# 509. Fibonacci Number

The **Fibonacci numbers**, commonly denoted `F(n)` form a sequence, called the **Fibonacci sequence**, such that each number is the sum of the two preceding ones, starting from `0` and `1`. That is,

```
F(0) = 0,   F(1) = 1
F(N) = F(N - 1) + F(N - 2), for N > 1.
```

Given `N`, calculate `F(N)`.

**Example 1:**

```
Input: 2
Output: 1
Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.
```

**Example 2:**

```
Input: 3
Output: 2
Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.
```

**Example 3:**

```
Input: 4
Output: 3
Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.
```

**Note:**

0 ≤ `N` ≤ 30.

```cpp
// Top-Down Dynamic Programming with Memoization
int helper(int N, vector<int>& memo) { // time: O(N); space: O(N)
    if (N < 2) return N;
    if (memo[N] != -1) return memo[N];
    memo[N] = helper(N - 1, memo) + helper(N - 2, memo);
    return memo[N];
}
int fib(int N) {
    vector<int> memo(N + 1, -1);
    return helper(N, memo);
}
```

```cpp
// Bottom-Up Dynamic Programming
int fib(int N) { // time: O(n); space: O(n)
    if (N == 0) return 0;
    vector<int> F(N + 1, 0);
    F[1] = 1;
    for (int i = 2; i <= N; ++i) {
        F[i] = F[i - 2] + F[i - 1];
    }
    return F.back();
}
```

```cpp
// Space Optimized Bottom-Up Dynamic Programming
int fib(int N) { // time: O(n); space: O(1)
    if (N == 0) return 0;
    int a = 0, b = 1;
    for (int i = 2; i <= N; ++i) {
        int c = a + b;
        a = b;
        b = c;
    }
    return b;
}
```