Interpolation Search

Interpolation search is an improved variant of binary search. This search algorithm works on the probing position of the required value. For this algorithm to work properly, the data collection should be in a sorted form and equally distributed.

Binary search has a huge advantage of time complexity over linear search. Linear search has worst-case complexity of Ο(n) whereas binary search has Ο(log n).

There are cases where the location of target data may be known in advance. For example, in case of a telephone directory, if we want to search the telephone number of Morphius. Here, linear search and even binary search will seem slow as we can directly jump to memory space where the names start from ‘M’ are stored.

Positioning in Binary Search

In binary search, if the desired data is not found then the rest of the list is divided in two parts, lower and higher. The search is carried out in either of them.

Even when the data is sorted, binary search does not take advantage to probe the position of the desired data.

Position Probing in Interpolation Search

Interpolation search finds a particular item by computing the probe position. Initially, the probe position is the position of the middle most item of the collection.

If a match occurs, then the index of the item is returned. To split the list into two parts, we use the following method −

mid = Lo + ((Hi - Lo) / (A[Hi] - A[Lo])) * (X - A[Lo])
where −
   A    = list
   Lo   = Lowest index of the list
   Hi   = Highest index of the list
   A[n] = Value stored at index n in the list

If the middle item is greater than the item, then the probe position is again calculated in the sub-array to the right of the middle item. Otherwise, the item is searched in the subarray to the left of the middle item. This process continues on the sub-array as well until the size of subarray reduces to zero.

Runtime complexity of interpolation search algorithm is Ο(log (log n)) as compared to Ο(log n) of BST in favorable situations.

Algorithm

As it is an improvisation of the existing BST algorithm, we are mentioning the steps to search the ‘target’ data value index, using position probing −

Step 1 − Start searching data from middle of the list.
Step 2 − If it is a match, return the index of the item, and exit.
Step 3 − If it is not a match, probe position.
Step 4 − Divide the list using probing formula and find the new midle.
Step 5 − If data is greater than middle, search in higher sub-list.
Step 6 − If data is smaller than middle, search in lower sub-list.
Step 7 − Repeat until match.

Pseudocode

A → Array list
N → Size of A
X → Target Value
Procedure Interpolation_Search()
   Set Lo  →  0
   Set Mid → -1
   Set Hi  →  N-1
   While X does not match
      if Lo equals to Hi OR A[Lo] equals to A[Hi]
         EXIT: Failure, Target not found
      end if
      Set Mid = Lo + ((Hi - Lo) / (A[Hi] - A[Lo])) * (X - A[Lo])
      if A[Mid] = X
         EXIT: Success, Target found at Mid
      else
         if A[Mid] < X
            Set Lo to Mid+1
         else if A[Mid] > X
            Set Hi to Mid-1
         end if
      end if
   End While
End Procedure

Interpolation search is an improved variant of binary search. This search algorithm works on the probing position of the required value. For this algorithm to work properly, the data collection should be in sorted and equally distributed form.

It’s runtime complexity is log₂(log₂ n).

Interpolation Search Program in C

#include<stdio.h>
#define MAX 10
// array of items on which linear search will be conducted.
int list[MAX] = { 10, 14, 19, 26, 27, 31, 33, 35, 42, 44 };
int find(int data) {
   int lo = 0;
   int hi = MAX - 1;
   int mid = -1;
   int comparisons = 1;
   int index = -1;
   while(lo <= hi) {
      printf("\nComparison %d  \n" , comparisons ) ;
      printf("lo : %d, list[%d] = %d\n", lo, lo, list[lo]);
      printf("hi : %d, list[%d] = %d\n", hi, hi, list[hi]);
      comparisons++;
      // probe the mid point
      mid = lo + (((double)(hi - lo) / (list[hi] - list[lo])) * (data - list[lo]));
      printf("mid = %d\n",mid);
      // data found
      if(list[mid] == data) {
         index = mid;
         break;
      } else {
         if(list[mid] < data) {
            // if data is larger, data is in upper half
            lo = mid + 1;
         } else {
            // if data is smaller, data is in lower half
            hi = mid - 1;
         }
      }
   }
   printf("\nTotal comparisons made: %d", --comparisons);
   return index;
}
int main() {
   //find location of 33
   int location = find(33);
   // if element was found
   if(location != -1)
      printf("\nElement found at location: %d" ,(location+1));
   else
      printf("Element not found.");
   return 0;
}

If we compile and run the above program, it will produce the following result −

Output

Comparison 1
lo : 0, list[0] = 10
hi : 9, list[9] = 44
mid = 6
Total comparisons made: 1
Element found at location: 7


You can change the search value and execute the program to test it.