grid47 Exploring patterns and algorithms
Oct 30, 2024
4 min read
Solution to LeetCode 78: Subsets Problem
You are given an integer array nums containing unique elements. Find all possible subsets (the power set) of the array nums. The solution set should not contain duplicate subsets, and the result can be returned in any order.
Problem
Approach
Steps
Complexity
Input: The input consists of an integer array `nums` with unique elements.
Example: nums = [1, 2, 3]
Constraints:
• 1 <= nums.length <= 10
• -10 <= nums[i] <= 10
• All elements of nums are unique.
Output: Return all possible subsets (the power set) of the array `nums`, ensuring no duplicate subsets are included. The order of the subsets does not matter.
• The output should be a list of subsets where each subset is a list of integers.
Goal: Generate all possible subsets (the power set) of the given array `nums` using a backtracking approach.
Steps:
• Start with an empty subset and recursively explore both options: including the current element or excluding it.
• For each element in the array, make a choice whether to add it to the current subset or skip it.
• Each time an element is included or excluded, recursively call the function to build all possible subsets.
Goal: The problem constraints ensure the input size and the uniqueness of elements in the input array.
Steps:
• 1 <= nums.length <= 10
• -10 <= nums[i] <= 10
• All elements in nums are unique.
Assumptions:
• The array `nums` contains unique integers.
• The solution set should include all possible subsets without duplicates.
• Input: nums = [1, 2, 3]
• Explanation: The output includes all subsets, ranging from the empty subset to the full set itself.
• Input: nums = [4]
• Explanation: There are two subsets: the empty subset `[]` and the subset `[4]`.
Approach: The approach involves using backtracking to explore both possibilities for each element (include or exclude it) and build all possible subsets.
Observations:
• We can generate all subsets by recursively exploring both inclusion and exclusion of elements in the array.
• Backtracking works well for generating subsets, as it allows for efficiently exploring all combinations by making decisions at each step.
Steps:
• Define a helper function to generate subsets recursively.
• For each element, explore both options: include the element in the subset or exclude it.
• Add the current subset to the result when all elements are considered.
Empty Inputs:
• If `nums` is an empty array, the only subset is the empty subset `[]`.
Large Inputs:
• For larger arrays, ensure the solution handles the exponential number of subsets efficiently.
Special Values:
• If `nums` contains just one element, the output will be two subsets: the empty subset `[]` and the subset with the single element.
Constraints:
• The function should efficiently handle arrays up to the maximum size of 10 elements.
vector<vector<int>> subsets(vector<int>& nums) {
vector<vector<int>> result = {{}};
for (intnum : nums) {
int n = result.size();
for (int i =0; i < n; ++i) {
result.push_back(result[i]);
result.back().push_back(num);
}
}
return result;
}
1 : Function Declaration
vector<vector<int>> subsets(vector<int>& nums) {
Declares a function `subsets` that takes a vector of integers `nums` as input and returns a vector of vectors representing all possible subsets.
2 : Variable Initialization
vector<vector<int>> result = {{}};
Initializes a vector `result` to store the generated subsets, starting with an empty set.
3 : Loop Iteration
for (intnum : nums) {
Iterates over each number `num` in the input vector `nums`.
4 : Variable Initialization
int n = result.size();
Stores the current size of the `result` vector in `n`.
5 : Loop Iteration
for (int i =0; i < n; ++i) {
Iterates over the existing subsets in the `result` vector.
6 : Vector Operation
result.push_back(result[i]);
Creates a new subset by copying the current subset `result[i]` and adds it to the `result` vector.
7 : Vector Operation
result.back().push_back(num);
Adds the current number `num` to the newly created subset.
8 : Return
return result;
Returns the `result` vector containing all the generated subsets.
Best Case: O(2^n)
Average Case: O(2^n)
Worst Case: O(2^n)
Description: The time complexity is O(2^n) because there are 2^n subsets of a set with n elements.
Best Case: O(n)
Worst Case: O(n)
Description: The space complexity is O(n) due to the recursive call stack and storage for the current subset being built.
