Searching Algorithms Questions Long
Exponential interpolation search is a searching algorithm that is used to find the position of a target value within a sorted array. It is an improved version of the binary search algorithm, which reduces the number of comparisons required to find the target value.
The time complexity of exponential interpolation search can be analyzed as follows:
1. Best-case time complexity:
In the best-case scenario, the target value is found at the first position itself. In this case, the time complexity of exponential interpolation search would be O(1), as only one comparison is required.
2. Average-case time complexity:
The average-case time complexity of exponential interpolation search is O(log(log(n))), where 'n' represents the size of the array. This is because the algorithm uses exponential interpolation to estimate the position of the target value. It starts by comparing the target value with the element at the first position, and if they match, the search is complete. Otherwise, it estimates the probable position of the target value using interpolation. The interpolation formula used is:
pos = low + ((target - arr[low]) * (high - low)) / (arr[high] - arr[low])
Here, 'low' and 'high' represent the current range of the array being searched. The algorithm then compares the target value with the element at the estimated position. If they match, the search is complete. Otherwise, it updates the range based on whether the target value is smaller or larger than the estimated element, and repeats the process until the target value is found or the range becomes empty.
The interpolation step reduces the range of the search space exponentially, hence the name "exponential interpolation search". This results in a faster search compared to binary search, especially when the elements in the array are uniformly distributed.
3. Worst-case time complexity:
The worst-case time complexity of exponential interpolation search is O(n), which occurs when the target value is not present in the array. In this case, the algorithm will keep updating the range and estimating the position using interpolation until the range becomes empty. This can lead to a linear search, resulting in a time complexity of O(n).
It is important to note that the time complexity mentioned above assumes that the array is sorted. If the array is not sorted, an additional step of sorting the array would be required, which would have a time complexity of O(n log(n)).
In conclusion, the time complexity of exponential interpolation search is O(1) in the best-case scenario, O(log(log(n))) in the average-case scenario, and O(n) in the worst-case scenario.