What is the concept of Fibonacci search?

The concept of Fibonacci search is a searching algorithm that is based on the Fibonacci sequence. It is an efficient and effective method for searching elements in a sorted array.

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, starting from 0 and 1. The sequence begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

The Fibonacci search algorithm works by dividing the array into two parts, similar to binary search. However, instead of dividing the array into two equal halves, Fibonacci search divides the array into two parts using Fibonacci numbers as the dividing points.

Here are the steps involved in the Fibonacci search algorithm:

1. Initialize the Fibonacci numbers: Start with two Fibonacci numbers, F(k-2) = 0 and F(k-1) = 1, where k is the smallest Fibonacci number greater than or equal to the length of the array.

2. Compare the key element with the middle element of the array. If they are equal, the search is successful, and the index of the element is returned.

3. If the key element is smaller than the middle element, the array is divided into two parts. The first part will have a length equal to F(k-2), and the second part will have a length equal to F(k-1). The first part becomes the new array, and the process is repeated from step 2.

4. If the key element is larger than the middle element, the array is divided into two parts. The first part will have a length equal to F(k-1), and the second part will have a length equal to F(k-2). The second part becomes the new array, and the process is repeated from step 2.

5. Repeat steps 2 to 4 until the key element is found or the array is exhausted.

The Fibonacci search algorithm has a time complexity of O(log n), making it more efficient than linear search but slightly slower than binary search. It is particularly useful when the array size is large and the elements are uniformly distributed.

In conclusion, the concept of Fibonacci search is a searching algorithm that divides the array using Fibonacci numbers as dividing points, making it an efficient and effective method for searching elements in a sorted array.