509. Fibonacci Number.java

E
tags: Math, DP, Memoization

#### Memoization
- fib[n] = fibonacci(n - 1) + fibonacci(n - 2);

#### DP array.
- 滚动数组, 简化DP

#### recursively calculate
- recursively calculate fib(n - 1) + fib(n - 2). 公式没问题, 但是时间太长, timeout.


```
/*
Find the Nth number in Fibonacci sequence.

A Fibonacci sequence is defined as follow:

The first two numbers are 0 and 1.
The i th number is the sum of i-1 th number and i-2 th number.
The first ten numbers in Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...


Example
Given 1, return 0

Given 2, return 1

Given 10, return 34

Note
The Nth fibonacci number won't exceed the max value of signed 32-bit integer in the test cases.

Tags Expand 
Enumeration Mathematics Non Recursion


*/

// Memoization
class Solution {
    int[] fib = null;
    public int fibonacci(int n) {
        if (fib == null) {
            fib = new int[n + 1];
        }
        if (n <= 2) {
            return n - 1;
        }
        if (fib[n] != 0) {
            return fib[n];
        }
        return fib[n] = fibonacci(n - 1) + fibonacci(n - 2);
    }
}

/*
    Recap 3.28.2016.
    Rolling array, instead of initiating array.
*/
class Solution {
    public int fibonacci(int n) {
        if (n <= 1) {
            return 0;
        }
        int first = 0;
        int second = 1;
        for (int i = 2; i < n; i++) {
            int temp = second;
            second = first + second;
            first = temp;
        }
        return second;
    }
}


/*
Thoughts:
Bottomup with for loop.
1. If non-recursion, do for loop for that n
2. Note: this specfiic problem is not 0-based. it's 1-based.
3. return fib[n]
*/
class Solution {
    public int fibonacci(int n) {
        if (n <= 1) {
            return 0;
        }
        int[] fib = new int[n];
        fib[0] = 0;
        fib[1] = 1;
        for (int i = 2; i < n; i++) {
            fib[i] = fib[i - 1] + fib[i - 2];
        }
        return fib[n - 1];
    }
}


/*
    Recursive. Long time complexity
    Timeout
*/
class Solution {
    public int fibonacci(int n) {
        if (n <= 1) {
            return 0;
        } 
        if (n == 2) {
            return 1;
        }
        return fibonacci(n - 1) + fibonacci(n - 2);
    }
}

```