Question

The _________ search algorithm repeatedly divides the portion of an array being searched in half.

Short answer

#Question# Using the Binary Search algorithm, find the index of the target value 7 in the given sorted array: [1, 5, 7, 10, 12, 15, 20].

Step-by-step solution

1. Understand the problem

For the Binary Search algorithm, we will be searching for a specific element in a sorted array. It adopts a divide and conquer strategy by repeatedly dividing the search interval in half. If the target value matches the middle element of the interval, we found the desired element. If the target value is less or greater than the middle element, the left or right half of the interval will be searched again, respectively.

2. Identify the Input and Output

The input for the Binary Search algorithm is a sorted array and a target value to search. The output is the index of the target element if found in the array, or -1 otherwise.

3. Perform a binary search on an example

Let's apply the Binary Search algorithm on the following example: Sorted array: [1, 5, 7, 10, 12, 15, 20] Target value: 10 Step by step solution: Step 1: Find the middle element of the array. Middle index = (0 + 6) / 2 = 3 (start index = 0, end index = 6) Middle element = Array[3] = 10 Step 2: Check if the middle element is equal to the target value. Middle element (10) == Target value (10) Step 3: The middle element is equal to the target value, so the index of the target element is the middle index. Index of target element = 3

4. Conclude the process

By following the Binary Search algorithm steps, we found that the target value (10) is located at index 3 in the sorted array. If the target value was not found in the array, we would return -1.

Key concepts

Search Algorithm

A search algorithm is a step-by-step procedure used to locate a specific item within a collection of data, such as an array or list. Different algorithms are designed based on the nature of the data and the speed of execution. One popular search algorithm is the Binary Search.

Binary Search is efficient because it systematically narrows down the search space. Instead of looking through each element one by one, it focuses on reducing the interval where the desired element might be located. This is done by continually checking the middle element of the search interval.

Let's look at how Binary Search works:

• First, identify the middle element of the sorted array or list.
• Compare it with the target value.
• If it matches, you've found the item; if not, decide whether to continue searching the left or right portion.

By halving the search space with each step, Binary Search is much faster than a linear search, especially in large datasets.

Sorted Array

A sorted array is a sequence of elements arranged in a particular order, usually numerical or alphabetical. Sorting is essential for certain algorithms, such as Binary Search, because it imposes structure on the data.

Why Use a Sorted Array?

• Efficiency: Searching and retrieving data is faster in sorted arrays, because you can skip unnecessary comparisons.
• Organization: Sorted data is easier to manage and visualize, beneficial for both human and computer processing.

In the context of Binary Search, having a sorted array is a prerequisite. Without sorting, Binary Search would not know which half of the interval to eliminate next, making the process ineffective. Challenges may arise in sorting large datasets initially, but the benefits in search efficiencies often outweigh these initial costs.

Divide and Conquer

Divide and conquer is a powerful strategy used in algorithm design, which involves breaking down a problem into smaller, more manageable sub-problems. Each sub-problem is solved individually, and combined to address the larger original problem.

In Binary Search, divide and conquer is applied by splitting the array into halves:

• First, you locate the middle element and compare it to your target value.
• If the middle element is not the target, Binary Search divides the current interval in two.
• This process repeats, continually halving the interval that needs to be searched.

The divide and conquer strategy is efficient because it reduces the problem size exponentially. Each division significantly shortens the amount of data that needs sorting, dramatically decreasing the time needed to find the solution. This approach is not only efficient in searching algorithms but also widely applied in sorting and optimization problems.