Question

Sort the following list using the bubble sort algorithm as discussed in this chapter. Show the list after each iteration of the outer for loop. \\[ 46,58,16,25,83,98,8,70,5,62 \\]

Short answer

The sorted list is: \[ 5, 8, 16, 25, 46, 58, 62, 70, 83, 98 \].

Step-by-step solution

Initial List

Start with the original list: \[ 46, 58, 16, 25, 83, 98, 8, 70, 5, 62 \]

First Iteration

During the first pass through the list, compare each pair of adjacent items and swap them if they are in the wrong order. The largest element moves to the end.Intermediate lists:- Compare 46 and 58: No swap - Compare 58 and 16: Swap \(58, 16\)- Compare 58 and 25: Swap \(58, 25\)- Compare 58 and 83: No swap- Compare 83 and 98: No swap- Compare 98 and 8: Swap \(98, 8\)- Compare 98 and 70: Swap \(98, 70\)- Compare 98 and 5: Swap \(98, 5\)- Compare 98 and 62: Swap \(98, 62\)List after first iteration:\[ 46, 16, 25, 58, 83, 8, 70, 5, 62, 98 \]

Second Iteration

Repeat the process, compare and swap:Intermediate lists:- Compare 46 and 16: Swap \(46, 16\)- Compare 46 and 25: Swap \(46, 25\)- Compare 46 and 58: No swap- Compare 58 and 83: No swap- Compare 83 and 8: Swap \(83, 8\)- Compare 83 and 70: Swap \(83, 70\)- Compare 83 and 5: Swap \(83, 5\)- Compare 83 and 62: Swap \(83, 62\)List after second iteration:\[ 16, 25, 46, 58, 8, 70, 5, 62, 83, 98 \]

Third Iteration

Continue the comparisons and swaps:Intermediate lists:- Compare 16 and 25: No swap- Compare 25 and 46: No swap- Compare 46 and 58: No swap- Compare 58 and 8: Swap \(58, 8\)- Compare 58 and 70: No swap- Compare 70 and 5: Swap \(70, 5\)- Compare 70 and 62: Swap \(70, 62\)List after third iteration:\[ 16, 25, 46, 8, 58, 5, 62, 70, 83, 98 \]

Fourth Iteration

Further sorting with comparisons and swaps:Intermediate lists:- Compare 16 and 25: No swap- Compare 25 and 46: No swap- Compare 46 and 8: Swap \(46, 8\)- Compare 46 and 58: No swap- Compare 58 and 5: Swap \(58, 5\)- Compare 58 and 62: No swapList after fourth iteration:\[ 16, 25, 8, 46, 5, 58, 62, 70, 83, 98 \]

Fifth Iteration

Continue sorting by comparing and swapping:Intermediate lists:- Compare 16 and 25: No swap- Compare 25 and 8: Swap \(25, 8\)- Compare 25 and 46: No swap- Compare 46 and 5: Swap \(46, 5\)- Compare 46 and 58: No swapList after fifth iteration:\[ 16, 8, 25, 5, 46, 58, 62, 70, 83, 98 \]

Sixth Iteration

Repeat the process again:Intermediate lists:- Compare 16 and 8: Swap \(16, 8\)- Compare 16 and 25: No swap- Compare 25 and 5: Swap \(25, 5\)List after sixth iteration:\[ 8, 16, 5, 25, 46, 58, 62, 70, 83, 98 \]

Seventh Iteration

Nearly sorted, continue swapping:Intermediate lists:- Compare 8 and 16: No swap- Compare 16 and 5: Swap \(16, 5\)List after seventh iteration:\[ 8, 5, 16, 25, 46, 58, 62, 70, 83, 98 \]

Eighth Iteration

Finalizing the sort with one last swap:Intermediate lists:- Compare 8 and 5: Swap \(8, 5\)List after eighth iteration:\[ 5, 8, 16, 25, 46, 58, 62, 70, 83, 98 \]

Completion of Sorting

After the eighth iteration, the list is fully sorted in ascending order and no more swaps are needed. Final List:\[ 5, 8, 16, 25, 46, 58, 62, 70, 83, 98 \]

Key concepts

Algorithm Analysis

Understanding how algorithms function and evaluating their efficiency is crucial in the world of computer science. Algorithm analysis helps to determine the resources necessary for a given sorting algorithm in terms of time and space. Specifically, the Bubble Sort algorithm is an intuitive and straightforward approach for arranging elements in a list. However, it's important to analyze various factors to comprehend its impact:

• Time Complexity: Bubble sort compares adjacent elements in the list, swapping them if needed. The process requires nested loops resulting in a time complexity of \( O(n^2) \), where \( n \) is the number of elements. This is not the most efficient for large datasets, as it takes longer as the list grows.
• Space Complexity: One advantage of the bubble sort is its small space complexity, \( O(1) \), as it requires no additional storage aside from temporary variables for swapping.
• Stability: Bubble sort is a stable sorting technique; it maintains the relative order of items with equal keys, which can be beneficial in scenarios where this property is needed.
• Adaptive Nature: Although bubble sort's basic form is not adaptive, modifications can be made to shorten pass cycles if no swaps occur, slightly improving efficiency in nearly sorted lists.

Sorting Techniques

Sorting is a fundamental operation in computer science making data easier to analyze and navigate. Bubble Sort stands out by its simplicity, perfect for understanding basic sorting concepts. The Bubble Sort method consists of repeatedly passing through the list, comparing adjacent elements, and swapping them if they are out of order. Each pass guarantees that the next largest unsorted element is placed in its correct position. Thus, the algorithm, although easy to follow, can be inefficient for large arrays, especially when compared to other sorting techniques like Quick Sort or Merge Sort. Key Characteristics of Bubble Sort:

• Comparison-based: It compares pairs of elements to decide their order.
• Iterative Approach: It involves many iterations over the data, making it simple but slow.
• In-place Sorting: Modifies the original data structure without needing additional arrays.
• Elementary in Nature: Designed to educate on sorting without complexities of other algorithms.

Bubble Sort proves a great introduction to sorting, preparing learners for more intricate algorithms.

Step-by-Step Solutions

Tackling a sorting problem step by step is a beneficial approach to solidify understanding, especially with Bubble Sort. This algorithm's step-by-step process showcases how fundamental sorting concepts work and interact. To solve using Bubble Sort, let's go through the mentioned process: 1. **Initial Observation:** Start with the entire list unsorted and take note of its order. 2. **First Pass:** Compare each adjacent pair, swapping them if needed, to move the largest unsorted element to the correct position. 3. **Subsequent Passes:** Repeat the process without including the sorted part of the list. 4. **Iterate:** Continue this until a pass requires no swaps, indicating that the list is sorted. Breaking it down this way emphasizes each decision made during sorting:

• Understanding pair comparison and when to swap.
• Recognizing how the sorted portion of the list grows with each pass.
• Ensuring lists are fully sorted when no swaps are made during a complete pass.

By offering a step-by-step solution, learners can visualize the sorting process more clearly and gain confidence in applying Bubble Sort to different sets of data.