使用递归解决问题

How do I thoroughly understand Recursive Algorithm and not feel like recursion is magic?

Recursion is magic! But any sufficiently advanced technology is indistinguishable from magic!

what is tail-recursion?

This results in the last statement being in the form of (return (recursive-function params)).

关于时间复杂度：

Are 2^n and n*2^n in the same time complexity?

The Three Laws of Recursion

Like the robots of Asimov, all recursive algorithms must obey three important laws:

1. A recursive algorithm must call itself, recursively.
2. A recursive algorithm must have a base case.
3. A recursive algorithm must change its state and move toward the base case.

A base case is the condition that allows the algorithm to stop recursing.

• A base case is typically a problem that is small enough to solve directly.
• In the factorial algorithm the base case is n=1.

We must arrange for a change of state that moves the algorithm toward the base case.

• A change of state means that some data that the algorithm is using is modified.
• Usually the data that represents our problem gets smaller in some way.
• In the factorial n decreases.

编写代码前的思考：

1. Break the problem I am trying to solve down into a problem that is one step simpler
2. Assume that my function will work to solve the simpler problem — really believe it beyond any doubt
3. Ask myself: Since I know I can solve the simpler problem, how would I solve the more complex problem?

One common mistake that I see people make when trying to develop a recursive algorithm to solve a problem is that they try to think about how to break the problem down all the way to the base case（递归条件）. 

I would like to emphasize that in order to develop the function above, I did not think about how I could break the problem down all the way to the base case. That is the function’s job, not yours. Instead, I only thought about the problem that is one step simpler than the problem I am really trying to solve, and then I wrote my recursive algorithm to build up from there to solve the real problem.

反转字符串 c++:

#include <iostream>
#include <string>
#include <vector>

using namespace std;

// 递归函数

string reverseString(vector<char>& vec){
    // 递归条件 
    if(vec.size()< 2) 
        return string(1,vec[0]); 

    // 递归链条 
    vector<char> vec0;
    vec0.push_back(vec[0]);
    vec.erase(vec.begin());

    return reverseString(vec) + reverseString(vec0);
}

int main(){
    vector<char> vec;
    string s = "helloworld";
    for(int i=0;i<(int)s.size();i++){
        vec.push_back(s[i]);
    }
    string ret = reverseString(vec);
    cout << ret << endl;

    return 0;
}