Question

Consider the following list: 28,18,21,10,25,30,12,71,32,58,15 This list is to be sorted using the insertion sort algorithm as described in this chapter. Show the resulting list after six passes of the sorting phase -that is, after six iterations of the for loop.

Short answer

After six passes: 10, 12, 18, 21, 25, 28, 30, 71, 32, 58, 15.

Step-by-step solution

Initial State

Start with the original list given: 28, 18, 21, 10, 25, 30, 12, 71, 32, 58, 15. This is the unsorted list that we'll sort using the insertion sort algorithm.

First Pass

Consider the second element 18 and insert it into the sorted sequence before it. - Compare 18 with 28. - Since 18 < 28, shift 28 to the right and place 18 at the beginning. Resulting list: 18, 28, 21, 10, 25, 30, 12, 71, 32, 58, 15.

Second Pass

Take the third element 21 and insert it into the sorted sequence before it (18, 28). - Compare 21 with 28; 21 < 28, so shift 28 to the right. - Compare 21 with 18; 21 > 18, so insert 21 between 18 and 28. Resulting list: 18, 21, 28, 10, 25, 30, 12, 71, 32, 58, 15.

Third Pass

Consider the fourth element 10 and insert it into the sorted sequence before it (18, 21, 28). - Compare 10 with 28, 21, and 18; shift each one to the right as all are greater than 10. - Insert 10 at the beginning. Resulting list: 10, 18, 21, 28, 25, 30, 12, 71, 32, 58, 15.

Fourth Pass

Take the fifth element 25 and find its position in the sorted sequence (10, 18, 21, 28). - Compare 25 with 28; 25 < 28, so shift 28 one position to the right. - 25 > 21, so insert 25 between 21 and 28. Resulting list: 10, 18, 21, 25, 28, 30, 12, 71, 32, 58, 15.

Fifth Pass

Consider the sixth element 30 and place it in its correct position in the sorted sequence (10, 18, 21, 25, 28). - Compare 30 with 28; 30 > 28. - Insert 30 after 28, which is already in its correct position. Resulting list: 10, 18, 21, 25, 28, 30, 12, 71, 32, 58, 15.

Sixth Pass

Take the seventh element 12 and find its spot in the sorted part (10, 18, 21, 25, 28, 30). - Compare 12 with 30, 28, 25, 21, and 18; shift all greater elements to the right. - Insert 12 after 10. Resulting list: 10, 12, 18, 21, 25, 28, 30, 71, 32, 58, 15.

Key concepts

Algorithm Analysis

Analyzing algorithms helps us understand their efficiency and performance for different inputs. Insertion sort has a clear methodology. It works by selecting each element and positioning it within a sorted subsection. For a list with "n" elements, insertion sort can take as few as \( O(n) \) operations in the best case, such as when the list is already sorted. However, in the average and worst cases, it requires \( O(n^2) \) comparisons and shifts. Why does this happen?- In each pass, elements can shift over.- Each insertion requires a search for the right position. When lists are reverse ordered, insertion sort operates at its worst.Despite this inefficiency on larger lists or unsorted data, insertion sort is efficient for sorting small data sets or for sorting partially sorted arrays.

Sorting Algorithms

Sorting algorithms are fundamental in computer science. They ensure data is in a specific order, which simplifies querying and decision-making processes. What makes a good sorting algorithm? - **Stability**: A stable sort keeps the order of equal elements unchanged. - **Adaptability**: Some adapt better to the initial data state. - **Complexity**: Time and space complexity determine an algorithm's efficiency. With insertion sort: - **Stability** is guaranteed, as it maintains the original order of equal elements. - **Adaptability** shines with small or partially sorted data but struggles with completely unsorted lists. Sorting algorithms either use comparison-based methods, like insertion sort, or non-comparison techniques, depending on the task.

Step-by-Step Problem Solving

Breaking problems into steps simplifies complex tasks. In insertion sort, we methodically handle one element at a time. Consider how its process works: 1. **Select an element**: It picks an element, starting from the second, as the first is considered sorted by itself. 2. **Compare and shift**: Each selected element is compared with the already-sorted part of the list, moving larger elements to the right. 3. **Insert**: Place the element in its correct position within the sorted subsection. This results in a growing sorted list. Walking through each pass carefully, as shown in the original exercise, demonstrates: - The list evolves gradually toward a fully sorted state. - After six passes, we see a significantly reordered sequence. Problem-solving in this manner ensures clarity and ease of understanding, especially for beginners learning sorting algorithms.