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Lesson 7, Topic 4
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# Exponential Search

##### Akash
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Exponential search is also known as doubling or galloping search. This mechanism is used to find the range where the search key may present. If L and U are the upper and lower bound of the list, then L and U both are the power of 2. For the last section, the U is the last position of the list. For that reason, it is known as exponential.

After finding the specific range, it uses the binary search technique to find the exact location of the search key.

#### The complexity of Exponential Search Technique

1. Time Complexity: O(1) for the best case. O(log2 i) for average or worst case. Where i is the location where search key is present.
2. Space Complexity: O(1)

#### Input and Output

```Input:
A sorted list of data:
10 13 15 26 28 50 56 88 94 127 159 356 480 567 689 699 780 850 956 995
The search key 780
Output:
Item found at location: 16```

#### Algorithm

binarySearch(array, start, end, key)

Input: An sorted array, start and end location, and the search key

Output: location of the key (if found), otherwise wrong location.

```Begin
if start <= end then
mid := start + (end - start) /2
if array[mid] = key then
return mid location
if array[mid] > key then
call binarySearch(array, mid+1, end, key)
else when array[mid] < key then
call binarySearch(array, start, mid-1, key)
else
return invalid location
End```

exponentialSearch(array, start, end, key)

Input: An sorted array, start and end location, and the search key

Output: location of the key (if found), otherwise wrong location.

```Begin
if (end – start) <= 0 then
return invalid location
i := 1
while i < (end - start) do
if array[i] < key then
i := i * 2 //increase i as power of 2
else
terminate the loop
done
call binarySearch(array, i/2, i, key)
End```

## Exponential Search Program in C

``````#include <stdio.h>

// Utility function to find minimum of two numbers
int min(int x, int y) {
return (x < y) ? x : y;
}

// Binary search algorithm to return the position of
// target x in the sub-array arr[low..high]
int BinarySearch(int arr[], int low, int high, int x)
{
// Base condition (search space is exhausted)
if (low > high)
return -1;

// we find the mid value in the search space and
// compares it with target value

int mid = (low + high)/2;	// overflow can happen
// int mid = low + (high - low)/2;

// Base condition (target value is found)
if (x == arr[mid])
return mid;

// discard all elements in the right search space
// including the mid element
else if (x < arr[mid])
return BinarySearch(arr, low,  mid - 1, x);

// discard all elements in the left search space
// including the mid element
else
return BinarySearch(arr, mid + 1,  high, x);
}

// Returns the position of target x in the given array of length n
int ExponentialSearch(int arr[], int n, int x)
{
int bound = 1;

// find the range in which the target x would reside
while (bound < n && arr[bound] < x)
bound *= 2;	// calculate the next power of 2

// call binary search on arr[bound/2 .. min(bound, n)]
return BinarySearch(arr, bound/2, min(bound, n), x);
}

// Exponential search algorithm
int main(void)
{
int arr[] = {2, 5, 6, 8, 9, 10};
int target = 9;

int n = sizeof(arr)/sizeof(arr[0]);
int index = ExponentialSearch(arr, n, target);

if (index != -1)
printf("Element found at index %d", index);
else

return 0;
}``````

Output :

``Element found at index 4``