Leetcode notes: Climbing Stairs (Fibonacci sequence problem)

I. Description of the topic

You are are climbing a stair case. It takes n steps to the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can your climb to the top?

The main idea is, known to have N-step staircase, can only climb 1-step or 2-step staircase, asked to climb to the nth staircase a total of several climbing method-_-| | The topic can be seen as, let F (n) indicate the number of methods to climb to the nth staircase, in order to climb to the nth staircase, there are the following two options:
• 1 steps forward from the F (n-1) step;
• 2 steps forward from the F (n-2) step;
then f (n) can be written as: F (n) = f (n-1) + f (n-2)

To the question of the Fibonacci sequence, there is a lot of online research on this content, the concept of Portal:
Http://baike.baidu.com/link?url=c2Bmk2jBGbI46qTIA-qKmdTkYBrVYYrejAHzf8BJRwCekIL4Sbx48fFCRkeGdul0

two. Topic Analysis

With respect to the Fibonacci sequence, you can use recursion or iteration to solve the problem, which can be written in the following recursive relationships:

It is obvious that iterative iteration is not the optimal solution, and when the value of f (n) is computed, all values of f (1), F (2),..., F (n-1) need to be computed, and there are many repetitive operations, such as the calculation of F (4) =f (3) +f (2), which needs to solve F (3) =f (2) +f (1). If you use an array to store all the calculated items, you can reduce the time complexity to O (n), and the space complexity is O (n).

In order to pursue the complexity of time, the direct use of Fibonacci formula, the derivation of the formula is as follows:

three. Sample Code

#include <iostream>
using namespace std;

Class Solution
{public
:
    //Time complexity O (1)
    int climbStairs1 (const int n)
    {
        const double sqrtnum = sqrt (5);
        return int (Floor ((POW (1 + sqrtnum)/2, n + 1)-pow ((1-sqrtnum)/2, n + 1))/sqrtnum);

    Time complexity O (n)
    int climbStairs2 (const int n)
    {int current
        = 1;
        int last = 0;
        for (int i = 1; I <= n; i++)
        {
            int temp = current;
            current = last;
            last = temp;
        }
        return current;
    }
;

Simple test Code:

#include "ClimbingStairs.h"
#include <iostream>

int main ()
{
    int n;
    cout << "How many stairs?" << "Input:";
    CIN >> N;
    Solution s;
    int result1 = S.CLIMBSTAIRS1 (n);
    int result2 = S.CLIMBSTAIRS2 (n);

    cout << "How many ways the finish line?" "RESULT1:" << result1 << Endl;
    cout << "How many ways the finish line?" "RESULT2:" << result2 << Endl;
    System ("pause");
    return 0;
}

Four. Summary

In fact, there is also a loophole in the use of the formula, because the formula uses floating-point operations, there are also physical books, and therefore can not guarantee the accuracy of the results. In the book "The Beauty of programming", we also give a solution to the strategy of divide and conquer. The algorithm can achieve the time complexity O (log2n), and online More posts have written seven kinds of Jiefepo of the method, but also to continue in-depth research AH.