Question

Design an algorithm for calculating the Fibonacci series of a given number. Is your solution efficient? Is your solution a polynomial or nonpolynomial one?

Short answer

The iterative algorithm is efficient with linear time complexity \( O(n) \) and considered a polynomial-time solution.

Step-by-step solution

Understand the Fibonacci Series

The Fibonacci series is a sequence where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Mathematically, it's defined by the recurrence relation \( F(n) = F(n-1) + F(n-2) \) with the seed values \( F(0) = 0 \) and \( F(1) = 1 \).

Decide on an Algorithm Approach

There are various methods to calculate the Fibonacci sequence: recursive, iterative, and dynamic programming. A straightforward recursive approach is simple but inefficient due to repeated calculations. We aim for an efficient solution, so we'll use the iterative approach.

Implement the Iterative Approach

Using the iterative method, we efficiently calculate Fibonacci numbers. Start with initializing two variables for the first two Fibonacci numbers, then use a loop to calculate subsequent numbers up to the given number. Here's a pseudocode: 1. Initialize \( a = 0 \) and \( b = 1 \).2. For each number from 2 to \( n \): - Calculate the next Fibonacci number using \( c = a + b \). - Update \( a \) to \( b \) and \( b \) to \( c \).3. \( b \) contains the \( n^{th} \) Fibonacci number at the end.

Evaluate the Efficiency of the Algorithm

This iterative approach runs in linear time, i.e., \( O(n) \), because it requires a single iteration through the sequence up to \( n \). It is more efficient than the naive recursive approach, which takes \( O(2^n) \). As the input size gets larger, this linear time algorithm is more scalable.

Assess the Polynomial Nature of the Solution

The iterative approach is a polynomial-time algorithm because its time complexity is \( O(n) \), which is a linear function of the input size \( n \). Polynomial time algorithms are generally considered efficient in computational complexity theory.

Key concepts

Fibonacci Series

The Fibonacci Series is a fascinating sequence of numbers that is both simple and complex. It starts with two numbers: 0 and 1. From there, each subsequent number is the sum of the two preceding numbers. This means that when you add 0 and 1, you get 1. Add the next two numbers, 1 and 1, to get 2. The sequence continues in this pattern: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Mathematically, you can define the series with the formula: \[ F(n) = F(n-1) + F(n-2) \]Here, \( F(n) \) is the nth Fibonacci number. The first two values are defined as \( F(0) = 0 \) and \( F(1) = 1 \). This definition allows you to generate the series endlessly. The Fibonacci series is not just limited to mathematics but can also be seen in nature; for example, in the arrangement of leaves on a stem, the branching of trees, and the spiral shells of sharks! Understanding this sequence is a fundamental part of learning about algorithms as it provides insights into both recursive and iterative problem-solving approaches.

Iterative Approach

The iterative approach is a method used to solve problems by reducing them to a series of steps or iterations. In the context of the Fibonacci series, the iterative approach is both straightforward and efficient. Instead of using a recursive method, which can be slow and requires more stack space, this approach iteratively computes each term in the series up to the desired Fibonacci number. Here's a simple breakdown of how it works:

• Start by initializing two variables: let's call them \( a \) and \( b \), starting as 0 and 1 respectively.
• For every number from 2 up to \( n \) (where \( n \) is the position in the Fibonacci sequence you want to find), calculate the next Fibonacci number by adding \( a \) and \( b \).
• Update the values: set \( a \) to the value of \( b \) and \( b \) to the new sum. This shifts the sequence forward for the next iteration.
• Continue this process until you reach the desired position \( n \).

At the end of these steps, the variable \( b \) will contain the \( n^{th} \) Fibonacci number. This approach is advantageous because it avoids the excessive calculations involved in a recursive solution, making it both faster and easier on computer resources.

Computational Complexity

Understanding computational complexity is crucial for assessing the efficiency of algorithms. It essentially describes how the runtime or space requirements of an algorithm grow as the size of the input grows. Algorithms with lower computational complexity are generally preferred because they perform better on larger inputs.When we talk about the Fibonacci series, the naive recursive method might initially seem like a simple approach, but its complexity grows exponentially as the input size increases because each call to find \( F(n) \) results in two more calls to find \( F(n-1) \) and \( F(n-2) \). This results in a time complexity of \( O(2^n) \), which is not efficient.On the other hand, the iterative approach, which we've discussed, dramatically improves performance. Its time complexity is \( O(n) \), meaning it grows linearly with the input size. This is because the algorithm only requires a single loop through the sequence up to \( n \), significantly reducing unnecessary calculations.This efficiency categorizes the iterative approach as a polynomial-time algorithm, which is generally deemed efficient in computational complexity theory. As a rule of thumb, when designing algorithms, if you can achieve either a constant \( O(1) \) or at least polynomial time complexity, your solution is likely to be scalable and effective for large inputs.