Generate All Subsets Powerset in DSA C++ - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a powerset of a set?

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

beginner

How many subsets does a set with n elements have?

A set with n elements has 2^n subsets in its powerset.

intermediate

What is the main idea behind generating all subsets using backtracking?

The main idea is to explore each element by either including it or excluding it, recursively building subsets step-by-step.

beginner

In C++, which STL container is commonly used to store subsets when generating a powerset?

The std::vector container is commonly used to store subsets because it allows dynamic resizing and easy element access.

intermediate

What is the time complexity of generating all subsets of a set with n elements?

The time complexity is O(n * 2^n) because there are 2^n subsets and each subset can take up to n time to build or copy.

How many subsets does the set {1, 2, 3} have?

A 9

B 6

C 3

D 8

A set with 3 elements has 2^3 = 8 subsets.

Which method is commonly used to generate all subsets of a set?

A Sorting

B Backtracking

C Binary Search

D Dynamic Programming

Backtracking explores all possibilities by including or excluding each element.

What is the first subset generated in powerset generation?

A Set with first element only

B Full set

C Empty set

D Single element set

The empty set is always included as the first subset in the powerset.

In C++, which data structure is best to store the current subset during recursion?

A std::vector

B std::map

C std::set

D std::queue

std::vector is used for dynamic storage and easy element addition/removal.

What is the total number of subsets for a set with 5 elements?

A 32

B 10

C 25

D 16

2^5 = 32 subsets exist for a set with 5 elements.

Explain how backtracking helps in generating all subsets of a set.

Think about choosing elements one by one and exploring both choices.

Describe the time complexity of generating the powerset and why it is that way.

Consider how many subsets and how long each takes to create.

Practice

1. What does the "Generate All Subsets (Powerset)" algorithm produce when given a set {1, 2}?

easy

A. Only subsets with two elements: {1, 2}

B. All possible subsets including empty set: {}, {1}, {2}, {1, 2}

C. Only subsets with one element: {1}, {2}

D. Only the empty set: {}

Solution

1. Step 1: Understand what subsets mean

   Subsets include all combinations of elements, from empty to full set.

2. Step 2: List all subsets of {1, 2}

   They are {}, {1}, {2}, and {1, 2}.

3. Final Answer:

   All possible subsets including empty set: {}, {1}, {2}, {1, 2} -> Option B

4. Quick Check:

   Subsets = all combinations [OK]

Hint: Remember powerset includes empty and full sets [OK]

Common Mistakes:

• Thinking subsets only have full or single elements
• Ignoring the empty subset
• Confusing subsets with permutations

2. Which of the following C++ code snippets correctly initializes an empty vector of vectors to store subsets?

easy

A. std::vector<int> subsets;

B. std::vector subsets = {{}};

C. std::vector subsets;

D. std::vector subsets<int>;

Solution

1. Step 1: Identify correct vector declaration

   We need a vector of vectors of int to store subsets.

2. Step 2: Check syntax correctness

   std::vector subsets; uses correct syntax: std::vector subsets; initializes empty vector of vectors.

3. Final Answer:

   std::vector subsets; -> Option C

4. Quick Check:

   Correct vector of vectors syntax [OK]

Hint: Use vector for subsets storage [OK]

Common Mistakes:

• Using vector<int> instead of vector
• Incorrect template syntax like vector subsets<int>
• Trying to initialize with empty braces incorrectly

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

std::vector<int> nums = {1, 2};
std::vector subsets = {{}};
for (int num : nums) {
    int n = subsets.size();
    for (int i = 0; i < n; i++) {
        std::vector<int> newSubset = subsets[i];
        newSubset.push_back(num);
        subsets.push_back(newSubset);
    }
}
for (auto &subset : subsets) {
    for (int x : subset) std::cout << x << " ";
    std::cout << "| ";
}

medium

A. | | 1 | 2 | 1 2 |

B. | | 1 2 | 1 | 2 |

C. | 1 | 2 | 1 2 |

D. | 1 2 | 1 | 2 |

Solution

1. Step 1: Understand initial subsets and loop

   Initially subsets = {{}}, empty subset only. For each num, we add new subsets by adding num to existing subsets.

2. Step 2: Trace iterations

   For num=1: subsets become {}, {1}. For num=2: subsets become {}, {1}, {2}, {1, 2}.

3. Final Answer:

   | | 1 | 2 | 1 2 | -> Option A

4. Quick Check:

   All subsets printed with spaces and separators [OK]

Hint: Start with empty subset and add elements iteratively [OK]

Common Mistakes:

• Missing the empty subset in output
• Printing subsets in wrong order
• Confusing push_back inside loop

4. Identify the error in this C++ code snippet for generating subsets:

std::vector<int> nums = {1, 2};
std::vector subsets;
for (int num : nums) {
    int n = subsets.size();
    for (int i = 0; i < n; i++) {
        std::vector<int> newSubset = subsets[i];
        newSubset.push_back(num);
        subsets.push_back(newSubset);
    }
}

medium

A. subsets vector is not initialized with empty subset before loop

B. Loop variable i should start from 1, not 0

C. newSubset.push_back(num) should be outside inner loop

D. nums vector should be empty to generate subsets

Solution

1. Step 1: Check initialization of subsets

   subsets is empty initially, so subsets.size() is 0, inner loop never runs.

2. Step 2: Identify missing empty subset

   We must start subsets with empty subset {{}} to build new subsets correctly.

3. Final Answer:

   subsets vector is not initialized with empty subset before loop -> Option A

4. Quick Check:

   Empty subset needed to start building subsets [OK]

Hint: Always start subsets with empty subset {{}} [OK]

Common Mistakes:

• Not initializing subsets with empty subset
• Changing loop start index incorrectly
• Misplacing push_back calls

5. Given a vector nums = {1, 2, 2}, which code modification ensures the generated subsets contain no duplicate subsets?

hard

A. Generate subsets normally; duplicates are impossible with vector<int>.

B. Use a set to store subsets after generation to remove duplicates.

C. Ignore duplicate elements in nums vector before generating subsets.

D. Sort nums first and skip adding duplicates during subset generation.

Solution

1. Step 1: Understand duplicates in input

   Input has duplicate 2's, so subsets like {2} can appear multiple times.

2. Step 2: Use sorting and skip duplicates

   Sorting nums helps detect duplicates; skip adding subsets with same element consecutively to avoid duplicates.

3. Final Answer:

   Sort nums first and skip adding duplicates during subset generation. -> Option D

4. Quick Check:

   Sorting + skipping duplicates avoids repeated subsets [OK]

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

Common Mistakes:

• Assuming duplicates never occur
• Removing duplicates from input loses valid subsets
• Using set after generation is less efficient