Ternary Search

Ternary search is a divide 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. In this, 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.

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:

1. First, we compare the key with the element at mid1. If found equal, we return mid1.
2. If not, then we compare the key with the element at mid2. If found equal, we return mid2.
3. If not, then we check whether the key is less than the element at mid1. If yes, then recur to the first part.
4. If not, then we check whether the key is greater than the element at mid2. If yes, then recur to the third part.
5. If not, then we recur to the second (middle) part.

Example: The code below shows the recursive implementation of ternary search:

C++

 // C++ program to illustrate // recursive approach to ternary search #include using namespace std;    // Function to perform Ternary Search 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 int 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

 // C program to illustrate // recursive approach to ternary search    #include    // Function to perform Ternary Search 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 int 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

 // 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

 # Python3 program to illustrate # recursive approach to ternary search import math as mt    # Function to perform Ternary Search def 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 code l, r, p = 0, 9, 5    # Get the array # Sort the array if not sorted 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 print("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 print("Index of", key, "is", p)    # This code is contributed by  # Mohit kumar 29

C#

 // CSharp program to illustrate  // recursive approach to ternary search  using 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

 = \$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 result echo "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 result echo "Index of ", \$key,       " is ", (int)\$p, "\n";    // This code is contributed by Arnab Kundu ?>

Output:

Index of 5 is 4
Index of 50 is -1

Iterative Approach

The code below shows the iterative approach to ternary search:

C

 // C program to illustrate // iterative approach to ternary search    #include    // Function to perform Ternary Search 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 int 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

 // 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);     } }

Python3

 # Python 3 program to illustrate iterative # approach to ternary search    # Function to perform Ternary Search def 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 sorted ar = [1,2,3,4,5,6,7,8,9,10]    # Starting index l = 0    # Length of list r = 9    # Checking for 5 # Key to be searched in the list key = 5    # Search the key using ternary search p = ternarySearch(l,r,key,ar)    # Print the result print("Index of",key,"is",p)    # Checking for 50 # Key to be searched in the list key = 50    # Search the key using ternary search p = ternarySearch(l, r, key, ar)    # Print the result print("Index of",key,"is",p)    # This code has been contributed by Sujal Motagi

C#

 // C# program to illustrate the iterative // approach to ternary search using 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

Output:

Index of 5 is 4
Index of 50 is -1

Time Complexity: , where n is the size of the array.

Uses: In finding the maximum or minimum of a unimodal function.

Hackerearth Problems on Ternary search

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.