The name of this searching algorithm may be misleading as it works in O(Log n) time. The name comes from the way it searches an element.
Given a sorted array an element x to be searched, find position of x in the array:
Input: arr[ ] = {10, 20, 40, 45, 55} x = 45
Output: Element found at index 3
Input: arr[ ] = {10, 15, 25, 45, 55} x = 15
Output: Element found at index 1
Exponential search involves two steps:
- Find range where element is present
- Do Binary Search in above found range.
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How to find the range where element may be present?
The idea is to start with sub-array size 1 compare its last element with x, then try size 2, then 4 and so on until last element of a sub-array is not greater.
Once we find an index i (after repeated doubling of i), we know that the element must be present between i/2 and i (Why i/2? because we could not find a greater value in previous iteration)
C++ Implementation:
c++
// C++ program to find an element x in a
// sorted array using Exponential search.
#include <bits/stdc++.h>
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;
}
Output :
Element is present at index 3
Time Complexity : O(Log n)
Auxiliary Space : The above implementation of Binary Search is recursive and requires O(Log n) space. With iterative Binary Search, we need only O(1) space.
Applications of Exponential Search:
- Exponential Binary Search is particularly useful for unbounded searches, where size of array is infinite. Please refer Unbounded Binary Search for an example.
- It works better than Binary Search for bounded arrays also when the element to be searched is closer to the first element.
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