Ternary Search

Like the binary search, it also separates the lists into sub-lists. This procedure divides the list into three parts using two intermediate mid values. As the lists are divided into more subdivisions, so it reduces the time to search a key value.

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.

The complexity of Ternary Search Technique

1. Time Complexity: O(log3 n)
2. Space Complexity: O(1)

Input and Output

Input:
A sorted list of data: 12 25 48 52 67 79 88 93
The search key 52
Output:
Item found at location: 3

Algorithm

ternarySearch(array, start, end, key)

Input: An sorted array, start and end location, and the search key

Output: location of the key (if found), otherwise wrong location.

Begin
   if start <= end then
      midFirst := start + (end - start) /3
      midSecond := midFirst + (end - start) / 3
      if array[midFirst] = key then
         return midFirst
      if array[midSecond] = key then
         return midSecond
      if key < array[midFirst] then
         call ternarySearch(array, start, midFirst-1, key)
      if key > array[midSecond] then
         call ternarySearch(array, midFirst+1, end, key)
      else
         call ternarySearch(array, midFirst+1, midSecond-1, key)
   else
      return invalid location
End

Ternary Search Program in C

// C program to illustrate 
// recursive approach to ternary search 
  
#include <stdio.h> 
  
// 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); 
} 

Output :

Index of 5 is 4
Index of 50 is -1

Iterative Approach of Ternary Search in C

// C program to illustrate 
// iterative approach to ternary search 
  
#include <stdio.h> 
  
// 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); 
} 

Output :

Index of 5 is 4
Index of 50 is -1