Why Ternary Search?
Maximize your search capabilities and reduce time complexity with the introduction of the Ternary Search algorithm. One can look at this article as unlocking the power of efficient searching with the lesser-known, but highly effective algorithm known as the Ternary Search.
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. the algorithm involves ‘N’ steps. The searchable range would be the size = 3^N. Inversely, the number of steps needed to find the element is the log of the size of the collection. So the runtime is O(log N base 3).
The time complexity for ternary search is O (log N base 3 ) on average.
Best case time complexity is O(1), and worst-case complexity is O (log N base 3).
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.
Example:
Recursive Implementation of Ternary Search
C++
#include <bits/stdc++.h>
using namespace std;
int ternarySearch(int l, int r, int key, int ar[])
{
if (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
return ternarySearch(l, mid1 - 1, key, ar);
}
else if (key > ar[mid2]) {
return ternarySearch(mid2 + 1, r, key, ar);
}
else {
return ternarySearch(mid1 + 1, mid2 - 1, key, ar);
}
}
return -1;
}
int main()
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
cout << "Index of " << key
<< " is " << p << endl;
key = 50;
p = ternarySearch(l, r, key, ar);
cout << "Index of " << key
<< " is " << p << endl;
}
|
C
#include <stdio.h>
int ternarySearch(int l, int r, int key, int ar[])
{
if (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
return ternarySearch(l, mid1 - 1, key, ar);
}
else if (key > ar[mid2]) {
return ternarySearch(mid2 + 1, r, key, ar);
}
else {
return ternarySearch(mid1 + 1, mid2 - 1, key, ar);
}
}
return -1;
}
int main()
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
printf("Index of %d is %d\n", key, p);
key = 50;
p = ternarySearch(l, r, key, ar);
printf("Index of %d is %d", key, p);
}
|
Java
class GFG {
static int ternarySearch(int l, int r, int key, int ar[])
{
if (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
return ternarySearch(l, mid1 - 1, key, ar);
}
else if (key > ar[mid2]) {
return ternarySearch(mid2 + 1, r, key, ar);
}
else {
return ternarySearch(mid1 + 1, mid2 - 1, key, ar);
}
}
return -1;
}
public static void main(String args[])
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
System.out.println("Index of " + key + " is " + p);
key = 50;
p = ternarySearch(l, r, key, ar);
System.out.println("Index of " + key + " is " + p);
}
}
|
Python3
import math as mt
def ternarySearch(l, r, key, ar):
if (r >= l):
mid1 = l + (r - l) //3
mid2 = r - (r - l) //3
if (ar[mid1] == key):
return mid1
if (ar[mid2] == key):
return mid2
if (key < ar[mid1]):
return ternarySearch(l, mid1 - 1, key, ar)
elif (key > ar[mid2]):
return ternarySearch(mid2 + 1, r, key, ar)
else:
return ternarySearch(mid1 + 1,
mid2 - 1, key, ar)
return -1
l, r, p = 0, 9, 5
ar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
l = 0
r = 9
key = 5
p = ternarySearch(l, r, key, ar)
print("Index of", key, "is", p)
key = 50
p = ternarySearch(l, r, key, ar)
print("Index of", key, "is", p)
|
C#
using System;
class GFG {
static int ternarySearch(int l, int r, int key, int[] ar)
{
if (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
return ternarySearch(l, mid1 - 1, key, ar);
}
else if (key > ar[mid2]) {
return ternarySearch(mid2 + 1, r, key, ar);
}
else {
return ternarySearch(mid1 + 1, mid2 - 1, key, ar);
}
}
return -1;
}
public static void Main()
{
int l, r, p, key;
int[] ar = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
Console.WriteLine("Index of " + key + " is " + p);
key = 50;
p = ternarySearch(l, r, key, ar);
Console.WriteLine("Index of " + key + " is " + p);
}
}
|
PHP
<?php
function ternarySearch($l, $r, $key, $ar)
{
if ($r >= $l)
{
$mid1 = (int)($l + ($r - $l) / 3);
$mid2 = (int)($r - ($r - $l) / 3);
if ($ar[$mid1] == $key)
{
return $mid1;
}
if ($ar[$mid2] == $key)
{
return $mid2;
}
if ($key < $ar[$mid1])
{
return ternarySearch($l, $mid1 - 1,
$key, $ar);
}
else if ($key > $ar[$mid2])
{
return ternarySearch($mid2 + 1, $r,
$key, $ar);
}
else
{
return ternarySearch($mid1 + 1, $mid2 - 1,
$key, $ar);
}
}
return -1;
}
$ar = array( 1, 2, 3, 4, 5,
6, 7, 8, 9, 10 );
$l = 0;
$r = 9;
$key = 5;
$p = ternarySearch($l, $r, $key, $ar);
echo "Index of ", $key,
" is ", (int)$p, "\n";
$key = 50;
$p = ternarySearch($l, $r, $key, $ar);
echo "Index of ", $key,
" is ", (int)$p, "\n";
?>
|
Javascript
<script>
function ternarySearch(l, r, key, ar)
{
if (r >= l) {
let mid1 = l + parseInt((r - l) / 3, 10);
let mid2 = r - parseInt((r - l) / 3, 10);
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
return ternarySearch(l, mid1 - 1, key, ar);
}
else if (key > ar[mid2]) {
return ternarySearch(mid2 + 1, r, key, ar);
}
else {
return ternarySearch(mid1 + 1, mid2 - 1, key, ar);
}
}
return -1;
}
let l, r, p, key;
let ar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
document.write("Index of " + key + " is " + p + "</br>");
key = 50;
p = ternarySearch(l, r, key, ar);
document.write("Index of " + key + " is " + p);
</script>
|
Output: Index of 5 is 4
Index of 50 is -1
Time Complexity: O(log3n)
Auxiliary Space: O(log3n)
Iterative Approach of Ternary Search
C++
#include <iostream>
using namespace std;
int ternarySearch(int l, int r, int key, int ar[])
{
while (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
r = mid1 - 1;
}
else if (key > ar[mid2]) {
l = mid2 + 1;
}
else {
l = mid1 + 1;
r = mid2 - 1;
}
}
return -1;
}
int main()
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
cout << "Index of "<<key<<" is " << p << endl;
key = 50;
p = ternarySearch(l, r, key, ar);
cout << "Index of "<<key<<" is " << p;
}
|
C
#include <stdio.h>
int ternarySearch(int l, int r, int key, int ar[])
{
while (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
r = mid1 - 1;
}
else if (key > ar[mid2]) {
l = mid2 + 1;
}
else {
l = mid1 + 1;
r = mid2 - 1;
}
}
return -1;
}
int main()
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
printf("Index of %d is %d\n", key, p);
key = 50;
p = ternarySearch(l, r, key, ar);
printf("Index of %d is %d", key, p);
}
|
Java
class GFG {
static int ternarySearch(int l, int r, int key, int ar[])
{
while (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
r = mid1 - 1;
}
else if (key > ar[mid2]) {
l = mid2 + 1;
}
else {
l = mid1 + 1;
r = mid2 - 1;
}
}
return -1;
}
public static void main(String args[])
{
int l, r, p, key;
int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
System.out.println("Index of " + key + " is " + p);
key = 50;
p = ternarySearch(l, r, key, ar);
System.out.println("Index of " + key + " is " + p);
}
}
|
Python3
def ternarySearch(l, r, key, ar):
while r >= l:
mid1 = l + (r-l) // 3
mid2 = r - (r-l) // 3
if key == ar[mid1]:
return mid1
if key == ar[mid2]:
return mid2
if key < ar[mid1]:
r = mid1 - 1
elif key > ar[mid2]:
l = mid2 + 1
else:
l = mid1 + 1
r = mid2 - 1
return -1
ar = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
l = 0
r = 9
key = 5
p = ternarySearch(l, r, key, ar)
print("Index of", key, "is", p)
key = 50
p = ternarySearch(l, r, key, ar)
print("Index of", key, "is", p)
|
C#
using System;
public class GFG {
static int ternarySearch(int l, int r,
int key, int[] ar)
{
while (r >= l) {
int mid1 = l + (r - l) / 3;
int mid2 = r - (r - l) / 3;
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
r = mid1 - 1;
}
else if (key > ar[mid2]) {
l = mid2 + 1;
}
else {
l = mid1 + 1;
r = mid2 - 1;
}
}
return -1;
}
public static void Main(String[] args)
{
int l, r, p, key;
int[] ar = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
Console.WriteLine("Index of " + key + " is " + p);
key = 50;
p = ternarySearch(l, r, key, ar);
Console.WriteLine("Index of " + key + " is " + p);
}
}
|
Javascript
<script>
function ternarySearch(l, r, key, ar)
{
while (r >= l) {
let mid1 = l + parseInt((r - l) / 3, 10);
let mid2 = r - parseInt((r - l) / 3, 10);
if (ar[mid1] == key) {
return mid1;
}
if (ar[mid2] == key) {
return mid2;
}
if (key < ar[mid1]) {
r = mid1 - 1;
}
else if (key > ar[mid2]) {
l = mid2 + 1;
}
else {
l = mid1 + 1;
r = mid2 - 1;
}
}
return -1;
}
let l, r, p, key;
let ar = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];
l = 0;
r = 9;
key = 5;
p = ternarySearch(l, r, key, ar);
document.write("Index of " + key + " is " + p + "</br>");
key = 50;
p = ternarySearch(l, r, key, ar);
document.write("Index of " + key + " is " + p);
</script>
|
PHP
<?php
function ternarySearch(int $l, int $r, int $key, array $ar): int
{
while ($r >= $l) {
$mid1 = $l + (int) (($r - $l) / 3);
$mid2 = $r - (int) (($r - $l) / 3);
if ($ar[$mid1] == $key) {
return $mid1;
}
if ($ar[$mid2] == $key) {
return $mid2;
}
if ($key < $ar[$mid1]) {
$r = $mid1 - 1;
} elseif ($key > $ar[$mid2]) {
$l = $mid2 + 1;
} else {
$l = $mid1 + 1;
$r = $mid2 - 1;
}
}
return -1;
}
$ar = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];
$l = 0;
$r = 9;
$key = 5;
$p = ternarySearch($l, $r, $key, $ar);
echo "Index of $key is $p\n";
$key = 50;
$p = ternarySearch($l, $r, $key, $ar);
echo "Index of $key is $p\n";
?>
|
Output: 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)
Binary search Vs Ternary Search
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.
Advantages:
- Ternary Search has a time complexity of O(log3n), which is more efficient than linear search and comparable to binary search.
- Number of comparisons get reduced.
- Works well for large datasets.
- Fits well with optimization problems.
- Ternary Search is a non-recursive algorithm, so it does not require additional memory to store function call stack, thus it’s space efficient.
Disadvantages:
- Ternary Search is only applicable to ordered lists or arrays, and cannot be used on unordered or non-linear data sets.
- Requires an in depth understanding of recursion.
- Implementation is not easy.
- Ternary search is not suitable for non-continuous function as it is based on dividing the search space into 3 parts.
When to use:
- When you have a large ordered array or list and need to find the position of a specific value.
- When you need to find the maximum or minimum value of a function.
- When you need an alternative algorithm for binary search with an efficient time complexity.
- When you are interested in reducing the number of comparisons.
Summary:
- Ternary Search is a divide-and-conquer algorithm that is used to find the position of a specific value in a given array or list.
- It works by dividing the array into three parts and recursively performing the search operation on the appropriate part until the desired element is found.
- The algorithm has a time complexity of O(log3n) and is more efficient than a linear search, but less commonly used than other search algorithms like binary search.
- It’s important to note that the array to be searched must be sorted for Ternary Search to work correctly.