: O(n * sum)

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

Common Mistake

[X] Wrong: "The time depends only on the number of elements, so it is O(n)."

[OK] Correct: The time also depends on the sum value because the inner loop runs up to half the sum, so ignoring sum underestimates the time.

Interview Connect

Understanding this time complexity helps you explain how dynamic programming solves subset problems efficiently compared to checking all subsets.

Self-Check

"What if we used recursion with memoization instead of this bottom-up approach? 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 contains any duplicate elements

B. If the array is sorted in ascending order

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

D. If the array contains only positive numbers

Solution

1. Step 1: Understand the problem statement

   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: Identify the key requirement

   We need to check if there exists a subset whose sum is half of the total sum of the array.

3. Final Answer:

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

4. Quick Check:

   Partition Equal Subset Sum = Split array equally [OK]

Hint: Check if total sum is even first to proceed [OK]

Common Mistakes:

• Thinking the problem is about sorting
• Checking for duplicates instead of sums
• Assuming all numbers must be positive

2. Which of the following is the correct way to declare a boolean array in C for dynamic programming in Partition Equal Subset Sum?

easy

A. boolean dp[sum+1];

B. int dp[sum+1];

C. bool dp = new bool[sum+1];

D. bool dp[sum+1];

Solution

1. Step 1: Identify correct C syntax for boolean arrays

   In C, bool is available via stdbool.h and arrays are declared as bool dp[size];.

2. Step 2: Check options for correctness

   bool dp[sum+1]; uses correct syntax. int dp[sum+1]; uses int, which is not boolean. bool dp = new bool[sum+1]; uses Java/C++ style dynamic allocation, not valid in C. boolean dp[sum+1]; uses 'boolean' which is not a C keyword.

3. Final Answer:

   bool dp[sum+1]; -> Option D

4. Quick Check:

   Use bool dp[size]; in C [OK]

Hint: Use stdbool.h and declare bool arrays with bool dp[size]; [OK]

Common Mistakes:

• Using 'boolean' instead of 'bool' in C
• Trying Java-style dynamic allocation in C
• Using int instead of bool for DP array

3. What will be the output of the following code snippet for array nums = {1, 5, 11, 5}?

#include 
#include 

bool canPartition(int* nums, int numsSize) {
    int sum = 0;
    for (int i = 0; i < numsSize; i++) sum += nums[i];
    if (sum % 2 != 0) return false;
    int target = sum / 2;
    bool dp[target + 1];
    for (int i = 0; i <= target; i++) dp[i] = false;
    dp[0] = true;
    for (int i = 0; i < numsSize; i++) {
        for (int j = target; j >= nums[i]; j--) {
            dp[j] = dp[j] || dp[j - nums[i]];
        }
    }
    return dp[target];
}

int main() {
    int nums[] = {1, 5, 11, 5};
    printf("%d\n", canPartition(nums, 4));
    return 0;
}

medium

A. 1

B. 0

C. Compilation error

D. Runtime error

Solution

1. Step 1: Calculate total sum and check parity

   Sum = 1 + 5 + 11 + 5 = 22, which is even, so proceed.

2. Step 2: Use DP to check subset sum equals 11

   The DP array tracks sums achievable. The subset {11} and {1,5,5} both sum to 11, so dp[11] becomes true.

3. Final Answer:

   1 -> Option A

4. Quick Check:

   Subset sum exists = 1 [OK]

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

Common Mistakes:

• Not checking if total sum is even
• Incorrect DP initialization
• Looping forward instead of backward in DP

4. Identify the bug in this Partition Equal Subset Sum code snippet:

bool canPartition(int* nums, int numsSize) {
    int sum = 0;
    for (int i = 0; i < numsSize; i++) sum += nums[i];
    if (sum % 2 != 0) return false;
    int target = sum / 2;
    bool dp[target + 1];
    for (int i = 0; i < target; i++) dp[i] = false;
    dp[0] = true;
    for (int i = 0; i < numsSize; i++) {
        for (int j = target; j >= nums[i]; j--) {
            dp[j] = dp[j] || dp[j - nums[i]];
        }
    }
    return dp[target];
}

medium

A. Loop initializing dp should run till target inclusive

B. The sum calculation is incorrect

C. The inner loop should run forward, not backward

D. dp[0] should be false initially

Solution

1. Step 1: Check dp array initialization loop

   The loop runs from 0 to target-1, missing dp[target] initialization, which should be set to false.

2. Step 2: Correct the loop to include target index

   Change loop condition to i <= target to initialize all dp elements properly.

3. Final Answer:

   Loop initializing dp should run till target inclusive -> Option A

4. Quick Check:

   Initialize dp array fully including dp[target] [OK]

Hint: Initialize dp array fully including last index [OK]

Common Mistakes:

• Forgetting to initialize dp[target]
• Running inner loop forward causing wrong results
• Setting dp[0] to false instead of true

5. Given an array nums = {2, 2, 3, 5}, which of the following modifications will correctly determine if it can be partitioned into two subsets with equal sum using dynamic programming?

hard

A. Change the DP array to track counts of subsets instead of booleans

B. Use a 2D DP array where dp[i][j] means if sum j is possible with first i elements

C. Sort the array first and then apply the existing 1D DP approach

D. Only check if the largest element is less than half the total sum

Solution

1. Step 1: Understand limitations of 1D DP for some cases

   1D DP works but 2D DP is clearer and handles subset sums by tracking elements used.

2. Step 2: Use 2D DP where dp[i][j] shows if sum j is possible using first i elements

   This approach explicitly considers each element and sum, making it easier to debug and understand.

3. Step 3: Sorting or counting subsets is not necessary or sufficient

   Sorting doesn't affect subset sums. Counting subsets is more complex and not required for yes/no answer.

4. Final Answer:

   Use a 2D DP array where dp[i][j] means if sum j is possible with first i elements -> Option B

5. Quick Check:

   2D DP tracks sums per element [OK]

Hint: Use 2D DP for clear subset sum tracking [OK]

Common Mistakes:

• Sorting array expecting it to solve partition
• Counting subsets instead of checking existence
• Only checking largest element against half sum