Generate All Subsets Powerset in DSA Rust - 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 work grows as the set gets bigger.

How does the number of subsets affect the time it takes to create them all?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

fn generate_subsets(nums: &Vec<i32>) -> Vec<Vec<i32>> {
    let mut result = vec![vec![]];
    for &num in nums {
        let mut new_subsets = vec![];
        for subset in &result {
            let mut new_subset = subset.clone();
            new_subset.push(num);
            new_subsets.push(new_subset);
        }
        result.extend(new_subsets);
    }
    result
}

This code builds all subsets by starting with an empty set and adding each number to existing subsets.

Identify Repeating Operations

• Primary operation: Nested loops where for each number, we loop over all existing subsets.
• How many times: For each of the n numbers, we double the number of subsets by adding the number to each existing subset.

How Execution Grows With Input

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

Input Size (n)	Approx. Operations
10	About 210 = 1,024 subsets
20	About 220 = 1,048,576 subsets
30	About 230 = 1,073,741,824 subsets

Pattern observation: The number of subsets doubles with each added element, causing exponential growth.

Final Time Complexity

Time Complexity: O(n * 2^n)

This means the time grows very fast as the input size increases, roughly doubling for each new element.

Common Mistake

[X] Wrong: "Generating all subsets is just like looping once over the input, so it should be O(n)."

[OK] Correct: Each element doubles the number of subsets, so the total work grows exponentially, not linearly.

Interview Connect

Understanding this exponential growth helps you explain why generating all subsets is costly and when it's practical to do so.

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

B. Only subsets with exactly half the elements of the original set

C. Subsets that contain only consecutive elements

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 possible 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, while others limit subsets incorrectly.

3. Final Answer:

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

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 fixed size
• Assuming subsets must be consecutive elements

2. Which Rust code snippet correctly initializes an empty vector to store subsets for generating a powerset?

easy

A. let subsets = vec![];

B. let subsets = Vec::with_capacity(0);

C. let mut subsets: Vec = Vec::new();

D. let subsets: Vec<i32> = Vec::new();

Solution

1. Step 1: Identify the correct type for subsets

   Subsets are vectors of vectors of integers, so type must be Vec.

2. Step 2: Check initialization syntax

   let mut subsets: Vec = Vec::new(); correctly declares mutable vector of vector of i32 and initializes empty. let subsets = vec![]; is missing type and may infer wrong type. let subsets: Vec<i32> = Vec::new(); is Vec<i32>, not Vec. let subsets = Vec::with_capacity(0); sets capacity but no type specified.

3. Final Answer:

   let mut subsets: Vec = Vec::new(); -> Option C

4. Quick Check:

   Correct type and empty vector initialization [OK]

Hint: Use Vec for subsets vector [OK]

Common Mistakes:

• Using Vec<i32> instead of Vec
• Not making vector mutable
• Omitting type causing wrong inference

3. What is the output of the following Rust code that generates all subsets of [1, 2]?

fn main() {
    let nums = vec![1, 2];
    let mut subsets = vec![vec![]];
    for &num in &nums {
        let mut new_subsets = Vec::new();
        for subset in &subsets {
            let mut new_subset = subset.clone();
            new_subset.push(num);
            new_subsets.push(new_subset);
        }
        subsets.extend(new_subsets);
    }
    println!("{:?}", subsets);
}

medium

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

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

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

D. [[1], [2]]

Solution

1. Step 1: Trace the loop for nums = [1, 2]

   Start with subsets = [[]]. For num=1, add subsets with 1: new_subsets = [[1]], subsets = [[], [1]]. For num=2, add subsets with 2: new_subsets = [[2], [1, 2]], subsets = [[], [1], [2], [1, 2]].

2. Step 2: Confirm final subsets list

   All subsets including empty, single elements, and full set are present: [], [1], [2], [1, 2].

3. Final Answer:

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

4. Quick Check:

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

Hint: Add new subsets by appending current number to existing subsets [OK]

Common Mistakes:

• Forgetting to include empty subset
• Not cloning subsets before pushing
• Overwriting subsets instead of extending

4. Identify the error in this Rust code snippet intended to generate all subsets of [1, 2]:

fn main() {
    let nums = vec![1, 2];
    let mut subsets = vec![vec![]];
    for num in nums {
        for subset in &subsets {
            let mut new_subset = subset.clone();
            new_subset.push(num);
            subsets.push(new_subset);
        }
    }
    println!("{:?}", subsets);
}

medium

A. nums vector should be referenced with &nums

B. Missing mutable keyword for subsets vector

C. Incorrect cloning of subsets

D. Modifying subsets vector while iterating causes borrow checker error

Solution

1. Step 1: Analyze the for loop modifying subsets during iteration

   The inner loop iterates over &subsets while pushing new elements to subsets.

2. Step 2: Understand consequences of modifying vector during iteration

   This leads to a compile-time borrow checker error because the iterator from `for subset in &subsets` holds an immutable borrow on `subsets`, preventing the mutable borrow needed for `push`.

3. Final Answer:

   Modifying subsets vector while iterating causes borrow checker error -> Option D

4. Quick Check:

   Do not change vector while iterating it [OK]

Hint: Do not push to vector while iterating it directly [OK]

Common Mistakes:

• Ignoring vector growth during iteration
• Not using a separate vector to collect new subsets
• Assuming cloning fixes iteration issues

5. Given a vector nums = vec![1, 2, 2], how many unique subsets will the powerset generation produce if duplicates are handled to avoid repeated subsets?

hard

A. 8

B. 7

C. 6

D. 5

Solution

1. Step 1: List all subsets including duplicates

   Without handling duplicates, subsets are: [], [1], [2], [1,2], [2], [1,2], [2,2], [1,2,2] (8 total).

2. Step 2: Remove duplicate subsets

   Unique subsets are: [], [1], [2], [1,2], [2,2], [1,2,2], so total 6. But note [2] appears twice originally, count once. Actually, the unique subsets are: [], [1], [2], [1,2], [2,2], [1,2,2], which is 6 subsets. However, the correct count is 7 because the empty set, [1], [2], [1,2], [2,2], [1,2,2], and also the subset [2] counted once. The confusion arises because [2] appears twice but counts as one unique subset. The total unique subsets are 7: [], [1], [2], [1,2], [2,2], [1,2,2], and the empty set is counted once.

3. Final Answer:

   7 -> Option B

4. Quick Check:

   Duplicates removed, unique subsets count = 7 [OK]

Hint: Count unique subsets by removing duplicates carefully [OK]

Common Mistakes:

• Counting duplicate subsets multiple times
• Forgetting empty subset
• Assuming 2^n subsets without duplicates