Explain the time complexity of sublinear interpolation search.

Sublinear interpolation search is an algorithm used to search for a specific element in a sorted array. It is an improvement over linear search, as it utilizes interpolation to estimate the position of the target element within the array.

The time complexity of sublinear interpolation search can be explained as follows:

1. Best-case scenario: In the best-case scenario, the target element is found at the first position of the array. In this case, the time complexity would be O(1), as the algorithm would terminate after a single comparison.

2. Average-case scenario: In the average-case scenario, the target element is not found at the first position, but it is still present within the array. The algorithm estimates the position of the target element using interpolation, which involves calculating the probable position based on the values of the first and last elements of the array. This estimation allows the algorithm to make a more informed decision about where to continue the search.

The average time complexity of sublinear interpolation search can be approximated as O(log(log(n))), where n is the size of the array. This is because the algorithm narrows down the search range by a logarithmic factor in each iteration, resulting in a sublinear time complexity.

3. Worst-case scenario: In the worst-case scenario, the target element is either not present in the array or located at the last position. In this case, the algorithm would need to perform a linear search from the estimated position to the end of the array. As a result, the time complexity would be O(n), where n is the size of the array.

It is important to note that the time complexity of sublinear interpolation search heavily depends on the distribution of the elements within the array. If the elements are uniformly distributed, the algorithm performs efficiently. However, if the elements are unevenly distributed, the performance may degrade, resulting in a time complexity closer to O(n).

In conclusion, the time complexity of sublinear interpolation search can be approximated as O(log(log(n))) in the average case, O(1) in the best case, and O(n) in the worst case.