Subsets Algorithm
The Subsets Algorithm is a combinatorial technique used to systematically generate all possible subsets of a given set. It is essential in many areas of computer science and mathematics, including combinatorics, set theory, and graph theory. The algorithm is based on the idea that each element in the set can either be present or absent in a subset, resulting in 2^n possible subsets, where n is the number of elements in the set. The algorithm creates an exhaustive list of these subsets, allowing the user to analyze and manipulate them as needed.
One common method of implementing the Subsets Algorithm is through a recursive approach. The base case occurs when the input set is empty, in which case the algorithm returns the empty set as the only possible subset. For non-empty sets, the algorithm removes one element from the set and recursively generates all subsets of the remaining elements. Then, it creates two versions of each subset: one with the removed element included and one without it. This process continues until all possible combinations have been generated, resulting in a complete list of subsets. Another common method is using a binary counter approach, where each subset is represented by a binary string of length n, with 1s and 0s indicating the presence or absence of an element in the subset. The algorithm iterates through all possible binary strings of length n, generating the corresponding subset for each one. Regardless of the specific implementation, the Subsets Algorithm plays a crucial role in solving a wide range of problems in computer science and mathematics.
class Solution {
public:
vector<vector<int> > subsets(vector<int>& S) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
vector<vector<int>> result;
vector<int> subset;
sort(S.begin(), S.end());
DFS(subset, 0, S, result);
return move(result);
}
void DFS(vector<int>& subset, int step, vector<int>& S, vector<vector<int>>& result) {
if (step == S.size()) {
result.push_back(subset);
return;
}
subset.push_back(S[step]);
DFS(subset, step + 1, S, result);
subset.pop_back();
DFS(subset, step + 1, S, result);
}
};
