Question

In a binary search, after three comparisons have been made, only ____________ of the array will be left to search.

Short answer

Answer: After three comparisons in a binary search, 1/8 of the array will be left to search.

Step-by-step solution

Understanding Binary Search

In a binary search, we first sort the array in ascending order and then start by comparing the target value with the element in the middle of the array. If the target value is equal to this element, the search is successful. If the target value is less than the middle element, the search continues in the left half of the array; otherwise, the search continues in the right half of the array. With each comparison, the array search space is reduced by half.

Calculate the Remaining Array to Search

After each comparison in a binary search, half of the current array is eliminated from the search space. So, we need to calculate how much of the array is left after three comparisons. 1. After the first comparison: \( \frac{1}{2} \cdot Original\_array\_size \) 2. After the second comparison: \( \frac{1}{2} \cdot \frac{1}{2} \cdot Original\_array\_size = \frac{1}{4} \cdot Original\_array\_size \) 3. After the third comparison: \( \frac{1}{2} \cdot \frac{1}{4} \cdot Original\_array\_size = \frac{1}{8} \cdot Original\_array\_size \)

Final Answer

After three comparisons have been made in a binary search, only 1/8 of the array will be left to search.

Key concepts

Algorithm Efficiency

When we talk about algorithm efficiency, particularly in the context of search algorithms, we're discussing how effectively an algorithm can perform a task with minimal resources. Binary search is often praised for its efficiency. Why? Because it drastically reduces the number of elements to check in each iteration.

In a binary search, every comparison eliminates half of the remaining search space. This means that even for large datasets, the number of required steps remains manageable, thanks to this halving approach. That's the power of efficiency at work.

• This efficiency is measured in time complexity, typically for binary search, it is $O(\log n)$, where $n$ is the number of elements in the array.
• By reducing the array size exponentially, it performs much faster than a linear search.

Efficient algorithms like binary search still require the array to be sorted first, as unsorted arrays make it impossible to exploit the halve-and-conquer strategy efficiently.

Array Search

Searching within arrays is a fundamental task in computing, where binary search shines most prominently. But how does searching in arrays work? Let's delve deeper!

There are different search methods employed for finding an element:

• Linear Search: A straightforward approach that checks every element, one at a time, against the target. It's simple but inefficient for large arrays, with a time complexity of $O(n)$.
• Binary Search: As discussed, it exploits a divide-and-conquer mechanism on sorted arrays, quickly pinning down the target with $O(\log n)$ complexity.

Binary search exemplifies an intelligent approach: knowing that the array is sorted, it doesn't just randomly access elements but rather strategically divides the array, providing quick results.

Search Algorithms

Search algorithms help us to locate desired elements within a data structure efficiently. The choice of a search algorithm affects how quickly and effectively we can find our target.

Binary search is a classical example of such an algorithm and is known for its distinct strategy:

• It requires pre-sorted data to function correctly. If the array is unsorted, elements could be anywhere, nullifying the binary core advantage.
• By dividing the array repeatedly, it simplifies a seemingly daunting problem into smaller, more manageable parts.

Understanding search algorithms like binary search involves recognizing their structures and behaviors, capacity to efficiently reduce the search domain, and their conditions for use, mainly concerning sorted data. This knowledge enables us to apply the right tool to solve complex searching problems effectively.

Divide and Conquer Strategy

The divide and conquer strategy is a powerful technique that forms the foundation of many efficient algorithms, including binary search.

This strategy involves three basic steps:

• Divide: Split the problem into smaller subproblems.
• Conquer: Solve the subproblems recursively.
• Combine: Integrate the solutions of subproblems to solve the original problem.

In binary search, dividing and conquering entails slicing the array repetitively until the target is located. It leverages the problem's structure, repeatedly halving the array, ensuring that with every recursive step, the problem space is reduced by half.

This method not only hones in on the solution efficiently but models an approach applicable to other domains beyond sorting and searching, demonstrating vertical scalability of problem-solving methods in computer science.