Searching Algorithm 5: Exponential Search

Exponential search is an improvement to binary search. We use this algorithm when we have large amount of data.

In exponential search algorithm we use binary search. But the difference is, we get the range of elements to search from and give that as input to binary search.

Now how do we select the range of elements?

Initially we start with “1”. Then for each pass, we increase the gap 2 times everytime.

Once we get our range, we give that range as input to binary search.

int exponential_search(arr[], int size, int key)
{

    int bound = 1;
    while (bound < size && arr[bound] < key) 
    {
    //for every pass, increase the gap 2 times.
        bound *= 2;
    }

    return binary_search(arr, key, bound/2, min(bound + 1, size));
}

 

Let us understand this algorithm with the help of an example

Consider the array:

arr[] = {1, 2, 3, 4, 5, 6, 7, 8};

and the search key = 7;

Initially, the bound value will be 1 for pass 1, now check “bound < size && arr[bound] < key” ? It is true. because bound = 1 that is less than 8 and arr[1] < key.

Now, the bound value will be 2 for pass 2, now check “bound < size && arr[bound] < key” ? It is true. because bound = 2 that is less than 8 and arr[2] < key.

Now, the bound value will be 4 for pass 3, , now check “bound < size && arr[bound] < key” ? It is true. because bound = 4 that is less than 8 and arr[4] < key.

Now, the bound value will be 8 for pass 4, now check “bound < size && arr[bound] < key” ? It is false. because bound = 8 that is not less than 8. Hence exit the loop.

Then we perform binary search by calling binary_search(arr, key, bound/2, min(bound + 1, size)).

as bound is 8, bound/2 = 4.

min(8+1, 8) = 8

Now we will call binary search as binary_search(arr, 7,  4, 8);

Now we know the exact range of elements where our key element is present. Now we perform binary search on that range and return if we found the element or not.

#include<iostream>
#include <vector>

using namespace std;


void binarySearch(int arr[], int start, int end, int key) 
{ 
  if (end >= start) 
  { 
    int mid = start + (end - start)/2; 

    if (arr[mid] == key)
    {
      cout<<"Element found"<<endl;
      return; 
    }

    if (arr[mid] > key) 
      return binarySearch(arr, start, mid-1, key); 

    return binarySearch(arr, mid+1, end, key); 
  }

  cout<<"Element NOT found"<<endl;
  return; 
} 



void exponential_search(int arr[], int size, int key) 
{ 
  if (arr[0] == key)
  {
    cout<<"Element found"<<endl;
    return; 
  }

  int bound = 1; 
  while (bound < size && arr[bound] <= key)
  {
    cout<<"Bound = "<<bound<<endl;
    bound = bound*2; 
  }

  binarySearch(arr, bound/2,  min(bound + 1, size), key);
  return; 
} 


int main() 
{  
    int arr[] = {1, 2, 3, 4, 5, 6, 7, 8};
    int len = 8;
    int key = 7;

    exponential_search(arr, len, key);

    return 0; 
}

Element found

The time complexity of the algorithm will be O(log i) at worst case.

Further Reading:

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