What is the difference between exponential search and binary search?

Exponential search and binary search are both searching algorithms used to find a specific element in a sorted array. However, they differ in their approach and efficiency.

1. Approach:
- Exponential search starts by comparing the target element with the first element of the array. If they match, the search is complete. Otherwise, it doubles the position and continues to compare the target element with the element at that position until a larger element is found or the end of the array is reached. Once a larger element is found, it performs a binary search within the range of the previous and current positions.
- Binary search, on the other hand, starts by comparing the target element with the middle element of the array. If they match, the search is complete. Otherwise, it divides the array into two halves and continues the search in the appropriate half by comparing the target element with the middle element of that half. This process is repeated until the target element is found or the search range becomes empty.

2. Efficiency:
- Exponential search has a time complexity of O(log i), where i is the position of the target element. It performs a binary search on a range of size i/2 to i, which reduces the number of comparisons required compared to a traditional binary search.
- Binary search has a time complexity of O(log n), where n is the size of the array. It continuously divides the search range in half, resulting in a logarithmic time complexity.

3. Use cases:
- Exponential search is useful when the target element is likely to be found closer to the beginning of the array. It is particularly efficient for unbounded or infinite arrays where the size is unknown.
- Binary search is efficient for searching in large arrays where the target element is evenly distributed or likely to be found in the middle or towards the end of the array.

In summary, the main difference between exponential search and binary search lies in their approach and efficiency. Exponential search performs a linear search to find a range, followed by a binary search within that range, while binary search continuously divides the array in half. Exponential search is more efficient when the target element is closer to the beginning of the array or when the array size is unknown, while binary search is more efficient for evenly distributed or larger arrays.