Binary Search's Efficient Cousin: Ternary Search Algorithm
Ternary search is a divide-and-conquer search algorithm that is particularly useful for evaluating real-valued functions where the function has a single local minimum or maximum. Unlike binary search, which is best suited for searching a sorted array, ternary search is effective for unimodal functions or arrays.
The algorithm works by repeatedly narrowing the search space by checking two middle points and discarding one-third of the range based on the values at those points. This process continues until the search space is small enough to identify the minimum or maximum with certainty.
Key Components and Steps
The ternary search algorithm involves several key components and steps. Here's a breakdown:
- Initialisation: The variable is used to store the answer and is initialised to -1. The variables , , , and are initialised based on the boundaries of the search space.
- Middle Index Calculation: is calculated as , and is calculated as .
- Updating Variables: In each iteration of the loop, the variables , , , and are updated based on the comparison of the array values at and . The is updated to either or depending on the comparison results.
- Termination Condition: The loop in Ternary Search runs while is not greater than . Once the search space is small enough, the algorithm terminates, and holds the position of the minimum or maximum.
Time and Space Complexity
The time complexity of Ternary Search is O(2 × logn), which is comparable to binary search in terms of efficiency. The auxiliary space complexity of Ternary Search is O(1), meaning it requires very little additional memory.
Differences with Binary Search
While both binary search and ternary search are divide-and-conquer algorithms, they have some key differences. Ternary search divides the search space into three equal parts, which can be slightly slower in practice due to the extra comparisons. On the other hand, binary search is faster and more commonly used for searching in sorted data with a time complexity of O(log2n).
Use Cases
Ternary search is particularly useful for optimization tasks like locating peak values in unimodal functions. It's also effective for solving problems like finding the bitonic point in a bitonic sequence and certain geometric or numeric optimization problems. In summary, ternary search is a valuable tool for finding the minimum or maximum of a unimodal array or function, especially when binary search is not applicable.
[1] For more information, refer to the original sources or relevant resources on ternary search.
- Algorithms, such as ternary search and binary search, are essential components in computer science and technology, particularly in the fields of education-and-self-development and science.
- Ternary search, unlike binary search, is not only an algorithm for evaluating a sorted array but also for unimodal functions or arrays, requiring math for division and calculation of middle points in the range.
- The Array data structure as well as the Trie data structure may benefit from the implementation of ternary search algorithms for finding minimum or maximum values.
- The time complexity of ternary search (O(2 × logn)) and binary search (O(log2n)) shows their efficiency in divide-and-conquer algorithms, while ternary search may be slightly slower in practice with more comparisons needed compared to binary search.
