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Lesson 7, Topic 5
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# Ternary Search

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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);
}
}

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;
}
}

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``````