Partition Equal Subset Sum in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Partition Equal Subset Sum problem?

To determine if a given array of positive integers can be split into two subsets with equal sums.

beginner

Why do we check if the total sum of the array is even in the Partition Equal Subset Sum problem?

Because if the total sum is odd, it's impossible to split the array into two equal sum subsets.

intermediate

What technique is commonly used to solve the Partition Equal Subset Sum problem efficiently?

Dynamic Programming, specifically a subset sum approach to check if a subset with sum equal to half of total sum exists.

intermediate

In the dynamic programming solution, what does the boolean DP table represent?

DP[i][j] is true if a subset of the first i numbers has a sum equal to j; otherwise false.

intermediate

What is the time complexity of the dynamic programming solution for Partition Equal Subset Sum?

O(n * sum/2), where n is the number of elements and sum is the total sum of the array.

What should you do first when solving Partition Equal Subset Sum?

A Check if the total sum of the array is even

B Sort the array

C Find the maximum element

D Reverse the array

If the total sum is odd, it's impossible to split into two equal subsets.

What does a true value in DP[i][j] mean in the dynamic programming table?

A Sum j is impossible with first i elements

B Subset of first j elements sums to i

C Subset of first i elements sums to j

D Sum i is impossible with first j elements

DP[i][j] means a subset of first i elements can sum to j.

If the total sum is 20, what sum should we look for in the subset sum problem?

A 5

B 20

C 15

D 10

We look for a subset with sum equal to half the total sum, which is 10.

Which algorithmic approach is best suited for Partition Equal Subset Sum?

A Dynamic Programming

B Greedy

C Divide and Conquer

D Backtracking only

Dynamic Programming efficiently solves the subset sum problem.

What happens if no subset sums to half the total sum?

A The array can be partitioned anyway

B The array cannot be partitioned into equal subsets

C The array is empty

D The array has duplicates

If no subset sums to half, equal partition is impossible.

Explain the steps to solve the Partition Equal Subset Sum problem using dynamic programming.

Think about subset sum problem and how to build the DP table.

Describe why the Partition Equal Subset Sum problem can be reduced to a subset sum problem.

Focus on the relationship between partition and subset sums.

Practice

1. What does the Partition Equal Subset Sum problem ask you to determine about a given array of integers?

easy

A. If the array is sorted in ascending order

B. If the array contains any duplicate elements

C. If the array contains only positive numbers

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

Solution

1. Step 1: Understand the problem statement

   The problem asks whether the 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 positivity, but about equal sum partitioning.

3. Final Answer:

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

4. Quick Check:

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

Hint: Focus on splitting array into two equal sum parts [OK]

Common Mistakes:

• Confusing with checking duplicates
• Thinking about sorting order
• Assuming all numbers must be positive

2. Which of the following C# code snippets correctly initializes a boolean array named dp of size sum + 1 for the Partition Equal Subset Sum problem?

easy

A. bool[] dp = new bool[sum];

B. bool[] dp = new bool(sum + 1);

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

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

Solution

1. Step 1: Understand array initialization syntax in C#

   Arrays are initialized with square brackets and the size inside new[]. For size sum+1, use new bool[sum + 1].

2. Step 2: Check each option

   bool[] dp = new bool[sum]; misses +1, bool dp = new bool[sum + 1]; uses wrong type declaration, bool[] dp = new bool(sum + 1); uses parentheses which is invalid.

3. Final Answer:

   bool[] dp = new bool[sum + 1]; -> Option C

4. Quick Check:

   Correct C# array syntax = bool[] dp = new bool[sum + 1]; [OK]

Hint: Use square brackets and new keyword for arrays in C# [OK]

Common Mistakes:

• Forgetting +1 for array size
• Using parentheses instead of square brackets
• Declaring array as single bool variable

3. What will be the output of the following C# code snippet for the input array nums = new int[] {1, 5, 11, 5};?

bool CanPartition(int[] nums) {
    int sum = 0;
    foreach (int num in nums) sum += num;
    if (sum % 2 != 0) return false;
    int target = sum / 2;
    bool[] dp = new bool[target + 1];
    dp[0] = true;
    foreach (int num in nums) {
        for (int i = target; i >= num; i--) {
            dp[i] = dp[i] || dp[i - num];
        }
    }
    return dp[target];
}

Console.WriteLine(CanPartition(nums));

medium

A. true

B. false

C. Compile-time error

D. Runtime exception

Solution

1. Step 1: Calculate total sum and check parity

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

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

   Target = 11. The dp array tracks if sum i is achievable. The code finds that subset {11} or {5,5,1} sums to 11, so dp[11] = true.

3. Final Answer:

   true -> Option A

4. Quick Check:

   Sum even and subset sum found = true [OK]

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

Common Mistakes:

• Not checking if sum is even
• Incorrect dp array update direction
• Returning false prematurely

4. Identify the bug in the following C# code snippet for Partition Equal Subset Sum and select the correct fix:

bool CanPartition(int[] nums) {
    int sum = 0;
    foreach (int num in nums) sum += num;
    if (sum % 2 != 0) return false;
    int target = sum / 2;
    bool[] dp = new bool[target];
    dp[0] = true;
    foreach (int num in nums) {
        for (int i = target; i >= num; i--) {
            dp[i] = dp[i] || dp[i - num];
        }
    }
    return dp[target];
}

medium

A. Initialize dp array with false values explicitly

B. Change dp array size to target + 1

C. Change loop to iterate from 0 to target

D. Remove the dp[0] = true assignment

Solution

1. Step 1: Check dp array size and indexing

   dp array is declared with size target, but dp[target] is accessed, causing out-of-range error.

2. Step 2: Fix array size to include target index

   Change dp size to target + 1 to safely access dp[target].

3. Final Answer:

   Change dp array size to target + 1 -> Option B

4. Quick Check:

   Array size must cover all indices used [OK]

Hint: Array size must be one more than max index accessed [OK]

Common Mistakes:

• Ignoring array index out of range
• Changing loop direction unnecessarily
• Removing dp[0] initialization

5. Given the array nums = new int[] {2, 2, 3, 5}, which of the following statements is true regarding the Partition Equal Subset Sum problem?

hard

A. The array cannot be partitioned into two subsets with equal sum

B. The array can be partitioned into two subsets with equal sum: {2,3} and {2,5}

C. The array can be partitioned into two subsets with equal sum: {2,2,3} and {5}

D. The array can be partitioned into two subsets with equal sum: {3,5} and {2,2}

Solution

1. Step 1: Calculate total sum of array elements

   Sum = 2 + 2 + 3 + 5 = 12, which is even, so partition might be possible.

2. Step 2: Check possible subsets for half sum (6)

   Half sum = 6. Possible subsets: {2,2,2} no (only two 2s), {3,3} no (only one 3), {5,1} no 1 missing. No subset sums to 6 exactly.

3. Step 3: Verify given options

   The array can be partitioned into two subsets with equal sum: {2,3} and {2,5} sums: {2,3}=5 and {2,5}=7 (not equal). The array can be partitioned into two subsets with equal sum: {2,2,3} and {5} sums: {2,2,3}=7 and {5}=5 (not equal). The array can be partitioned into two subsets with equal sum: {3,5} and {2,2} sums: {3,5}=8 and {2,2}=4 (not equal).

4. Final Answer:

   The array cannot be partitioned into two subsets with equal sum -> Option A

5. Quick Check:

   Sum even but no subset sums to half sum = false [OK]

Hint: Check if any subset sums exactly to half total sum [OK]

Common Mistakes:

• Assuming even sum means partition always possible
• Incorrectly summing subsets
• Ignoring subset element counts