Dynamic programming (2)-Fibonacci (recursive and non-recursive methods)

Fibonacci sequence

• Question: The value of the nth item in the sequence

• Abstract state F(i): the value of the i-th item in the sequence

• Transfer equation: F(i): F(i-1)+F(i-2)

• Initial state: We need two terms in the transfer equation to run,

  1. F(0) = 0
  2. F(1) = 1
  3. Return F(n)

The first type: recursion

public class Solution {
    
    
    public int Fibonacci(int n) {
    
    
        if(n == 0)return 0;
        if(n == 1 || n == 2)return 1;
        int f3 = Fibonacci(n-1)+Fibonacci(n-2);
        return f3;
    }
}

The second type: non-recursive

public class TestDemo1 {
    
    
    public static void main(String[] args) {
    
    
        //斐波那契非递归(输出第n项的值)
        Scanner scan = new Scanner(System.in);
        int n = scan.nextInt();
        System.out.println(Fibonacci(n));
    }
    public static int Fibonacci(int n) {
    
    
        if(n == 0)return 0;
        if(n == 1 || n == 2)return 1;
        int[] arr = new int[n];
        arr[0] = 1;
        arr[1] = 1;
        for (int i = 2;i < n;i++){
    
    
            arr[i] = arr[i-1]+arr[i-2];
        }
        return arr[n-1];
    }
 }