Generate All Subsets Powerset in DSA C# - Time & Space Complexity

Time Complexity: Generate All Subsets Powerset

O(n * 2^n)

Understanding Time Complexity

When we generate all subsets (powerset) of a set, we want to know how the work grows as the set gets bigger.

We ask: How many steps does it take to list every subset?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public IList<IList<int>> Subsets(int[] nums) {
    var result = new List<IList<int>>();
    Generate(0, new List<int>(), nums, result);
    return result;
}

private void Generate(int index, List<int> current, int[] nums, IList<IList<int>> result) {
    if (index == nums.Length) {
        result.Add(new List<int>(current));
        return;
    }
    Generate(index + 1, current, nums, result);
    current.Add(nums[index]);
    Generate(index + 1, current, nums, result);
    current.RemoveAt(current.Count - 1);
}
    

This code generates all subsets of the input array by choosing to include or exclude each element recursively.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls that branch twice at each element.
• How many times: For n elements, there are 2 choices (include/exclude) per element, leading to 2^n calls.

How Execution Grows With Input

Each element doubles the number of subsets, so the work grows very fast.

Input Size (n)	Approx. Operations
10	About 1,024 subsets
20	About 1,048,576 subsets
30	About 1,073,741,824 subsets

Pattern observation: Doubling the input size roughly squares the work, showing exponential growth.

Final Time Complexity

Time Complexity: O(n * 2^n)

This means the time grows exponentially with the number of elements, multiplied by the size of each subset created.

Common Mistake

[X] Wrong: "Generating all subsets is just O(n) because we visit each element once."

[OK] Correct: Each element creates two branches (include or exclude), so the total subsets grow exponentially, not linearly.

Interview Connect

Understanding this exponential growth helps you explain why some problems need careful handling and why generating all subsets is costly but sometimes necessary.

Self-Check

"What if we only wanted subsets of a fixed size k? How would the time complexity change?"

Practice

1. What does the term powerset mean in the context of subsets?

easy

A. Only the subsets with even number of elements

B. All possible subsets of a given set including empty and full set

C. Subsets that contain only one element

D. Subsets that exclude the empty set

Solution

1. Step 1: Understand the definition of powerset

   The powerset of a set is the collection of all subsets, including the empty set and the set itself.

2. Step 2: Compare options with definition

   All possible subsets of a given set including empty and full set correctly describes all subsets including empty and full set. Other options limit subsets incorrectly.

3. Final Answer:

   All possible subsets of a given set including empty and full set -> Option B

4. Quick Check:

   powerset = all subsets [OK]

Hint: Powerset means every subset possible, including empty [OK]

Common Mistakes:

• Thinking powerset excludes empty set
• Confusing powerset with subsets of fixed size
• Ignoring the full set as a subset

2. Which of the following C# code snippets correctly initializes an empty list of subsets for generating a powerset?

easy

A. var subsets = new List();

B. var subsets = new List<int>();

C. var subsets = new List();

D. var subsets = new List<int>(); subsets.Add(new List<int>());

Solution

1. Step 1: Identify the correct data structure for subsets

   Each subset is a list of integers, so the collection of subsets should be a list of lists of integers.

2. Step 2: Check each option's type

   var subsets = new List(); uses List, which is correct. var subsets = new List<int>(); is List<int> (single list), var subsets = new List(); is List (array instead of list), var subsets = new List<int>(); subsets.Add(new List<int>()); mixes types incorrectly.

3. Final Answer:

   var subsets = new List(); -> Option A

4. Quick Check:

   List of list<int> for subsets [OK]

Hint: Use List to hold all subsets [OK]

Common Mistakes:

• Using List<int> instead of List
• Confusing arrays with lists
• Adding elements before initializing list

3. What is the output of the following C# code snippet?

var nums = new int[] {1, 2};
var subsets = new List { new List<int>() };
foreach (var num in nums) {
  int count = subsets.Count;
  for (int i = 0; i < count; i++) {
    var newSubset = new List<int>(subsets[i]) { num };
    subsets.Add(newSubset);
  }
}
foreach (var subset in subsets) {
  Console.Write(string.Join(",", subset) + "|");
}

medium

A. | |1|2|

B. |1|2|1,2|

C. | |1|2|1,2|

D. |1|2|

Solution

1. Step 1: Trace initial subsets list

   Initially subsets contains one empty list: [[]].

2. Step 2: Process first number 1

   For num=1, we add subsets: [1] to subsets. Now subsets = [[], [1]].

3. Step 3: Process second number 2

   For num=2, we add subsets: [2] and [1,2]. Now subsets = [[], [1], [2], [1,2]].

4. Step 4: Print all subsets

   Empty subset prints as "|", so output is "| |1|2|1,2|".

5. Final Answer:

   | |1|2|1,2| -> Option C

6. Quick Check:

   All subsets printed with empty first [OK]

Hint: Empty subset prints as empty string before others [OK]

Common Mistakes:

• Ignoring empty subset in output
• Not copying subsets before adding new element
• Miscounting subsets during iteration

4. Identify the error in the following C# code for generating subsets:

var nums = new int[] {1, 2};
var subsets = new List();
foreach (var num in nums) {
  int count = subsets.Count;
  for (int i = 0; i < count; i++) {
    var newSubset = new List<int>(subsets[i]) { num };
    subsets.Add(newSubset);
  }
}

medium

A. subsets list is not initialized with empty subset

B. Using List<int> instead of List

C. Incorrect loop range in inner for loop

D. Adding elements to subsets inside foreach causes runtime error

Solution

1. Step 1: Check initialization of subsets

   subsets is empty initially, so count is 0 and inner loop never runs.

2. Step 2: Understand consequence

   Without starting with empty subset, no new subsets are generated.

3. Final Answer:

   subsets list is not initialized with empty subset -> Option A

4. Quick Check:

   Empty subset needed to start [OK]

Hint: Always start subsets with empty list to generate powerset [OK]

Common Mistakes:

• Not adding empty subset before loop
• Assuming subsets auto-initialize
• Modifying list while iterating without base case

5. Given a list nums = new List<int> {1, 2, 2}, which modification to the standard powerset generation code will correctly generate all unique subsets without duplicates?

hard

A. Use a HashSet to store subsets as strings to avoid duplicates

B. Add all subsets without any check; duplicates will be filtered later

C. Use a Dictionary to map counts and generate subsets

D. Sort nums and skip adding duplicate elements in inner loop

Solution

1. Step 1: Understand duplicate problem

   Input has duplicates (2 appears twice), so normal powerset code generates duplicate subsets.

2. Step 2: Identify correct approach

   Sorting nums allows skipping duplicates during subset generation to avoid repeated subsets.

3. Step 3: Evaluate options

   Sort nums and skip adding duplicate elements in inner loop correctly describes sorting and skipping duplicates. Use a HashSet to store subsets as strings to avoid duplicates is workaround but less efficient. Add all subsets without any check; duplicates will be filtered later allows duplicates. Use a Dictionary to map counts and generate subsets is unrelated.

4. Final Answer:

   Sort nums and skip adding duplicate elements in inner loop -> Option D

5. Quick Check:

   Sort and skip duplicates to avoid repeats [OK]

Hint: Sort input and skip duplicates while generating subsets [OK]

Common Mistakes:

• Not sorting input before generating subsets
• Trying to remove duplicates after generation
• Using complex data structures unnecessarily