# Fibonacci – Normal vs Dynamic programming Huge Time complexity Difference

Dynamic programming, guarantees to find the optimal solution of a problem if the solution exists. It basically follows these steps,

- Divide the main complex problems into sub-problems
- Saves the sub-problem result found
- Apply the saved result rather than re-computing the solution again.

Let’s understand the differences it provides (time complexity difference) with dynamic programming approach,

**Fibonacci Program – Normal & Dynamic Programming:**

```
import java.time.LocalDateTime;
public class Fibonacci {
public static void main(String[] args) {
// 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
int n = 10;
// normal
long ldtStart = LocalDateTime.now().getNano();
int fibo = nthFibo(n);
System.out.println(fibo);
long ldtEnd = LocalDateTime.now().getNano();
System.out.println("Total Execution Time: " + (ldtEnd - ldtStart));
// with dynamic programming
long ldtStartDP = LocalDateTime.now().getNano();
int fiboDP = nthFiboDP(n);
System.out.println(fiboDP);
long ldtEndDP = LocalDateTime.now().getNano();
System.out.println("Total Execution Time for DP: " + (ldtEndDP - ldtStartDP));
}
private static int nthFibo(int n) {
if (n <= 1) {
return n;
}
return nthFibo(n - 1) + nthFibo(n - 2);
}
private static int nthFiboDP(int n) {
if (n <= 1) {
return n;
}
int[] dp = new int[n + 1];
dp[0] = 0;
dp[1] = 1;
for (int i = 2; i <= n; i++) {
dp[i] = dp[i - 1] + dp[i - 2];
}
return dp[n];
}
}
```

If I run the above code, then my output is, (here execution time is very less in both the approaches)

55**Total Execution Time: 0**55

**Total Execution Time for DP: 0**

Now with the n value as = 20,

6765**Total Execution Time: 999700**6765

**Total Execution Time for DP: 0**

Now with the n value as n =30,

832040**Total Execution Time: 3998800**832040

**Total Execution Time for DP: 0**

And strange problem here is if the n value is greater than 30 then time complexities getting higher and higher and at n = 100 literly in my machine nothing works for few mins and completely hangs out.

Now, in order to address all these stuff’s in a efficient way, dynamic programming helps really in a greater extend.

Hope this simple program helps you to understand how dynamic programming brings much better improvements from the time complexities perspective.

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