Question

Consider the following sorted list of names. Arturo, Elsa, JoAnn, John, José, Lee, Snyder, Tracy a. Use binary search to decide whether Elsa is in this list. What names will be compared with Elsa? b. Use binary search to decide whether Tracy is in this list. What names will be compared with Tracy? c. Use binary search to decide whether Emile is in this list. What names will be compared with Emile?

Short answer

Elsa is found; compared: John, Elsa. Tracy is found; compared: John, Snyder, Tracy. Emile not found; compared: John, Elsa, JoAnn.

Step-by-step solution

Understanding Binary Search

Binary search is a method for finding a target value within a sorted list. It repeatedly divides the list in half, comparing the middle element to the target value, and deciding which half to search next based on the comparison.

Locate Elsa with Binary Search

1. Initial list: Arturo, Elsa, JoAnn, John, José, Lee, Snyder, Tracy (8 elements). 2. Middle element (index 3) is John. Compare Elsa with John. 3. Since Elsa < John, search the left half: Arturo, Elsa, JoAnn. 4. New middle element (index 1) is Elsa. Compare Elsa with Elsa. 5. Match found. Elsa is in the list.

Locate Tracy with Binary Search

1. Initial list: Arturo, Elsa, JoAnn, John, José, Lee, Snyder, Tracy (8 elements). 2. Middle element (index 3) is John. Compare Tracy with John. 3. Since Tracy > John, search the right half: José, Lee, Snyder, Tracy. 4. New middle element of the right half (index 6) is Snyder. Compare Tracy with Snyder. 5. Since Tracy > Snyder, search the right half: Tracy. 6. Only element left is Tracy. Match found. Tracy is in the list.

Locate Emile with Binary Search

1. Initial list: Arturo, Elsa, JoAnn, John, José, Lee, Snyder, Tracy (8 elements). 2. Middle element (index 3) is John. Compare Emile with John. 3. Since Emile < John, search the left half: Arturo, Elsa, JoAnn. 4. New middle element (index 1) is Elsa. Compare Emile with Elsa. 5. Since Emile > Elsa, search the right half: JoAnn. 6. Only element left is JoAnn. Compare Emile with JoAnn. 7. Since Emile < JoAnn and no more elements are available on the left, Emile is not in the list.

Key concepts

Search Algorithms

Search algorithms are methods designed to find a particular element in a dataset. They are essential in computer science for handling data efficiently. One powerful search algorithm is the **binary search**. This method is specifically designed for sorted lists, offering a more efficient alternative to linear search, which checks each element one by one until it finds the target.
Binary search works by dividing the list into two halves repeatedly. You start by comparing the middle element of the list to your target value. If it matches, you've found your element. If it's greater than your target, you eliminate the right half; if it's smaller, you eliminate the left half. This dramatically reduces the search space with each step, especially impressive considering you only navigate through a small segment of the list.
This efficiency relies heavily on the list being sorted initially, making it a go-to approach when dealing with pre-organized data.

Sorted Lists

Sorted lists are sequences where the elements are arranged based on a certain order, commonly numeric or lexical (alphabetical for words). **Sorting** before searching optimizes data handling and makes search algorithms like binary search applicable.
For binary search to function, a sorted list is crucial because it relies on the ability to predict where elements could reside based on order. Without sorting, binary search would yield incorrect results, as there's no guaranteed relationship between elements to guide the search.
Sorting algorithms prepare lists for efficient searches and can be implemented via various methods, like quicksort, mergesort, or bubblesort. Once sorted, lists allow faster data retrieval, making tasks that involve locating data quicker and less computationally expensive.

Algorithm Analysis

Algorithm analysis involves studying an algorithm's performance, which is crucial for determining its efficiency and resource usage in practical applications. Binary search is known for its efficiency in time complexity, commonly expressed as **O(log n)**. This notation means the time it takes to complete the search increases logarithmically with the number of items, rather than linearly.
Comparing an algorithm's time complexity provides insights into how well it can handle large data sets. For binary search, each decision cuts the list in half, promoting faster conclusions. By contrast, linear search has a time complexity of **O(n)**, meaning the time to complete the search grows directly with the list size.
Understanding algorithm analysis helps developers choose the right approach based on needs, factoring in limitations like time, space, and input size. Advancements in computing largely depend on efficient algorithms that handle increasingly massive datasets with speed and precision.