Generate All Subsets Powerset in DSA Python - Time & Space Complexity

Time Complexity: Generate All Subsets Powerset

O(n * 2^n)

Understanding Time Complexity

When we generate all subsets of a set, we want to know how the time needed grows as the set gets bigger.

We ask: How does the work increase when the input size increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def generate_subsets(nums):
    result = []
    def backtrack(start, path):
        result.append(path[:])
        for i in range(start, len(nums)):
            path.append(nums[i])
            backtrack(i + 1, path)
            path.pop()
    backtrack(0, [])
    return result

This code finds all possible subsets of the input list by exploring each choice step-by-step.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls that explore subsets by adding or skipping elements.
• How many times: Each element can be either included or excluded, so the recursion explores 2ⁿ paths.

How Execution Grows With Input

Each new element doubles the number of subsets to explore.

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: The work doubles with each added element, growing very fast.

Final Time Complexity

Time Complexity: O(n * 2n)

This means the time grows very fast because we explore every subset, and each subset can take up to n steps to build.

Common Mistake

[X] Wrong: "Generating subsets is just O(n) because we loop through the list once."

[OK] Correct: We do more than one loop; we explore all combinations, which grow exponentially, not linearly.

Interview Connect

Understanding this complexity helps you explain why some problems take longer and shows you can reason about exponential growth clearly.

Self-Check

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

Practice

1. What does the powerset of a set represent in terms of subsets?

easy

A. Only subsets with even number of elements

B. Only the full set itself

C. Only subsets with odd number of elements

D. All possible subsets including empty and full set

Solution

1. Step 1: Understand the definition of powerset

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

2. Step 2: Compare options with definition

   All possible subsets including empty and full set correctly states all possible subsets are included. Other options limit subsets incorrectly.

3. Final Answer:

   All possible subsets including empty and full set -> Option D

4. Quick Check:

   Powerset = All subsets [OK]

Hint: Remember powerset means every subset possible [OK]

Common Mistakes:

• Thinking powerset excludes empty set
• Confusing powerset with subsets of certain size
• Assuming powerset only includes full set

2. Which of the following Python code snippets correctly initializes an empty list to store subsets for generating a powerset?

easy

A. subsets = set()

B. subsets = {}

C. subsets = []

D. subsets = ()

Solution

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

   Subsets are typically stored in a list because order and duplicates matter for powerset generation.

2. Step 2: Analyze each option

   subsets = [] uses an empty list []. subsets = {} uses a dictionary {}, which is incorrect. subsets = () uses a tuple (), which is immutable. subsets = set() uses a set(), which cannot store duplicates or ordered subsets.

3. Final Answer:

   subsets = [] -> Option C

4. Quick Check:

   Use list [] for subsets storage [OK]

Hint: Use list [] to store multiple subsets [OK]

Common Mistakes:

• Using {} which creates a dictionary, not a list
• Using () which is immutable tuple
• Using set() which loses order and duplicates

3. What is the output of the following code?

def powerset(nums):
    result = [[]]
    for num in nums:
        result += [curr + [num] for curr in result]
    return result

print(powerset([1, 2]))

medium

A. [[], [1], [1, 2]]

B. [[], [1], [2], [1, 2]]

C. [[1], [2], [1, 2]]

D. [[], [2], [1, 2]]

Solution

1. Step 1: Trace initial result and loop iterations

   Start with result = [[]]. For num=1, add subsets: [] + [1] = [1], so result = [[], [1]]. For num=2, add subsets: [] + [2] = [2], [1] + [2] = [1, 2], so result = [[], [1], [2], [1, 2]].

2. Step 2: Confirm final output list

   The final list contains all subsets: empty, single elements, and full set.

3. Final Answer:

   [[], [1], [2], [1, 2]] -> Option B

4. Quick Check:

   All subsets of [1,2] = [[], [1], [2], [1, 2]] [OK]

Hint: Add new element to all existing subsets each loop [OK]

Common Mistakes:

• Missing empty subset [] in output
• Not combining previous subsets with new element
• Confusing order of subsets

4. Find the error in this powerset generation code:

def powerset(nums):
    result = []
    for num in nums:
        for curr in result:
            result.append(curr + [num])
    return result

print(powerset([1, 2]))

medium

A. result list is empty initially, so loop never runs

B. Appending inside loop modifies list during iteration

C. Missing return statement

D. Using list instead of set causes duplicates

Solution

1. Step 1: Check initial value of result

   result is initialized as empty list []. The inner loop for curr in result will not run because result is empty.

2. Step 2: Understand consequence on output

   Since inner loop never runs, no subsets are added. The function returns empty list [].

3. Final Answer:

   result list is empty initially, so loop never runs -> Option A

4. Quick Check:

   Empty result means no subsets generated [OK]

Hint: Initialize result with [[]] to start with empty subset [OK]

Common Mistakes:

• Not initializing result with [[]]
• Modifying list while iterating (can cause bugs)
• Forgetting to add empty subset initially

5. Given a list nums = [1, 2, 2], which code modification correctly generates the powerset without duplicate subsets?

hard

A. Sort nums and skip duplicates during subset generation

B. Use a set to store tuples of subsets to avoid duplicates

C. Use list comprehension without any checks

D. Remove duplicates from nums before generating subsets

Solution

1. Step 1: Understand the problem with duplicates

   Input has duplicates (2 appears twice). Powerset generation can create duplicate subsets if duplicates are not handled.

2. Step 2: Analyze options for handling duplicates

   Use a set to store tuples of subsets to avoid duplicates uses a set of tuples which works but changes data type and order. Sort nums and skip duplicates during subset generation sorts nums and skips duplicates during recursion, which is standard and efficient. Use list comprehension without any checks ignores duplicates. Remove duplicates from nums before generating subsets removes duplicates, which changes the original set.

3. Final Answer:

   Sort nums and skip duplicates during subset generation -> Option A

4. Quick Check:

   Sort and skip duplicates to avoid repeated subsets [OK]

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

Common Mistakes:

• Removing duplicates changes original data meaning
• Using sets loses subset order
• Ignoring duplicates causes repeated subsets