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. 26,45,17,65,33,55,12,18

Short answer

Sorted list: 12, 17, 18, 26, 33, 45, 55, 65.

Step-by-step solution

Initial List

Start with the given list of numbers: 26, 45, 17, 65, 33, 55, 12, 18.

First Pass

In the first iteration of the outer loop, compare pairs of elements and swap them if needed, starting from the beginning of the list: - Compare 26 and 45, no swap. - Compare 45 and 17, swap → 26, 17, 45, 65, 33, 55, 12, 18. - Compare 45 and 65, no swap. - Compare 65 and 33, swap → 26, 17, 45, 33, 65, 55, 12, 18. - Compare 65 and 55, swap → 26, 17, 45, 33, 55, 65, 12, 18. - Compare 65 and 12, swap → 26, 17, 45, 33, 55, 12, 65, 18. - Compare 65 and 18, swap → 26, 17, 45, 33, 55, 12, 18, 65. List after first pass: 26, 17, 45, 33, 55, 12, 18, 65.

Second Pass

In the second iteration, again compare and swap elements: - Compare 26 and 17, swap → 17, 26, 45, 33, 55, 12, 18, 65. - Compare 26 and 45, no swap. - Compare 45 and 33, swap → 17, 26, 33, 45, 55, 12, 18, 65. - Compare 45 and 55, no swap. - Compare 55 and 12, swap → 17, 26, 33, 45, 12, 55, 18, 65. - Compare 55 and 18, swap → 17, 26, 33, 45, 12, 18, 55, 65. List after second pass: 17, 26, 33, 45, 12, 18, 55, 65.

Third Pass

Proceed with the third pass: - Compare 17 and 26, no swap. - Compare 26 and 33, no swap. - Compare 33 and 45, no swap. - Compare 45 and 12, swap → 17, 26, 33, 12, 45, 18, 55, 65. - Compare 45 and 18, swap → 17, 26, 33, 12, 18, 45, 55, 65. List after third pass: 17, 26, 33, 12, 18, 45, 55, 65.

Fourth Pass

Continue with the fourth pass: - Compare 17 and 26, no swap. - Compare 26 and 33, no swap. - Compare 33 and 12, swap → 17, 26, 12, 33, 18, 45, 55, 65. - Compare 33 and 18, swap → 17, 26, 12, 18, 33, 45, 55, 65. List after fourth pass: 17, 26, 12, 18, 33, 45, 55, 65.

Fifth Pass

Proceed with the fifth pass: - Compare 17 and 26, no swap. - Compare 26 and 12, swap → 17, 12, 26, 18, 33, 45, 55, 65. - Compare 26 and 18, swap → 17, 12, 18, 26, 33, 45, 55, 65. List after fifth pass: 17, 12, 18, 26, 33, 45, 55, 65.

Sixth Pass

In the sixth pass, only compare the first few elements: - Compare 17 and 12, swap → 12, 17, 18, 26, 33, 45, 55, 65. - Compare 17 and 18, no swap. List after sixth pass: 12, 17, 18, 26, 33, 45, 55, 65.

Seventh Pass (Optional)

The list is already sorted after the sixth pass, but a seventh pass verifies: - No swaps are necessary, confirming that the list is sorted. Final list: 12, 17, 18, 26, 33, 45, 55, 65.

Key concepts

Iteration in Sorting

When using the Bubble Sort Algorithm, iteration is key. It systematically processes through the list, starting from the very first element to the last one. Each iteration represents a complete pass through the list. During this pass, the adjacent elements are compared, which is crucial in identifying any misplaced items.

Every time an element is out of order compared to the one next to it, swapping is needed to correct this. As each pair of elements is evaluated, the heaviest items "bubble" to the end of the list. After every full iteration, at least one more element is correctly placed, reducing the unsorted section by one.

The process might seem repetitive, as it continues until no more swaps are required. This marks the moment when the list is fully sorted. Each loop through the list reinforces this pattern, demonstrating the power of iteration in simplifying the sorting problem.

Swapping Elements

Swapping elements is a fundamental part of the Bubble Sort Algorithm. It's the operation where two elements exchange their positions in the list. This happens when a smaller element is found after a larger one, and fixing their order can improve the list's arrangement.

Here's how swapping plays out in each step:

• Two adjacent items are compared.
• If the first is greater than the second, swapping occurs.
• This swapped pair will help inch the largest element to its final sorted position for each pass.

By moving these elements around, swapping helps organize elements by their correct rank in the order intended. It's a simple yet powerful operation that directly tackles the problem of misplaced elements, ultimately building toward an ordered list.

Comparison in Algorithms

Comparisons are the backbone of any sorting algorithm, including Bubble Sort. Each comparison assesses whether two elements are in the correct order or need to be swapped. This decision-making process based on pairwise evaluations is what drives the sorting.

In Bubble Sort, the number of comparisons increases as follows:

• Initially, every pair is checked.
• With each subsequent pass, fewer comparisons are needed because the last elements of the list are already sorted.
• The efficiency comes from reducing the range of comparison with every complete pass, as confirmed by the decreasing need for swaps.

Through comparisons, the algorithm identifies where swaps are necessary. It's an essential operation that, combined with swapping, results in a sorted array. A well-defined comparison ensures each element reaches its rightful place in the order of the list.