Question

Consider the following iterative solution of the Fibonacci problem:

python
Copy code
int iterativeFibonacci(int n) {
//iterative solution to the Fibonacci problem
// initialise base cases:
int previous = 1; // initially rabbit(1)
int current = 1; //Initially rabbit(2)
int next = 1; // results when n is 1 or 2
// compute next Fibonacci values when n >= 3
for (int i = 3; i <= n; i++) {
//current is rabbit(i-1), previous is rabbit(i-2)
next = current + previous; //rabbit(i)
previous = current;
current = next;
}// end for
return next;
}
Implement the iterative, recursive, and DP solutions for the Fibonacci problem and compare the time taken by each solution method for n=1000.

A) Write iterative, recursive, and DP solutions and compare their time complexities.
B) Write iterative, recursive, and DP solutions and compare their output values.
C) Write only the iterative solution and compare its time complexity.
D) Write only the recursive solution and compare its output values.

Answer 1

The iterative and DP solutions to the Fibonacci problem have a linear time complexity of O(n) and perform efficiently even for large inputs like n=1000. The recursive solution has an exponential time complexity of O(2^n) and is impractical for large values of n. All three methods produce the same Fibonacci numbers, but their performance varies greatly.

Step-by-step explanation:

The question asks to write and compare different methods for solving the Fibonacci sequence: iterative, recursive, and dynamic programming (DP). The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. Here are brief implementations of the three methods:

Iterative Solution

The provided code in the question is already an iterative solution for the Fibonacci problem, which effectively calculates the Fibonacci number for a given n in linear time complexity, O(n).

Recursive Solution

int recursiveFibonacci(int n) {
if(n <= 1) {
return n;
} else {
return recursiveFibonacci(n-1) + recursiveFibonacci(n-2);
}
}

This method has exponential time complexity, O(2^n), which is inefficient for large values of n.

DP Solution (Memoization)

int dpFibonacci(int n) {
int memo[n+1];
memo[0] = 0;
memo[1] = 1;
for(int i = 2; i <= n; i++) {
memo[i] = memo[i-1] + memo[i-2];
}
return memo[n];
}

The DP solution offers a more efficient approach with time complexity O(n), similar to the iterative solution but uses additional memory to store the results.

The iterative and DP solutions have the same time complexity and will yield the same results efficiently. However, the recursive method, without memoization, will have an exponential time complexity and is not practical for large values of n (such as n=1000). All methods will provide the same Fibonacci number for a given n, yet their performance differs significantly due to their time complexities.