Generate All Subsets Powerset in DSA Javascript - Execution Trace

Concept Flow - Generate All Subsets Powerset

Start with empty subset

↓

Pick next element?

No→Done: All subsets generated

Yes↓

Add element to current subset

↓

Save current subset

↓

Recurse with next element

↓

Backtrack: Remove element

↩Back to Pick next element?

Start with an empty subset. For each element, choose to include or exclude it, save the subset, and recurse until all subsets are generated.

Execution Sample

DSA Javascript

function generateSubsets(nums) {
  const result = [];
  function backtrack(start, subset) {
    result.push([...subset]);
    for (let i = start; i < nums.length; i++) {
      subset.push(nums[i]);
      backtrack(i + 1, subset);
      subset.pop();
    }
  }
  backtrack(0, []);
  return result;
}

Generates all subsets (powerset) of the input array nums using backtracking.

Execution Table

Step	Operation	Current Index (start)	Current Subset	Action	Resulting Subsets
1	Start backtrack	0	[]	Add [] to result	[[]]
2	Loop i=0	0	[]	Add nums[0]=1	[[], [1]]
3	Recurse backtrack	1	[1]	Add [1] to result	[[], [1]]
4	Loop i=1	1	[1]	Add nums[1]=2	[[], [1], [1,2]]
5	Recurse backtrack	2	[1,2]	Add [1,2] to result	[[], [1], [1,2]]
6	Loop i=2	2	[1,2]	Add nums[2]=3	[[], [1], [1,2], [1,2,3]]
7	Recurse backtrack	3	[1,2,3]	Add [1,2,3] to result	[[], [1], [1,2], [1,2,3]]
8	Loop ends	3	[1,2,3]	Backtrack: remove 3	[[], [1], [1,2], [1,2,3]]
9	Backtrack	2	[1,2]	Backtrack: remove 2	[[], [1], [1,2], [1,2,3]]
10	Loop i=2	1	[1]	Add nums[2]=3	[[], [1], [1,2], [1,2,3], [1,3]]
11	Recurse backtrack	3	[1,3]	Add [1,3] to result	[[], [1], [1,2], [1,2,3], [1,3]]
12	Loop ends	3	[1,3]	Backtrack: remove 3	[[], [1], [1,2], [1,2,3], [1,3]]
13	Backtrack	1	[1]	Backtrack: remove 1	[[], [1], [1,2], [1,2,3], [1,3]]
14	Loop i=1	0	[]	Add nums[1]=2	[[], [1], [1,2], [1,2,3], [1,3], [2]]
15	Recurse backtrack	2	[2]	Add [2] to result	[[], [1], [1,2], [1,2,3], [1,3], [2]]
16	Loop i=2	2	[2]	Add nums[2]=3	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3]]
17	Recurse backtrack	3	[2,3]	Add [2,3] to result	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3]]
18	Loop ends	3	[2,3]	Backtrack: remove 3	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3]]
19	Backtrack	2	[2]	Backtrack: remove 2	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3]]
20	Loop i=2	0	[]	Add nums[2]=3	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3], [3]]
21	Recurse backtrack	3	[3]	Add [3] to result	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3], [3]]
22	Loop ends	3	[3]	Backtrack: remove 3	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3], [3]]
23	Backtrack	0	[]	Loop ends, done	[[], [1], [1,2], [1,2,3], [1,3], [2], [2,3], [3]]

💡 All elements processed, all subsets generated

Variable Tracker

Variable	Start	After Step 2	After Step 4	After Step 6	After Step 10	After Step 14	After Step 20	Final
start	0	1	2	3	2	3	3	3
subset	[]	[1]	[1,2]	[1,2,3]	[1,3]	[2]	[3]	[]
result.length	0	2	3	4	5	6	8	8

Key Moments - 3 Insights

Why do we push a copy of the subset instead of the subset itself?

Because subsets are modified later by adding/removing elements, pushing a copy (using [...subset]) saves the current state. See Step 1 and Step 3 where subsets are added.

Why do we pop elements after recursion?

Popping removes the last added element to backtrack and explore subsets without that element. This is shown in Steps 8, 9, 12, 13, etc.

How does the recursion know when to stop?

When the start index reaches the length of nums, the loop ends and recursion stops, as in Step 23.

Visual Quiz - 3 Questions

Look at the execution table at Step 6. What is the current subset?

A [1, 2, 3]

B [1, 2]

C [1]

D []

💡 Hint

Check the 'Current Subset' column at Step 6 in the execution_table.

At which step does the subset [2, 3] get added to the result?

A Step 17

B Step 16

C Step 15

D Step 18

💡 Hint

Look for the 'Add [2,3] to result' action in the execution_table.

If we remove the pop() call after recursion, what happens to the subsets generated?

A No subsets will be generated

B Subsets will be correct but in different order

C Subsets will include duplicates and incorrect elements

D Only empty subset will be generated

💡 Hint

Refer to key_moments about why pop() is needed for backtracking.

Concept Snapshot

Generate All Subsets (Powerset):
- Use backtracking to explore include/exclude choices
- Start with empty subset and add it to results
- For each element, add it, recurse, then remove it (backtrack)
- Push a copy of subset to results to save current state
- Stop when all elements are processed

Full Transcript

This visualization shows how to generate all subsets of an array using backtracking. We start with an empty subset and at each step decide to include or exclude the next element. We save the current subset by pushing a copy to the result list. Then we recurse to process the next elements. After recursion, we remove the last added element to backtrack and explore other subsets. This continues until all elements are processed, generating the full powerset. The execution table tracks each step, the current subset, and the subsets saved so far. The variable tracker shows how the start index and subset change over time. Key moments clarify why we push copies and why popping is necessary. The quiz tests understanding of subset states and backtracking importance.

Practice

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

easy

A. The set itself without any subsets

B. All possible subsets of a set including the empty set

C. Only the subsets with exactly two elements

D. Subsets that contain only odd numbers

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 the definition

   All possible subsets of a set including the empty set correctly describes the powerset as all possible subsets including the empty set. Other options describe only parts or incorrect subsets.

3. Final Answer:

   All possible subsets of a set including the empty set -> Option B

4. Quick Check:

   powerset = all subsets [OK]

Hint: Powerset means every subset, even empty one [OK]

Common Mistakes:

• Thinking powerset excludes empty set
• Confusing powerset with subsets of fixed size
• Assuming powerset is just the original set

2. Which of the following JavaScript function signatures correctly represents a function to generate all subsets of an array arr?

easy

A. function generateSubsets(arr, subsets) { ... }

B. function generateSubsets(arr[]) { ... }

C. function generateSubsets() { ... }

D. function generateSubsets(arr) { ... }

Solution

1. Step 1: Check valid JavaScript function syntax

   function generateSubsets(arr) { ... } uses correct syntax with one parameter named arr. function generateSubsets(arr[]) { ... } uses invalid syntax with arr[]. function generateSubsets() { ... } has no parameters, which is incomplete for this task. function generateSubsets(arr, subsets) { ... } has two parameters which might be valid but is not the standard signature for generating subsets.

2. Step 2: Choose the most appropriate signature

   function generateSubsets(arr) { ... } is the standard and correct way to define a function that takes an array to generate subsets.

3. Final Answer:

   function generateSubsets(arr) { ... } -> Option D

4. Quick Check:

   Correct function syntax = function generateSubsets(arr) { ... } [OK]

Hint: Function parameters must be valid identifiers without brackets [OK]

Common Mistakes:

• Using array syntax in parameter list (arr[])
• Omitting required parameters
• Adding unnecessary parameters without explanation

3. What is the output of the following JavaScript code?

function subsets(nums) {
  let result = [[]];
  for (const num of nums) {
    const newSubsets = [];
    for (const subset of result) {
      newSubsets.push([...subset, num]);
    }
    result.push(...newSubsets);
  }
  return result;
}

console.log(subsets([1, 2]));

medium

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

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

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

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

Solution

1. Step 1: Trace the initial result and loop

   Initially, result = [[]]. For num=1, newSubsets = [[1]], result becomes [[], [1]]. For num=2, newSubsets = [[2], [1, 2]], result becomes [[], [1], [2], [1, 2]].

2. Step 2: Confirm final output

   The function returns [[], [1], [2], [1, 2]] which matches [[], [1], [2], [1, 2]] exactly.

3. Final Answer:

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

4. Quick Check:

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

Hint: Each new number doubles subsets by adding to existing ones [OK]

Common Mistakes:

• Missing empty subset in output
• Not adding new subsets correctly
• Confusing order of subsets

4. Identify the error in this JavaScript code that tries to generate all subsets of nums:

function getSubsets(nums) {
  let result = [];
  for (let num of nums) {
    let newSubsets = [];
    for (let subset of result) {
      newSubsets.push(subset.concat(num));
    }
    result = result.concat(newSubsets);
  }
  return result;
}

console.log(getSubsets([1, 2]));

medium

A. The initial result should contain the empty subset [[]]

B. Using concat instead of push causes error

C. The inner loop should iterate over nums, not result

D. Missing return statement

Solution

1. Step 1: Check initial value of result

   The code initializes result as an empty array []. It should start with [[]] to include the empty subset, otherwise no subsets are generated.

2. Step 2: Understand impact of initial value

   Without the empty subset, the inner loop never runs on first iteration, so no subsets are created. Adding [[]] fixes this.

3. Final Answer:

   The initial result should contain the empty subset [[]] -> Option A

4. Quick Check:

   Start with [[]] to generate subsets [OK]

Hint: Always start subsets with empty set [[]] [OK]

Common Mistakes:

• Starting with empty array instead of [[]]
• Confusing concat and push usage
• Looping over wrong array

5. Given an array nums = [1, 2, 2], which code modification correctly generates all unique subsets (no duplicates) using the powerset approach?

hard

A. Remove duplicates from nums before generating subsets

B. Use a Set to store subsets as strings to avoid duplicates

C. Sort nums first, then skip adding duplicate subsets during generation

D. Generate all subsets normally without any change

Solution

1. Step 1: Understand the problem with duplicates

   Array [1, 2, 2] has duplicates. Generating subsets normally will create duplicate subsets like [2] twice.

2. Step 2: Apply sorting and skipping duplicates

   Sorting nums makes duplicates adjacent. During subset generation, skip adding subsets that duplicate previous ones to avoid duplicates in result.

3. Final Answer:

   Sort nums first, then skip adding duplicate subsets during generation -> Option C

4. Quick Check:

   Sort and skip duplicates to get unique subsets [OK]

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

Common Mistakes:

• Ignoring duplicates causing repeated subsets
• Removing duplicates from input changes problem
• Using Set of strings can be inefficient or incorrect