回溯以查找所有子集

给定一组正整数，找到其所有子集。
例子：

Input: array = {1, 2, 3}
Output: // this space denotes null element. 
         1
         1 2
         1 2 3
         1 3
         2
         2 3
         3
Explanation: These are all the subsets that 
can be formed using the array.

Input: 1 2
Output: 
         1 
         2
         1 2
Explanation: These are all the subsets that 
can be formed using the array.

这里已经讨论了迭代解决方案：找到所有子集的迭代方法。本文旨在提供一种回溯方法。

方法：这个想法很简单，如果数组中有n个元素，则每个元素都有两个选择。将该元素包含在子集中或不包含它。
使用上述想法形成了对该问题的递归解决方案。

算法：

1. 创建一个采用以下参数的递归函数：输入数组，当前索引，输出数组或当前子集，如果需要存储所有子集，则需要数组向量，如果仅需要打印子集，则此空间可以忽略。
2. 如果当前索引等于数组的大小，则打印子集或输出数组，或者将输出数组插入数组的一个或多个向量中，然后返回。
3. 完全索引有两种选择。
4. 忽略当前元素，并使用当前子集和下一个索引(即i + 1)调用递归函数。
5. 将当前元素插入子集中，并使用当前子集和下一个索引(即i + 1)调用递归函数。

执行：

C++

// CPP program to find all subsets by backtracking.
#include 
using namespace std;
  
// In the array A at every step we have two
// choices for each element either  we can
// ignore the element or we can include the
// element in our subset
void subsetsUtil(vector& A, vector >& res,
                 vector& subset, int index)
{
    res.push_back(subset);
    for (int i = index; i < A.size(); i++) {
  
        // include the A[i] in subset.
        subset.push_back(A[i]);
  
        // move onto the next element.
        subsetsUtil(A, res, subset, i + 1);
  
        // exclude the A[i] from subset and triggers
        // backtracking.
        subset.pop_back();
    }
  
    return;
}
  
// below function returns the subsets of vector A.
vector > subsets(vector& A)
{
    vector subset;
    vector > res;
  
    // keeps track of current element in vector A;
    int index = 0;
    subsetsUtil(A, res, subset, index);
  
    return res;
}
  
// Driver Code.
int main()
{
    // find the subsets of below vector.
    vector array = { 1, 2, 3 };
  
    // res will store all subsets.
    // O(2 ^ (number of elements inside array))
    // because at every step we have two choices
    // either include or ignore.
    vector > res = subsets(array);
  
    // Print result
    for (int i = 0; i < res.size(); i++) {
        for (int j = 0; j < res[i].size(); j++)
            cout << res[i][j] << " ";
        cout << endl;
    }
  
    return 0;
}

Python3

# Python3 program to find all subsets
# by backtracking. 
  
# In the array A at every step we have two 
# choices for each element either we can 
# ignore the element or we can include the 
# element in our subset 
def subsetsUtil(A, subset, index):
    print(*subset)
    for i in range(index, len(A)): 
          
        # include the A[i] in subset. 
        subset.append(A[i])
          
        # move onto the next element. 
        subsetsUtil(A, subset, i + 1) 
          
        # exclude the A[i] from subset and 
        # triggers backtracking.
        subset.pop(-1) 
    return
  
# below function returns the subsets of vector A. 
def subsets(A):
    global res
    subset = []
      
    # keeps track of current element in vector A 
    index = 0
    subsetsUtil(A, subset, index) 
      
# Driver Code
  
# find the subsets of below vector. 
array = [1, 2, 3]
  
# res will store all subsets. 
# O(2 ^ (number of elements inside array)) 
# because at every step we have two choices 
# either include or ignore. 
subsets(array) 
  
# This code is contributed by SHUBHAMSINGH8410

输出：

1 
1 2 
1 2 3 
1 3 
2 
2 3 
3 

复杂度分析：

• 时间复杂度： O(2 ^ n)。
  对于每个索引，我都会产生两个递归情况，因此时间复杂度为O(2 ^ n)。
• 空间复杂度： O(n)。
  如果不存储输出数组，并且使用static和global变量存储输出字符串，则可以减少空间复杂度。