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:
- 创建一个使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。
- 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。
- 完全索引有两种选择。
- 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。
- 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。