Ternary Search

20 Jun, 2022
Ternary search is a decrease(by constant) and conquer algorithm that can be used to find an element in an array. It is similar to binary search where we divide the array into two parts but in this algorithm, we divide the given array into three parts and determine which has the key (searched element). We can divide the array into three parts by taking mid1 and mid2 which can be calculated as shown below. Initially, l and r will be equal to 0 and n-1 respectively, where n is the length of the array. 
It is same as the binary search. The only difference is that, it reduces the time complexity a bit more. Its time complexity is O(log n base 3) and that of binary search is O(log n base 2).
mid1 = l + (r-l)/3 mid2 = r – (r-l)/3 
Note: Array needs to be sorted to perform ternary search on it.
Steps to perform Ternary Search: 
First, we compare the key with the element at mid1. If found equal, we return mid1.If not, then we compare the key with the element at mid2. If found equal, we return mid2.If not, then we check whether the key is less than the element at mid1. If yes, then recur to the first part.If not, then we check whether the key is greater than the element at mid2. If yes, then recur to the third part.If not, then we recur to the second (middle) part.
First, we compare the key with the element at mid1. If found equal, we return mid1.
If not, then we compare the key with the element at mid2. If found equal, we return mid2.
If not, then we check whether the key is less than the element at mid1. If yes, then recur to the first part.
If not, then we check whether the key is greater than the element at mid2. If yes, then recur to the third part.
If not, then we recur to the second (middle) part.
Example:
Recursive Implementation of Ternary Search 
C++
C
Java
Python3
C#
PHP
Javascript
// C++ program to illustrate// recursive approach to ternary search#include <bits/stdc++.h>using namespace std;  // Function to perform Ternary Searchint ternarySearch(int l, int r, int key, int ar[]){    if (r >= l) {          // Find the mid1 and mid2        int mid1 = l + (r - l) / 3;        int mid2 = r - (r - l) / 3;          // Check if key is present at any mid        if (ar[mid1] == key) {            return mid1;        }        if (ar[mid2] == key) {            return mid2;        }          // Since key is not present at mid,        // check in which region it is present        // then repeat the Search operation        // in that region        if (key < ar[mid1]) {              // The key lies in between l and mid1            return ternarySearch(l, mid1 - 1, key, ar);        }        else if (key > ar[mid2]) {              // The key lies in between mid2 and r            return ternarySearch(mid2 + 1, r, key, ar);        }        else {              // The key lies in between mid1 and mid2            return ternarySearch(mid1 + 1, mid2 - 1, key, ar);        }    }      // Key not found    return -1;}  // Driver codeint main(){    int l, r, p, key;      // Get the array    // Sort the array if not sorted    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    cout << "Index of " << key         << " is " << p << endl;      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    cout << "Index of " << key         << " is " << p << endl;}  // This code is contributed// by Akanksha_Rai
// C program to illustrate// recursive approach to ternary search  #include <stdio.h>  // Function to perform Ternary Searchint ternarySearch(int l, int r, int key, int ar[]){    if (r >= l) {          // Find the mid1 and mid2        int mid1 = l + (r - l) / 3;        int mid2 = r - (r - l) / 3;          // Check if key is present at any mid        if (ar[mid1] == key) {            return mid1;        }        if (ar[mid2] == key) {            return mid2;        }          // Since key is not present at mid,        // check in which region it is present        // then repeat the Search operation        // in that region          if (key < ar[mid1]) {              // The key lies in between l and mid1            return ternarySearch(l, mid1 - 1, key, ar);        }        else if (key > ar[mid2]) {              // The key lies in between mid2 and r            return ternarySearch(mid2 + 1, r, key, ar);        }        else {              // The key lies in between mid1 and mid2            return ternarySearch(mid1 + 1, mid2 - 1, key, ar);        }    }      // Key not found    return -1;}  // Driver codeint main(){    int l, r, p, key;      // Get the array    // Sort the array if not sorted    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    printf("Index of %d is %d\n", key, p);      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    printf("Index of %d is %d", key, p);}
// Java program to illustrate// recursive approach to ternary search  class GFG {      // Function to perform Ternary Search    static int ternarySearch(int l, int r, int key, int ar[])    {        if (r >= l) {              // Find the mid1 and mid2            int mid1 = l + (r - l) / 3;            int mid2 = r - (r - l) / 3;              // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }              // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region              if (key < ar[mid1]) {                  // The key lies in between l and mid1                return ternarySearch(l, mid1 - 1, key, ar);            }            else if (key > ar[mid2]) {                  // The key lies in between mid2 and r                return ternarySearch(mid2 + 1, r, key, ar);            }            else {                  // The key lies in between mid1 and mid2                return ternarySearch(mid1 + 1, mid2 - 1, key, ar);            }        }          // Key not found        return -1;    }      // Driver code    public static void main(String args[])    {        int l, r, p, key;          // Get the array        // Sort the array if not sorted        int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };          // Starting index        l = 0;          // length of array        r = 9;          // Checking for 5          // Key to be searched in the array        key = 5;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        System.out.println("Index of " + key + " is " + p);          // Checking for 50          // Key to be searched in the array        key = 50;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        System.out.println("Index of " + key + " is " + p);    }}
# Python3 program to illustrate# recursive approach to ternary searchimport math as mt  # Function to perform Ternary Searchdef ternarySearch(l, r, key, ar):      if (r >= l):          # Find the mid1 and mid2        mid1 = l + (r - l) //3        mid2 = r - (r - l) //3          # Check if key is present at any mid        if (ar[mid1] == key):             return mid1                  if (ar[mid2] == key):             return mid2                  # Since key is not present at mid,        # check in which region it is present        # then repeat the Search operation        # in that region        if (key < ar[mid1]):               # The key lies in between l and mid1            return ternarySearch(l, mid1 - 1, key, ar)                  elif (key > ar[mid2]):               # The key lies in between mid2 and r            return ternarySearch(mid2 + 1, r, key, ar)                  else:               # The key lies in between mid1 and mid2            return ternarySearch(mid1 + 1,                                  mid2 - 1, key, ar)              # Key not found    return -1  # Driver codel, r, p = 0, 9, 5  # Get the array# Sort the array if not sortedar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]  # Starting indexl = 0  # length of arrayr = 9  # Checking for 5  # Key to be searched in the arraykey = 5  # Search the key using ternarySearchp = ternarySearch(l, r, key, ar)  # Print the resultprint("Index of", key, "is", p)  # Checking for 50  # Key to be searched in the arraykey = 50  # Search the key using ternarySearchp = ternarySearch(l, r, key, ar)  # Print the resultprint("Index of", key, "is", p)  # This code is contributed by # Mohit kumar 29
// CSharp program to illustrate// recursive approach to ternary searchusing System;  class GFG {      // Function to perform Ternary Search    static int ternarySearch(int l, int r, int key, int[] ar)    {        if (r >= l) {              // Find the mid1 and mid2            int mid1 = l + (r - l) / 3;            int mid2 = r - (r - l) / 3;              // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }              // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region              if (key < ar[mid1]) {                  // The key lies in between l and mid1                return ternarySearch(l, mid1 - 1, key, ar);            }            else if (key > ar[mid2]) {                  // The key lies in between mid2 and r                return ternarySearch(mid2 + 1, r, key, ar);            }            else {                  // The key lies in between mid1 and mid2                return ternarySearch(mid1 + 1, mid2 - 1, key, ar);            }        }          // Key not found        return -1;    }      // Driver code    public static void Main()    {        int l, r, p, key;          // Get the array        // Sort the array if not sorted        int[] ar = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };          // Starting index        l = 0;          // length of array        r = 9;          // Checking for 5          // Key to be searched in the array        key = 5;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        Console.WriteLine("Index of " + key + " is " + p);          // Checking for 50          // Key to be searched in the array        key = 50;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        Console.WriteLine("Index of " + key + " is " + p);    }}  // This code is contributed by Ryuga
<?php// PHP program to illustrate// recursive approach to ternary search  // Function to perform Ternary Searchfunction ternarySearch($l, $r, $key, $ar){    if ($r >= $l)    {          // Find the mid1 and mid2        $mid1 = (int)($l + ($r - $l) / 3);        $mid2 = (int)($r - ($r - $l) / 3);          // Check if key is present at any mid        if ($ar[$mid1] == $key)         {            return $mid1;        }        if ($ar[$mid2] == $key)        {            return $mid2;        }          // Since key is not present at mid,        // check in which region it is present        // then repeat the Search operation        // in that region        if ($key < $ar[$mid1])         {              // The key lies in between l and mid1            return ternarySearch($l, $mid1 - 1,                                      $key, $ar);        }        else if ($key > $ar[$mid2])         {              // The key lies in between mid2 and r            return ternarySearch($mid2 + 1, $r,                                      $key, $ar);        }        else        {              // The key lies in between mid1 and mid2            return ternarySearch($mid1 + 1, $mid2 - 1,                                            $key, $ar);        }    }      // Key not found    return -1;}  // Driver code  // Get the array// Sort the array if not sorted$ar = array( 1, 2, 3, 4, 5,              6, 7, 8, 9, 10 );  // Starting index$l = 0;  // length of array$r = 9;  // Checking for 5  // Key to be searched in the array$key = 5;  // Search the key using ternarySearch$p = ternarySearch($l, $r, $key, $ar);  // Print the resultecho "Index of ", $key,     " is ", (int)$p, "\n";  // Checking for 50  // Key to be searched in the array$key = 50;  // Search the key using ternarySearch$p = ternarySearch($l, $r, $key, $ar);  // Print the resultecho "Index of ", $key,      " is ", (int)$p, "\n";  // This code is contributed by Arnab Kundu?>
<script>      // JavaScript program to illustrate    // recursive approach to ternary search          // Function to perform Ternary Search    function ternarySearch(l, r, key, ar)    {        if (r >= l) {               // Find the mid1 and mid2            let mid1 = l + parseInt((r - l) / 3, 10);            let mid2 = r - parseInt((r - l) / 3, 10);               // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }               // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region               if (key < ar[mid1]) {                   // The key lies in between l and mid1                return ternarySearch(l, mid1 - 1, key, ar);            }            else if (key > ar[mid2]) {                   // The key lies in between mid2 and r                return ternarySearch(mid2 + 1, r, key, ar);            }            else {                   // The key lies in between mid1 and mid2                return ternarySearch(mid1 + 1, mid2 - 1, key, ar);            }        }           // Key not found        return -1;    }          let l, r, p, key;       // Get the array    // Sort the array if not sorted    let ar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    document.write("Index of " + key + " is " + p + "</br>");      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    document.write("Index of " + key + " is " + p);          </script>
Index of 5 is 4
Index of 50 is -1
 
Time Complexity: O(log3n)
Auxiliary Space: O(log3n)
Iterative Approach of Ternary Search 
C++
C
Java
Python3
C#
Javascript
// C++ program to illustrate// iterative approach to ternary search  #include <iostream>using namespace std;  // Function to perform Ternary Searchint ternarySearch(int l, int r, int key, int ar[])  {    while (r >= l) {          // Find the mid1 and mid2        int mid1 = l + (r - l) / 3;        int mid2 = r - (r - l) / 3;          // Check if key is present at any mid        if (ar[mid1] == key) {            return mid1;        }        if (ar[mid2] == key) {            return mid2;        }          // Since key is not present at mid,        // check in which region it is present        // then repeat the Search operation        // in that region          if (key < ar[mid1]) {              // The key lies in between l and mid1            r = mid1 - 1;        }        else if (key > ar[mid2]) {              // The key lies in between mid2 and r            l = mid2 + 1;        }        else {              // The key lies in between mid1 and mid2            l = mid1 + 1;            r = mid2 - 1;        }    }      // Key not found    return -1;}  // Driver codeint main(){    int l, r, p, key;      // Get the array    // Sort the array if not sorted    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    cout << "Index of "<<key<<" is " << p << endl;      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    cout << "Index of "<<key<<" is " << p;}
// C program to illustrate// iterative approach to ternary search  #include <stdio.h>  // Function to perform Ternary Searchint ternarySearch(int l, int r, int key, int ar[])  {    while (r >= l) {          // Find the mid1 and mid2        int mid1 = l + (r - l) / 3;        int mid2 = r - (r - l) / 3;          // Check if key is present at any mid        if (ar[mid1] == key) {            return mid1;        }        if (ar[mid2] == key) {            return mid2;        }          // Since key is not present at mid,        // check in which region it is present        // then repeat the Search operation        // in that region          if (key < ar[mid1]) {              // The key lies in between l and mid1            r = mid1 - 1;        }        else if (key > ar[mid2]) {              // The key lies in between mid2 and r            l = mid2 + 1;        }        else {              // The key lies in between mid1 and mid2            l = mid1 + 1;            r = mid2 - 1;        }    }      // Key not found    return -1;}  // Driver codeint main(){    int l, r, p, key;      // Get the array    // Sort the array if not sorted    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    printf("Index of %d is %d\n", key, p);      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    printf("Index of %d is %d", key, p);}
// Java program to illustrate// the iterative approach to ternary search  class GFG {      // Function to perform Ternary Search    static int ternarySearch(int l, int r, int key, int ar[])      {        while (r >= l) {              // Find the mid1  mid2            int mid1 = l + (r - l) / 3;            int mid2 = r - (r - l) / 3;              // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }              // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region              if (key < ar[mid1]) {                  // The key lies in between l and mid1                r = mid1 - 1;            }            else if (key > ar[mid2]) {                  // The key lies in between mid2 and r                l = mid2 + 1;            }            else {                  // The key lies in between mid1 and mid2                l = mid1 + 1;                r = mid2 - 1;            }        }          // Key not found        return -1;    }      // Driver code    public static void main(String args[])    {        int l, r, p, key;          // Get the array        // Sort the array if not sorted        int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };          // Starting index        l = 0;          // length of array        r = 9;          // Checking for 5          // Key to be searched in the array        key = 5;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        System.out.println("Index of " + key + " is " + p);          // Checking for 50          // Key to be searched in the array        key = 50;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        System.out.println("Index of " + key + " is " + p);    }}
# Python 3 program to illustrate iterative# approach to ternary search  # Function to perform Ternary Searchdef ternarySearch(l, r, key, ar):    while r >= l:                  # Find mid1 and mid2        mid1 = l + (r-l) // 3        mid2 = r - (r-l) // 3          # Check if key is at any mid        if key == ar[mid1]:            return mid1        if key == mid2:            return mid2          # Since key is not present at mid,         # Check in which region it is present        # Then repeat the search operation in that region        if key < ar[mid1]:            # key lies between l and mid1            r = mid1 - 1        elif key > ar[mid2]:            # key lies between mid2 and r            l = mid2 + 1        else:            # key lies between mid1 and mid2            l = mid1 + 1            r = mid2 - 1      # key not found    return -1  # Driver code  # Get the list# Sort the list if not sortedar = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]  # Starting indexl = 0  # Length of listr = 9  # Checking for 5# Key to be searched in the listkey = 5  # Search the key using ternary searchp = ternarySearch(l, r, key, ar)  # Print the resultprint("Index of", key, "is", p)  # Checking for 50# Key to be searched in the listkey = 50  # Search the key using ternary searchp = ternarySearch(l, r, key, ar)  # Print the resultprint("Index of", key, "is", p)  # This code has been contributed by Sujal Motagi
// C# program to illustrate the iterative// approach to ternary searchusing System;  public class GFG {      // Function to perform Ternary Search    static int ternarySearch(int l, int r,                             int key, int[] ar)      {        while (r >= l) {              // Find the mid1 and mid2            int mid1 = l + (r - l) / 3;            int mid2 = r - (r - l) / 3;              // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }              // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region              if (key < ar[mid1]) {                  // The key lies in between l and mid1                r = mid1 - 1;            }            else if (key > ar[mid2]) {                  // The key lies in between mid2 and r                l = mid2 + 1;            }            else {                  // The key lies in between mid1 and mid2                l = mid1 + 1;                r = mid2 - 1;            }        }          // Key not found        return -1;    }      // Driver code    public static void Main(String[] args)    {        int l, r, p, key;          // Get the array        // Sort the array if not sorted        int[] ar = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };          // Starting index        l = 0;          // length of array        r = 9;          // Checking for 5          // Key to be searched in the array        key = 5;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        Console.WriteLine("Index of " + key + " is " + p);          // Checking for 50          // Key to be searched in the array        key = 50;          // Search the key using ternarySearch        p = ternarySearch(l, r, key, ar);          // Print the result        Console.WriteLine("Index of " + key + " is " + p);    }}  // This code has been contributed by 29AjayKumar
<script>      // JavaScript program to illustrate the iterative    // approach to ternary search          // Function to perform Ternary Search    function ternarySearch(l, r, key, ar)       {        while (r >= l) {               // Find the mid1 and mid2            let mid1 = l + parseInt((r - l) / 3, 10);            let mid2 = r - parseInt((r - l) / 3, 10);               // Check if key is present at any mid            if (ar[mid1] == key) {                return mid1;            }            if (ar[mid2] == key) {                return mid2;            }               // Since key is not present at mid,            // check in which region it is present            // then repeat the Search operation            // in that region               if (key < ar[mid1]) {                   // The key lies in between l and mid1                r = mid1 - 1;            }            else if (key > ar[mid2]) {                   // The key lies in between mid2 and r                l = mid2 + 1;            }            else {                   // The key lies in between mid1 and mid2                l = mid1 + 1;                r = mid2 - 1;            }        }           // Key not found        return -1;    }          let l, r, p, key;       // Get the array    // Sort the array if not sorted    let ar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];      // Starting index    l = 0;      // length of array    r = 9;      // Checking for 5      // Key to be searched in the array    key = 5;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    document.write("Index of " + key + " is " + p + "</br>");      // Checking for 50      // Key to be searched in the array    key = 50;      // Search the key using ternarySearch    p = ternarySearch(l, r, key, ar);      // Print the result    document.write("Index of " + key + " is " + p);      </script>
Index of 5 is 4
Index of 50 is -1
 
Time Complexity: O(log3n), where n is the size of the array.
Auxiliary Space: O(1)
The time complexity of the binary search is more than  the ternary search but it does not mean that ternary search is better. In reality, the number of comparisons in ternary search much more which makes it slower than binary search. 
 
Uses: In finding the maximum or minimum of a unimodal function.Hackerearth Problems on Ternary search 
 
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