Question

For each of the following lists, perform a bubble sort, and show the list after each exchange. Compare the number of exchanges done here and in the Prectice Problem at the end of Soction \(3.33\). a. \(4,8,2,6\) b. \(12,3,6,8,2,5,7\) c. D, B, G, F, A, C, E, H

Short answer

List a: 3 swaps; List b: 13 swaps; List c: 12 swaps.

Step-by-step solution

Understand Bubble Sort

Bubble sort is a simple sorting algorithm that repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. This process is repeated until the list is sorted.

Bubble Sort on List a

Initial list: List: \(4, 8, 2, 6\)1. Compare 4 and 8, no swap needed. List: \(4, 8, 2, 6\)2. Compare 8 and 2, swap needed. List: \(4, 2, 8, 6\)3. Compare 8 and 6, swap needed. List: \(4, 2, 6, 8\)4. Compare 4 and 2, swap needed. List: \(2, 4, 6, 8\)5. Compare 4, 6, and 8, no swap needed. Number of exchanges: 3.

Bubble Sort on List b

Initial list: List: \(12, 3, 6, 8, 2, 5, 7\)1. Compare 12 and 3, swap needed. List: \(3, 12, 6, 8, 2, 5, 7\)2. Compare 12 and 6, swap needed. List: \(3, 6, 12, 8, 2, 5, 7\)3. Compare 12 and 8, swap needed. List: \(3, 6, 8, 12, 2, 5, 7\)4. Compare 12 and 2, swap needed. List: \(3, 6, 8, 2, 12, 5, 7\)5. Compare 12 and 5, swap needed. List: \(3, 6, 8, 2, 5, 12, 7\)6. Compare 12 and 7, swap needed. List: \(3, 6, 8, 2, 5, 7, 12\)7. Compare 6 and 8, no swap needed.8. Compare 8 and 2, swap needed. List: \(3, 6, 2, 8, 5, 7, 12\)9. Compare 8 and 5, swap needed. List: \(3, 6, 2, 5, 8, 7, 12\)10. Compare 8 and 7, swap needed. List: \(3, 6, 2, 5, 7, 8, 12\)11. Compare 6 and 2, swap needed. List: \(3, 2, 6, 5, 7, 8, 12\)12. Compare 6 and 5, swap needed. List: \(3, 2, 5, 6, 7, 8, 12\)13. Compare 3 and 2, swap needed. List: \(2, 3, 5, 6, 7, 8, 12\)Number of exchanges: 13.

Bubble Sort on List c

Initial list: List: D, B, G, F, A, C, E, H 1. Compare D and B, swap needed. List: B, D, G, F, A, C, E, H 2. Compare G and F, swap needed. List: B, D, F, G, A, C, E, H 3. Compare G and A, swap needed. List: B, D, F, A, G, C, E, H 4. Compare G and C, swap needed. List: B, D, F, A, C, G, E, H 5. Compare G and E, swap needed. List: B, D, F, A, C, E, G, H 6. Compare D and F, no swap needed. 7. Compare F and A, swap needed. List: B, D, A, F, C, E, G, H 8. Compare F and C, swap needed. List: B, D, A, C, F, E, G, H 9. Compare F and E, swap needed. List: B, D, A, C, E, F, G, H 10. Compare D and A, swap needed. List: B, A, D, C, E, F, G, H 11. Compare D and C, swap needed. List: B, A, C, D, E, F, G, H 12. Compare B and A, swap needed. List: A, B, C, D, E, F, G, H Number of exchanges: 12.

Compare Exhanges

List a required 3 exchanges, list b required 13 exchanges, and list c required 12 exchanges. Compare these numbers with the practice problem from Section 3.33.

Key concepts

Sorting Algorithms

Sorting algorithms are crucial in computer science as they arrange data in a specific order. This is essential for tasks such as searching and organizing information efficiently. Bubble Sort is an example of a basic sorting algorithm. It works by repeatedly stepping through the list, comparing adjacent elements, and swapping them if they are in the wrong order. This process continues until the list is sorted. Although simple to understand, Bubble Sort is not the most efficient algorithm for large datasets. It has a time complexity of \(O(n^2)\), meaning the time it takes to sort grows exponentially with the number of elements.
Despite this inefficiency, Bubble Sort is beneficial for educational purposes. It introduces key concepts such as comparisons and swaps, which form the basis of more complex algorithms. There are many sorting algorithms, each with strengths and weaknesses, such as Quick Sort for speed or Merge Sort for stability. Understanding the differences and the context in which each should be applied is crucial for efficient algorithmic design in programming.

Algorithm Analysis

Algorithm analysis involves evaluating the performance and efficiency of an algorithm. It is vital to determine how an algorithm will perform as the input size grows. Common metrics used in algorithm analysis include time complexity and space complexity. For instance, studying Bubble Sort involves recognizing that its time complexity is \(O(n^2)\). This means its performance suffers as the number of elements increases, due to a large number of comparisons and swaps needed.
Different types of algorithmic analyses provide insights into best, average, and worst-case scenarios. The best-case scenario for Bubble Sort, which is \(O(n)\), happens when the list is already mostly sorted. The average and worst cases, however, show more significant performance constraints. Algorithm analysis helps developers choose the most efficient algorithm for a particular problem by comparing these complexities and considering resource constraints.

Data Structures

Data structures are a method of organizing and storing data so that it can be accessed and modified efficiently. Common data structures include arrays, linked lists, stacks, and queues. In the context of sorting algorithms like Bubble Sort, arrays are often used. Arrays allow for easy manipulation of data since elements are stored in contiguous memory locations.
Sorting typically involves rearranging these elements within an array to introduce a specific order, such as ascending or descending. Knowing how data structures work and selecting the right one is crucial for the efficiency of sorting algorithms. Data structures impact the execution of an algorithm by influencing how data is stored and accessed. Bubble Sort operates directly on the data within the structure, highlighting the interplay between algorithms and data structures in computer science. Choosing an appropriate data structure for a given algorithm can significantly affect its performance and efficiency.