In which scenario is ternary search useful?

Ternary search is a search algorithm that divides the search space into three parts instead of two, like binary search. This makes it more efficient for certain types of problems.

When to Use Ternary Search:

Ternary search is particularly useful when the search space is large and the function being searched is unimodal. A unimodal function is one that has a single maximum or minimum value.

Examples of Unimodal Functions:

• Finding the maximum or minimum value of a function
• Finding the root of a polynomial equation
• Finding the optimal solution to an optimization problem

Advantages of Ternary Search:

• Faster than binary search: Ternary search reduces the number of comparisons required to find the desired value.
• Suitable for large search spaces: It is more efficient than binary search when the search space is large.
• Applicable to unimodal functions: It is specifically designed for functions that have a single maximum or minimum value.

How Ternary Search Works:

Ternary search works by dividing the search space into three equal parts. It then evaluates the function at the two endpoints of the middle third. Based on the results, it eliminates one of the three parts and continues the search in the remaining two parts.

Example:

Suppose we want to find the maximum value of the function f(x) = x^3 – 3x^2 + 2x + 1 in the interval [0, 1].

• Step 1: Divide the interval into three equal parts: [0, 1/3], [1/3, 2/3], [2/3, 1].
• Step 2: Evaluate f(x) at the endpoints of the middle third: f(1/3) and f(2/3).
• Step 3: Since f(1/3) > f(2/3), the maximum value cannot be in the interval [2/3, 1].
• Step 4: Repeat steps 1-3 for the remaining interval [0, 1/3].
• Step 5: Eventually, we find the maximum value at x = 1/3.

Conclusion:

Ternary search is a powerful search algorithm that is particularly useful for finding the maximum or minimum value of a unimodal function in a large search space. It is faster than binary search and can significantly reduce the number of comparisons required.