# 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 nad 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 nad 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 nad 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 nad 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 nad 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 nad 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);
}
}

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 nad 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 nad 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); ` `    ``} ` `} `

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