Comparing Time and Space Complexity
When comparing the time and space complexity of different approaches to solving the Fibonacci sequence using dynamic programming, we can observe the following:
Time Complexity: The bottom-up dynamic programming approach has a time complexity of O(n), where n is the index of the Fibonacci number we want to compute. This is because we only need to perform n computations to calculate the desired Fibonacci number.
Space Complexity: The bottom-up dynamic programming approach has a space complexity of O(1), which means it uses constant space. This is because we only need to store the last two Fibonacci numbers and update them as we iterate through the sequence.
Let's take a look at an example in C# to illustrate the time and space complexity of the different approaches:
1public class Fibonacci
2{
3 public static int GetFibonacciTimeComplexity(int n)
4 {
5 if (n == 0)
6 return 0;
7
8 if (n == 1)
9 return 1;
10
11 int[] fibonacci = new int[n + 1];
12 fibonacci[0] = 0;
13 fibonacci[1] = 1;
14
15 for (int i = 2; i <= n; i++)
16 {
17 fibonacci[i] = fibonacci[i - 1] + fibonacci[i - 2];
18 }
19
20 return fibonacci[n];
21 }
22
23 public static int GetFibonacciSpaceComplexity(int n)
24 {
25 if (n == 0)
26 return 0;
27
28 if (n == 1)
29 return 1;
30
31 int prev = 0;
32 int curr = 1;
33
34 for (int i = 2; i <= n; i++)
35 {
36 int temp = curr;
37 curr = prev + curr;
38 prev = temp;
39 }
40
41 return curr;
42 }
43}
44
45int fibonacciNumber = Fibonacci.GetFibonacciTimeComplexity(5);
46Console.WriteLine(fibonacciNumber);
47
48fibonacciNumber = Fibonacci.GetFibonacciSpaceComplexity(5);
49Console.WriteLine(fibonacciNumber);```
50
51In the above code, we have two methods: `GetFibonacciTimeComplexity` and `GetFibonacciSpaceComplexity`. The `GetFibonacciTimeComplexity` method calculates the Fibonacci number at a given index `n` using the bottom-up dynamic programming approach with O(n) time complexity. The `GetFibonacciSpaceComplexity` method calculates the Fibonacci number using the space-optimized bottom-up dynamic programming approach with O(1) space complexity.
52
53Feel free to modify the code and experiment with different values of `n` to see the time and space efficiency of the different approaches.xxxxxxxxxxConsole.WriteLine(fibonacciNumber);public class Fibonacci{ public static int GetFibonacciTimeComplexity(int n) { if (n == 0) return 0; if (n == 1) return 1; int[] fibonacci = new int[n + 1]; fibonacci[0] = 0; fibonacci[1] = 1; for (int i = 2; i <= n; i++) { fibonacci[i] = fibonacci[i - 1] + fibonacci[i - 2]; } return fibonacci[n]; } public static int GetFibonacciSpaceComplexity(int n) { if (n == 0) return 0; if (n == 1) return 1;