Interpolation Search

Last Updated : 05 Feb, 2025

Given a sorted array of n uniformly distributed values arr[], write a function to search for a particular element x in the array.
Linear Search finds the element in O(n) time, Jump Search takes O(n) time and Binary Search takes O(log n) time.
The Interpolation Search is an improvement over Binary Search for instances, where the values in a sorted array are uniformly distributed. Interpolation constructs new data points within the range of a discrete set of known data points. Binary Search always goes to the middle element to check. On the other hand, interpolation search may go to different locations according to the value of the key being searched. For example, if the value of the key is closer to the last element, interpolation search is likely to start search toward the end side.
To find the position to be searched, it uses the following formula.

// The idea of formula is to return higher value of pos
// when element to be searched is closer to arr[hi]. And
// smaller value when closer to arr[lo]

arr[] ==> Array where elements need to be searched

x ==> Element to be searched

lo ==> Starting index in arr[]

hi ==> Ending index in arr[]

$pos = lo + [ \frac{(x-arr[lo])*(hi-lo) }{ (arr[hi]-arr[Lo]) }]$

There are many different interpolation methods and one such is known as linear interpolation. Linear interpolation takes two data points which we assume as (x1,y1) and (x2,y2) and the formula is : at point(x,y).

This algorithm works in a way we search for a word in a dictionary. The interpolation search algorithm improves the binary search algorithm. The formula for finding a value is: K = data-low/high-low.

K is a constant which is used to narrow the search space. In the case of binary search, the value for this constant is: K=(low+high)/2.

The formula for pos can be derived as follows.

Let's assume that the elements of the array are linearly distributed. 

General equation of line : y = m*x + c.
y is the value in the array and x is its index.

Now putting value of lo,hi and x in the equation
arr[hi] = m*hi+c ----(1)
arr[lo] = m*lo+c ----(2)
x = m*pos + c     ----(3)

m = (arr[hi] - arr[lo] )/ (hi - lo)

subtracting eqxn (2) from (3)
x - arr[lo] = m * (pos - lo)
lo + (x - arr[lo])/m = pos
pos = lo + (x - arr[lo]) *(hi - lo)/(arr[hi] - arr[lo])

Algorithm
The rest of the Interpolation algorithm is the same except for the above partition logic.

Step1: In a loop, calculate the value of “pos” using the probe position formula.
Step2: If it is a match, return the index of the item, and exit.
Step3: If the item is less than arr[pos], calculate the probe position of the left sub-array. Otherwise, calculate the same in the right sub-array.
Step4: Repeat until a match is found or the sub-array reduces to zero.

Below is the implementation of the algorithm.

C++

// C++ program to implement interpolation
// search with recursion
#include <bits/stdc++.h>
using namespace std;

// If x is present in arr[0..n-1], then returns
// index of it, else returns -1.
int interpolationSearch(int arr[], int lo, int hi, int x)
{
    int pos;

    // Since array is sorted, an element present
    // in array must be in range defined by corner
    if (lo <= hi && x >= arr[lo] && x <= arr[hi]) {

        // Probing the position with keeping
        // uniform distribution in mind.
        pos = lo
              + (((double)(hi - lo) / (arr[hi] - arr[lo]))
                 * (x - arr[lo]));

        // Condition of target found
        if (arr[pos] == x)
            return pos;

        // If x is larger, x is in right sub array
        if (arr[pos] < x)
            return interpolationSearch(arr, pos + 1, hi, x);

        // If x is smaller, x is in left sub array
        if (arr[pos] > x)
            return interpolationSearch(arr, lo, pos - 1, x);
    }
    return -1;
}

// Driver Code
int main()
{

    // Array of items on which search will
    // be conducted.
    int arr[] = { 10, 12, 13, 16, 18, 19, 20, 21,
                  22, 23, 24, 33, 35, 42, 47 };

    int n = sizeof(arr) / sizeof(arr[0]);

    // Element to be searched
    int x = 18;
    int index = interpolationSearch(arr, 0, n - 1, x);

    // If element was found
    if (index != -1)
        cout << "Element found at index " << index;
    else
        cout << "Element not found.";

    return 0;
}

// This code is contributed by equbalzeeshan

// C program to implement interpolation search
// with recursion
#include <stdio.h>

// If x is present in arr[0..n-1], then returns
// index of it, else returns -1.
int interpolationSearch(int arr[], int lo, int hi, int x)
{
    int pos;
    // Since array is sorted, an element present
    // in array must be in range defined by corner
    if (lo <= hi && x >= arr[lo] && x <= arr[hi]) {
        // Probing the position with keeping
        // uniform distribution in mind.
        pos = lo
              + (((double)(hi - lo) / (arr[hi] - arr[lo]))
                 * (x - arr[lo]));

        // Condition of target found
        if (arr[pos] == x)
            return pos;

        // If x is larger, x is in right sub array
        if (arr[pos] < x)
            return interpolationSearch(arr, pos + 1, hi, x);

        // If x is smaller, x is in left sub array
        if (arr[pos] > x)
            return interpolationSearch(arr, lo, pos - 1, x);
    }
    return -1;
}

// Driver Code
int main()
{
    // Array of items on which search will
    // be conducted.
    int arr[] = { 10, 12, 13, 16, 18, 19, 20, 21,
                  22, 23, 24, 33, 35, 42, 47 };
    int n = sizeof(arr) / sizeof(arr[0]);

    int x = 18; // Element to be searched
    int index = interpolationSearch(arr, 0, n - 1, x);

    // If element was found
    if (index != -1)
        printf("Element found at index %d", index);
    else
        printf("Element not found.");
    return 0;
}

Java

// Java program to implement interpolation
// search with recursion
import java.util.*;

class GFG {

    // If x is present in arr[0..n-1], then returns
    // index of it, else returns -1.
    public static int interpolationSearch(int arr[], int lo,
                                          int hi, int x)
    {
        int pos;

        // Since array is sorted, an element
        // present in array must be in range
        // defined by corner
        if (lo <= hi && x >= arr[lo] && x <= arr[hi]) {

            // Probing the position with keeping
            // uniform distribution in mind.
            pos = lo
                  + (((hi - lo) / (arr[hi] - arr[lo]))
                     * (x - arr[lo]));

            // Condition of target found
            if (arr[pos] == x)
                return pos;

            // If x is larger, x is in right sub array
            if (arr[pos] < x)
                return interpolationSearch(arr, pos + 1, hi,
                                           x);

            // If x is smaller, x is in left sub array
            if (arr[pos] > x)
                return interpolationSearch(arr, lo, pos - 1,
                                           x);
        }
        return -1;
    }

    // Driver Code
    public static void main(String[] args)
    {

        // Array of items on which search will
        // be conducted.
        int arr[] = { 10, 12, 13, 16, 18, 19, 20, 21,
                      22, 23, 24, 33, 35, 42, 47 };

        int n = arr.length;

        // Element to be searched
        int x = 18;
        int index = interpolationSearch(arr, 0, n - 1, x);

        // If element was found
        if (index != -1)
            System.out.println("Element found at index "
                               + index);
        else
            System.out.println("Element not found.");
    }
}

// This code is contributed by equbalzeeshan

Python

# Python3 program to implement
# interpolation search
# with recursion

# If x is present in arr[0..n-1], then
# returns index of it, else returns -1.


def interpolationSearch(arr, lo, hi, x):

    # Since array is sorted, an element present
    # in array must be in range defined by corner
    if (lo <= hi and x >= arr[lo] and x <= arr[hi]):

        # Probing the position with keeping
        # uniform distribution in mind.
        pos = lo + ((hi - lo) // (arr[hi] - arr[lo]) *
                    (x - arr[lo]))

        # Condition of target found
        if arr[pos] == x:
            return pos

        # If x is larger, x is in right subarray
        if arr[pos] < x:
            return interpolationSearch(arr, pos + 1,
                                       hi, x)

        # If x is smaller, x is in left subarray
        if arr[pos] > x:
            return interpolationSearch(arr, lo,
                                       pos - 1, x)
    return -1

# Driver code


# Array of items in which
# search will be conducted
arr = [10, 12, 13, 16, 18, 19, 20,
       21, 22, 23, 24, 33, 35, 42, 47]
n = len(arr)

# Element to be searched
x = 18
index = interpolationSearch(arr, 0, n - 1, x)

if index != -1:
    print("Element found at index", index)
else:
    print("Element not found")

# This code is contributed by Hardik Jain

// C# program to implement 
// interpolation search
using System;

class GFG{

// If x is present in 
// arr[0..n-1], then 
// returns index of it, 
// else returns -1.
static int interpolationSearch(int []arr, int lo, 
                               int hi, int x)
{
    int pos;
    
    // Since array is sorted, an element
    // present in array must be in range
    // defined by corner
    if (lo <= hi && x >= arr[lo] && 
                    x <= arr[hi])
    {
        
        // Probing the position 
        // with keeping uniform 
        // distribution in mind.
        pos = lo + (((hi - lo) / 
                (arr[hi] - arr[lo])) * 
                      (x - arr[lo]));

        // Condition of 
        // target found
        if(arr[pos] == x) 
        return pos; 
        
        // If x is larger, x is in right sub array 
        if(arr[pos] < x) 
            return interpolationSearch(arr, pos + 1,
                                       hi, x); 
        
        // If x is smaller, x is in left sub array 
        if(arr[pos] > x) 
            return interpolationSearch(arr, lo, 
                                       pos - 1, x); 
    } 
    return -1;
}

// Driver Code 
public static void Main() 
{
    
    // Array of items on which search will 
    // be conducted. 
    int []arr = new int[]{ 10, 12, 13, 16, 18, 
                           19, 20, 21, 22, 23, 
                           24, 33, 35, 42, 47 };
                           
    // Element to be searched                       
    int x = 18; 
    int n = arr.Length;
    int index = interpolationSearch(arr, 0, n - 1, x);
    
    // If element was found
    if (index != -1)
        Console.WriteLine("Element found at index " + 
                           index);
    else
        Console.WriteLine("Element not found.");
}
}

// This code is contributed by equbalzeeshan

JavaScript

<script>
// Javascript program to implement Interpolation Search

// If x is present in arr[0..n-1], then returns
// index of it, else returns -1.

function interpolationSearch(arr, lo, hi, x){
  let pos;
  
  // Since array is sorted, an element present
  // in array must be in range defined by corner
  
  if (lo <= hi && x >= arr[lo] && x <= arr[hi]) {
    
    // Probing the position with keeping
    // uniform distribution in mind.
    pos = lo + Math.floor(((hi - lo) / (arr[hi] - arr[lo])) * (x - arr[lo]));;
    
    // Condition of target found
        if (arr[pos] == x){
          return pos;
        }
 
        // If x is larger, x is in right sub array
        if (arr[pos] < x){
          return interpolationSearch(arr, pos + 1, hi, x);
        }
 
        // If x is smaller, x is in left sub array
        if (arr[pos] > x){
          return interpolationSearch(arr, lo, pos - 1, x);
        }
    }
    return -1;
}

// Driver Code
let arr = [10, 12, 13, 16, 18, 19, 20, 21, 
           22, 23, 24, 33, 35, 42, 47];

let n = arr.length;

// Element to be searched
let x = 18
let index = interpolationSearch(arr, 0, n - 1, x);

// If element was found
if (index != -1){
   document.write(`Element found at index ${index}`)
}else{
   document.write("Element not found");
}

// This code is contributed by _saurabh_jaiswal
</script>

PHP

<?php
// PHP program to implement $erpolation search
// with recursion

// If x is present in arr[0..n-1], then returns
// index of it, else returns -1.
function interpolationSearch($arr, $lo, $hi, $x)
{
    // Since array is sorted, an element present
    // in array must be in range defined by corner
    if ($lo <= $hi && $x >= $arr[$lo] && $x <= $arr[$hi]) {
        // Probing the position with keeping
        // uniform distribution in mind.
        $pos = (int)($lo
                     + (((double)($hi - $lo)
                         / ($arr[$hi] - $arr[$lo]))
                        * ($x - $arr[$lo])));

        // Condition of target found
        if ($arr[$pos] == $x)
            return $pos;

        // If x is larger, x is in right sub array
        if ($arr[$pos] < $x)
            return interpolationSearch($arr, $pos + 1, $hi,
                                       $x);

        // If x is smaller, x is in left sub array
        if ($arr[$pos] > $x)
            return interpolationSearch($arr, $lo, $pos - 1,
                                       $x);
    }
    return -1;
}

// Driver Code
// Array of items on which search will
// be conducted.
$arr = array(10, 12, 13, 16, 18, 19, 20, 21, 22, 23, 24, 33,
             35, 42, 47);
$n = sizeof($arr);

$x = 47; // Element to be searched
$index = interpolationSearch($arr, 0, $n - 1, $x);

// If element was found
if ($index != -1)
    echo "Element found at index ".$index;
else
    echo "Element not found.";
return 0;
#This code is contributed by Susobhan Akhuli
?>

Output

Element found at index 4

Time Complexity: O(log₂(log₂ n)) for the average case, and O(n) for the worst case
Auxiliary Space Complexity: O(1)

Another approach:-

This is the iteration approach for the interpolation search.

Step1: In a loop, calculate the value of “pos” using the probe position formula.
Step2: If it is a match, return the index of the item, and exit.
Step3: If the item is less than arr[pos], calculate the probe position of the left sub-array. Otherwise, calculate the same in the right sub-array.
Step4: Repeat until a match is found or the sub-array reduces to zero.

Below is the implementation of the algorithm.

C++

// C++ program to implement interpolation search by using iteration approach
#include<bits/stdc++.h>
using namespace std;
  
int interpolationSearch(int arr[], int n, int x)
{
    // Find indexes of two corners
    int low = 0, high = (n - 1);
    // Since array is sorted, an element present
    // in array must be in range defined by corner
    while (low <= high && x >= arr[low] && x <= arr[high])
    {
        if (low == high)
        {if (arr[low] == x) return low;
        return -1;
        }
        // Probing the position with keeping
        // uniform distribution in mind.
        int pos = low + (((double)(high - low) /
            (arr[high] - arr[low])) * (x - arr[low]));
  
        // Condition of target found
        if (arr[pos] == x)
            return pos;
        // If x is larger, x is in upper part
        if (arr[pos] < x)
            low = pos + 1;
        // If x is smaller, x is in the lower part
        else
            high = pos - 1;
    }
    return -1;
}
  
// Main function
int main()
{
    // Array of items on whighch search will
    // be conducted.
    int arr[] = {10, 12, 13, 16, 18, 19, 20, 21,
                 22, 23, 24, 33, 35, 42, 47};
    int n = sizeof(arr)/sizeof(arr[0]);
  
    int x = 18; // Element to be searched
    int index = interpolationSearch(arr, n, x);
  
    // If element was found
    if (index != -1)
        cout << "Element found at index " << index;
    else
        cout << "Element not found.";
    return 0;
}
 //this code contributed by  Ajay Singh

Java

// Java program to implement interpolation
// search with recursion
import java.util.*;

class GFG {

    // If x is present in arr[0..n-1], then returns
    // index of it, else returns -1.
    public static int interpolationSearch(int arr[], int lo,
                                        int hi, int x)
    {
        int pos;

        if (lo <= hi && x >= arr[lo] && x <= arr[hi]) {

            // Probing the position with keeping
            // uniform distribution in mind.
            pos = lo
                + (((hi - lo) / (arr[hi] - arr[lo]))
                    * (x - arr[lo]));

            // Condition of target found
            if (arr[pos] == x)
                return pos;

            // If x is larger, x is in right sub array
            if (arr[pos] < x)
                return interpolationSearch(arr, pos + 1, hi,
                                        x);

            // If x is smaller, x is in left sub array
            if (arr[pos] > x)
                return interpolationSearch(arr, lo, pos - 1,
                                        x);
        }
        return -1;
    }

    // Driver Code
    public static void main(String[] args)
    {

        // Array of items on which search will
        // be conducted.
        int arr[] = { 10, 12, 13, 16, 18, 19, 20, 21,
                    22, 23, 24, 33, 35, 42, 47 };

        int n = arr.length;

        // Element to be searched
        int x = 18;
        int index = interpolationSearch(arr, 0, n - 1, x);

        // If element was found
        if (index != -1)
            System.out.println("Element found at index "
                            + index);
        else
            System.out.println("Element not found.");
    }
}

Python

# Python equivalent of above C++ code 
# Python program to implement interpolation search by using iteration approach
def interpolationSearch(arr, n, x): 
  
    # Find indexes of two corners 
    low = 0
    high = (n - 1) 
  
    # Since array is sorted, an element present 
    # in array must be in range defined by corner 
    while low <= high and x >= arr[low] and x <= arr[high]: 
        if low == high: 
            if arr[low] == x: 
                return low; 
            return -1; 
  
        # Probing the position with keeping 
        # uniform distribution in mind. 
        pos = int(low + (((float(high - low)/( arr[high] - arr[low])) * (x - arr[low])))) 
  
        # Condition of target found 
        if arr[pos] == x: 
            return pos 
  
        # If x is larger, x is in upper part 
        if arr[pos] < x: 
            low = pos + 1; 
  
        # If x is smaller, x is in lower part 
        else: 
            high = pos - 1; 
      
    return -1
  
# Main function
if __name__ == "__main__":
    # Array of items on whighch search will 
    # be conducted.
    arr = [10, 12, 13, 16, 18, 19, 20, 21,
           22, 23, 24, 33, 35, 42, 47]
    n = len(arr) 
  
    x = 18 # Element to be searched
    index = interpolationSearch(arr, n, x) 
  
    # If element was found
    if index != -1: 
        print ("Element found at index",index)
    else: 
        print ("Element not found")

// C# program to implement interpolation search by using
// iteration approach
using System;

class Program
{
    // Interpolation Search function
    static int InterpolationSearch(int[] arr, int n, int x)
    {
        int low = 0;
        int high = n - 1;
  
        while (low <= high && x >= arr[low] && x <= arr[high]) 
        {
            if (low == high) 
            {
                if (arr[low] == x) 
                    return low; 
                return -1; 
            }
  
            int pos = low + (int)(((float)(high - low) / (arr[high] - arr[low])) * (x - arr[low]));
  
            if (arr[pos] == x) 
                return pos; 
  
            if (arr[pos] < x) 
                low = pos + 1; 
  
            else 
                high = pos - 1; 
        }
  
        return -1;
    }
  
    // Main function
    static void Main(string[] args)
    {
        int[] arr = {10, 12, 13, 16, 18, 19, 20, 21, 22, 23, 24, 33, 35, 42, 47};
        int n = arr.Length;
  
        int x = 18;
        int index = InterpolationSearch(arr, n, x);
  
        if (index != -1) 
            Console.WriteLine("Element found at index " + index);
        else 
            Console.WriteLine("Element not found");
    }
}

// This code is contributed by Susobhan Akhuli

JavaScript

// JavaScript program to implement interpolation search by using iteration approach

function interpolationSearch(arr, n, x) {
// Find indexes of two corners
let low = 0;
let high = n - 1;

// Since array is sorted, an element present
// in array must be in range defined by corner
while (low <= high && x >= arr[low] && x <= arr[high]) {
    if (low == high) {
        if (arr[low] == x) {
            return low;
        }
        return -1;
    }

    // Probing the position with keeping
    // uniform distribution in mind.
    let pos = Math.floor(low + (((high - low) / (arr[high] - arr[low])) * (x - arr[low])));

    // Condition of target found
    if (arr[pos] == x) {
        return pos;
    }

    // If x is larger, x is in upper part
    if (arr[pos] < x) {
        low = pos + 1;
    }

    // If x is smaller, x is in lower part
    else {
        high = pos - 1;
    }
}

return -1;
}
// Main function
let arr = [10, 12, 13, 16, 18, 19, 20, 21, 22, 23, 24, 33, 35, 42, 47];
let n = arr.length;

let x = 18; // Element to be searched
let index = interpolationSearch(arr, n, x);

// If element was found
if (index != -1) {
console.log("Element found at index", index);
} else {
console.log("Element not found");
}

Output

Element found at index 4

Time Complexity: O(log2(log2 n)) for the average case, and O(n) for the worst case
Auxiliary Space Complexity: O(1)

Exponential Search

Aayu Sachdev

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