递归详解

在计算机科学领域, 递归是用于处理一类具有相同子问题处理方式的问题;
是数学归纳法, 数学递推公式在计算机中的应用

The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
--- by Niklaus Wirth, Algorithms + Data Structures = Programs, 1976

用递归的方法来解决问题的关键, 在于准确找到子问题

递归的简单应用

1. 阶乘

计算 \(n! = n *(n-1)* \dots *2* 1\)

即令 \(f(n) = n * f(n-1)\)

void Recursion_Factorial(int n)
{
    if(n == 1) return n;
    return n * Recursion_Factorial(n-1);
}

2. 寻找所有子集

template<class T>
void Recursion_Subset(vector<T> dataSet, vector<vector<T>>& result)
{
	if (dataSet.empty()) { result.push_back({}); }
	else
	{
		T data = dataSet.back();
		dataSet.pop_back();
		Recursion_Subset(dataSet, result);
		int size = result.size();
		for (int i = 0; i < size; i++)
		{
			result.push_back(result[i]);
			result.back().push_back(data);
		}
	}
	return;
}