Generate All Subsets Powerset in DSA Rust - Execution Trace

Concept Flow - Generate All Subsets Powerset

Start with empty subset

↓

Pick next element?

No→Add current subset to result

Yes↓

Include element in subset

→Recurse

↓

Exclude element from subset

→Recurse

↓

Backtrack to previous state

↩Back to Pick next element?

We start with an empty subset and at each step decide to include or exclude the next element, recursively building all subsets until all elements are processed.

Execution Sample

DSA Rust

fn generate_subsets(nums: &[i32], index: usize, current: &mut Vec<i32>, result: &mut Vec<Vec<i32>>) {
    if index == nums.len() {
        result.push(current.clone());
        return;
    }
    current.push(nums[index]);
    generate_subsets(nums, index + 1, current, result);
    current.pop();
    generate_subsets(nums, index + 1, current, result);
}

This Rust function recursively generates all subsets of the input array by including or excluding each element.

Execution Table

Step	Operation	Index	Current Subset	Action	Resulting Subsets
1	Start recursion	0	[]	Decide to include nums[0]=1	[]
2	Include element	1	[1]	Decide to include nums[1]=2	[]
3	Include element	2	[1, 2]	Decide to include nums[2]=3	[]
4	Include element	3	[1, 2, 3]	Index == length, add subset	[[1, 2, 3]]
5	Backtrack	3	[1, 2]	Exclude nums[2]=3	[[1, 2, 3]]
6	Exclude element	3	[1, 2]	Index == length, add subset	[[1, 2, 3], [1, 2]]
7	Backtrack	2	[1]	Exclude nums[1]=2	[[1, 2, 3], [1, 2]]
8	Exclude element	2	[1]	Decide to include nums[2]=3	[[1, 2, 3], [1, 2]]
9	Include element	3	[1, 3]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3]]
10	Backtrack	3	[1]	Exclude nums[2]=3	[[1, 2, 3], [1, 2], [1, 3]]
11	Exclude element	3	[1]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3], [1]]
12	Backtrack	1	[]	Exclude nums[0]=1	[[1, 2, 3], [1, 2], [1, 3], [1]]
13	Exclude element	1	[]	Decide to include nums[1]=2	[[1, 2, 3], [1, 2], [1, 3], [1]]
14	Include element	2	[2]	Decide to include nums[2]=3	[[1, 2, 3], [1, 2], [1, 3], [1]]
15	Include element	3	[2, 3]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3]]
16	Backtrack	3	[2]	Exclude nums[2]=3	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3]]
17	Exclude element	3	[2]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2]]
18	Backtrack	2	[]	Exclude nums[1]=2	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2]]
19	Exclude element	2	[]	Decide to include nums[2]=3	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2]]
20	Include element	3	[3]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3]]
21	Backtrack	3	[]	Exclude nums[2]=3	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3]]
22	Exclude element	3	[]	Index == length, add subset	[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3], []]
23	End recursion	-	-	-	All subsets generated

💡 Recursion ends when index reaches length of input array, all subsets collected.

Variable Tracker

Variable	Start	After Step 4	After Step 6	After Step 11	After Step 17	After Step 22	Final
index	0	3	3	3	3	3	3
current subset	[]	[1, 2, 3]	[1, 2]	[1]	[2]	[]	[]
resulting subsets count	0	1	2	4	6	8	8

Key Moments - 3 Insights

Why do we add the current subset to the result only when index equals the length?

Because at that point, we have made decisions for all elements, so the current subset is complete. See steps 4, 6, 9, 11, 15, 17, 20, 22 in the execution_table.

Why do we push and then pop elements from the current subset?

We push to include an element before recursion and pop to backtrack after exploring that path. This ensures the current subset reflects the correct elements at each step. See steps 2 and 5 for example.

How does the recursion explore all subsets?

At each index, recursion branches into two: one including the element and one excluding it, covering all combinations. This is shown by the alternating include/exclude actions in the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 9, what is the current subset?

A [1, 2]

B [1, 3]

C [2, 3]

D [1, 2, 3]

💡 Hint

Check the 'Current Subset' column at step 9 in the execution_table.

At which step does the recursion first add the empty subset to the result?

A Step 22

B Step 11

C Step 4

D Step 17

💡 Hint

Look for when 'Current Subset' is [] and 'Action' is 'Index == length, add subset' in the execution_table.

If we remove the pop operation after including an element, how would the subsets change?

A Subsets would be empty

B Subsets would be the same

C Subsets would include repeated elements and be incorrect

D Subsets would only include full length subsets

💡 Hint

Refer to the variable_tracker showing how 'current subset' changes after push and pop.

Concept Snapshot

Generate All Subsets (Powerset):
- Use recursion with index to track position
- At each step, include or exclude current element
- When index == length, add current subset to result
- Use push before recurse include, pop after to backtrack
- Result contains all 2^n subsets of input array

Full Transcript

This visualization shows how to generate all subsets of a list using recursion. We start with an empty subset and at each element decide to include or exclude it. The recursion explores both choices, building subsets step by step. When we reach the end of the list, the current subset is complete and added to the results. We use push and pop operations to add and remove elements as we explore different branches. The execution table tracks each step, showing the current subset and the growing list of subsets. Key moments clarify why subsets are added only at the end and how backtracking works. The quiz tests understanding of the subset states and the importance of backtracking. This method generates all 2^n subsets efficiently and clearly.

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