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## Exponential Search
Exponential search is an algorithm for searching in a sorted list. For a sorted list of size $n$, binary search works in $\log{(n)}$, but can we do better that this?
Exponential search is an algorithm for searching in a sorted list. For a sorted list of size $n$, binary search works in $\log{(n)}$, but can we do better than this?
Apparantely, no. But what if we use some mechanism to determine a smaller index-range, in which the element we are searching(say $X$) belongs?
Apparently, no. But what if we use some mechanism to determine a smaller index-range, in which the element we are searching(say $X$) belongs?
For example, we are searching for $8$ in the list ${2,4,5,8,9,10}$. Now, if we use binary search then the searching index-range will be $1$ to $6$. But we can see that if we binary search in the index-range $3$ to $6$ then also we will be able to find $8$.
Note that the number of comparisons for smaller index-range is lesser than the ordinary binary search, this is the main idea behind **Exponential search**.
But how do we determine such index-range for a given element?
But how do we determine such an index-range for a given element?
As the name of the algorithm suggests, we are going to use exponentiation. We find an index of form $2^k$(starting from $k$=$0$), such that the element at that index is greater than $X$.
As the name of the algorithm suggests, we are going to use exponentiation. We find an index of the form $2^k$(starting from $k$=$0$), such that the element at that index is greater than $X$.
After that we do binary search over the index-range $2^{k-1}$ to $2^k$. It is easy to see that $X$ is in the index-range $2^{k-1}$ to $2^k$, because element at index $2^{k-1}$ is lesser then $X$ and element at index $2^k$ is greater than $X$.
After that, we do binary search over the index-range $2^{k-1}$ to $2^k$. It is easy to see that $X$ is in the index-range $2^{k-1}$ to $2^k$, because element at index $2^{k-1}$ is lesser then $X$ and element at index $2^k$ is greater than $X$.
Let's have an example,
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### When to use exponential search?
Worst case time-complexity of exponential search is $\mathcal{O}(\log(n))$.
When the element is near to the front of the list, exponential search performs better than binary search for the case of very large list.
When the element is near to the front of the list, exponential search performs better than binary search for the case of a very large list.
For an example purpose, Let say for below array we are searching for 3,`[1,3,4,6,10,14,18,20,22,23,25,28,30,33,36,39,40,41]`. Then it is easy to see that binary search certainly takes more comparisons then exponential search.