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Exponential search algorithm (also called doubling search, galloping search, Struzik search) is a search algorithm, created by *Jon Bentley* and *Andrew Chi-Chih Yao* in 1976, for searching sorted, unbounded/infinite lists.

### Algorithm

Exponential search involves two basic steps:

- Find range where element is present
- Execute Binary Search algorithm in above found range.

** How to find the range where element may be present? **

The idea is to start with sub-list of size 1. Ccompare the last element of the list with the target element, then try size 2, then 4 and so on until last element of the list is not greater.

Once we find a location `loc`

(after repeated doubling of list size), we know that the element must be present between `loc/2`

and `loc`

.

### Complexity

- Worst case time complexity:
`O(log i)`

where i is the index of the element being searched. - Average case time complexity:
`O(log i)`

- Best case time complexity:
`O(1)`

- Space complexity:
`O(1)`

### Implementations

Implementation of exponential search algorithm in 9 languages that includes `C`

, `C++`

, `Java`

, `Go`

, `JavaScript`

, `PHP`

, `Python`

, `Rust`

and `Swift`

.

- C
- C++

### C

```
#include <stdio.h>
int
min(int a, int b)
{
return (a < b ? a : b);
}
int
binarySearch(int arr[], int l, int r, int x)
{
if (r >= l) {
int mid = l + (r - l) / 2;
if (arr[mid] == x)
return (mid);
if (arr[mid] > x)
return (binarySearch(arr, l, mid - 1, x));
return (binarySearch(arr, mid + 1, r, x));
}
return (-1);
}
int
exponentialSearch(int arr[], int n, int x)
{
if (arr[0] == x)
return (0);
int i = 1;
while (i < n && arr[i] <= x)
i = i*2;
return (binarySearch(arr, i / 2, min(i, n), x));
}
int
main()
{
int n;
printf("Enter size of Array \n");
scanf("%d", &n);
int arr[n], i;
printf("Enter %d integers in ascending order \n", n);
for(i = 0; i < n; i++)
scanf("%d", &arr[i]);
int x;
printf("Enter integer to be searched \n");
scanf("%d", &x);
int result = exponentialSearch(arr, n, x);
if (result == -1)
printf("%d is not present in array \n", x);
else
printf("%d is present at index %d \n", x, result);
return (0);
}
```

### C++

```
// C++ program to find an element x in a
// sorted array using Exponential search.
#include <cstdio>
#define min(a, b) ((a) < (b) ? (a) : (b))
using namespace std;
int binarySearch(int arr[], int, int, int);
// Returns position of first ocurrence of
// x in array
int exponentialSearch(int arr[], int n, int x)
{
// If x is present at firt location itself
if (arr[0] == x)
return 0;
// Find range for binary search by
// repeated doubling
int i = 1;
while (i < n && arr[i] <= x)
i = i*2;
// Call binary search for the found range.
return binarySearch(arr, i/2, min(i, n), x);
}
// A recursive binary search function. It returns
// location of x in given array arr[l..r] is
// present, otherwise -1
int binarySearch(int arr[], int l, int r, int x)
{
if (r >= l)
{
int mid = l + (r - l)/2;
// If the element is present at the middle
// itself
if (arr[mid] == x)
return mid;
// If element is smaller than mid, then it
// can only be present n left subarray
if (arr[mid] > x)
return binarySearch(arr, l, mid-1, x);
// Else the element can only be present
// in right subarray
return binarySearch(arr, mid+1, r, x);
}
// We reach here when element is not present
// in array
return -1;
}
// Driver code
int main(void)
{
int arr[] = {2, 3, 4, 10, 40};
int n = sizeof(arr)/ sizeof(arr[0]);
int x = 10;
int result = exponentialSearch(arr, n, x);
(result == -1)? printf("Element is not present in array")
: printf("Element is present at index %d",
result);
return 0;
}
```

### Applications

- Exponential Binary Search is useful for unbounded searches where size of array is infinite.
- It works better than Binary Search for bounded arrays when the element to be searched is closer to the beginning of the array.