Searching Algorithms Questions Long
The Fibonacci search algorithm is a searching technique that is based on the Fibonacci sequence. It is an efficient search algorithm that can be used to search for an element in a sorted array.
The time complexity of the Fibonacci search algorithm is O(log n), where n is the number of elements in the array. This makes it a very efficient searching algorithm, especially for large arrays.
The Fibonacci search algorithm works by dividing the array into two parts using Fibonacci numbers. It starts by initializing two Fibonacci numbers, let's say Fm and Fm+1, such that Fm+1 is the smallest Fibonacci number greater than or equal to n. Then, it compares the key element with the element at index Fm. If the key is smaller, it narrows down the search to the subarray before Fm. If the key is larger, it narrows down the search to the subarray after Fm. This process is repeated until the key element is found or the subarray size becomes 1.
The reason behind the time complexity of O(log n) is that at each step, the size of the subarray is reduced by approximately half. This reduction is achieved by using Fibonacci numbers, which have a property that Fm+1/Fm approaches the golden ratio (approximately 1.618) as m approaches infinity. This property ensures that the size of the subarray is reduced by a constant factor at each step, leading to a logarithmic time complexity.
In conclusion, the time complexity of the Fibonacci search algorithm is O(log n), making it an efficient searching algorithm for sorted arrays.