Question

Assume the following list of keys: 25,32,20,15,45,4,18,91,62,88,66 This list is to be sorted using the insertion sort algorithm as described in this chapter for array-based lists. Show the resulting list after seven passes of the sorting phase - that is, after seven iterations of the for loop.

Short answer

After seven passes: 4,15,18,20,25,32,45,91,62,88,66.

Step-by-step solution

Initialize and Start Sorting

Start with the list: 25,32,20,15,45,4,18,91,62,88,66. The insertion sort begins by treating the first element (25) as a sorted subsection. Begin the second pass by comparing the second element with the first.

First Pass

The second element 32 is compared with 25. Since 32 is greater than 25, they remain in their positions. List after the first pass: 25,32,20,15,45,4,18,91,62,88,66.

Second Pass

Take the third element 20 and compare it with the elements on its left. It is less than 32 and 25, so it is shifted left past both. Insert 20 at the beginning: 20,25,32,15,45,4,18,91,62,88,66.

Third Pass

Consider the fourth element 15. It's compared with the elements before it. 15 is less than 32, 25, and 20. Move them rightward and insert 15 in its place: 15,20,25,32,45,4,18,91,62,88,66.

Fourth Pass

Take the fifth element 45. Compare it with the elements before. 45 is greater than 32, so no shifts are needed. List remains: 15,20,25,32,45,4,18,91,62,88,66.

Fifth Pass

Consider the sixth element 4, which is smaller than all preceding elements. Move all elements to the right to insert 4 at the start: 4,15,20,25,32,45,18,91,62,88,66.

Sixth Pass

Look at the seventh element 18. It's smaller than 45 but greater than 15. Shift 25, 32, and 45 to the right and insert 18: 4,15,18,20,25,32,45,91,62,88,66.

Seventh Pass

Finally, consider the eighth element 91, which is in the correct position as it is greater than 45. No changes are made. List remains: 4,15,18,20,25,32,45,91,62,88,66.

Key concepts

Sorting Algorithms

In the world of computer science, sorting algorithms are essential for managing data efficiently. These algorithms organize data in a specific order, usually ascending or descending. Sorting plays a crucial role in optimizing the performance of other algorithms that require sorted data sets. For instance, search algorithms like binary search rely heavily on sorted arrays.

Sorting algorithms come in various types, including bubble sort, merge sort, quick sort, and insertion sort. Each type has its own advantages and complexities. The choice of a sorting algorithm depends on several factors, such as the size of the dataset and the desired performance characteristics. Insertion sort, which the original exercise highlights, is a simple and intuitive algorithm suitable for small datasets. It works by gradually building a sorted sequence one element at a time.

Understanding sorting algorithms provides a foundation for computational thinking and helps students develop skills in problem-solving and data manipulation.

Array-Based Lists

Array-based lists serve as a fundamental data structure in most programming languages. They allow for the storage and manipulation of data, making them vital for various operations, including sorting. An array is essentially a collection of items stored at contiguous memory locations, accessible through indexing.

In the context of sorting, arrays provide a straightforward method to examine and manipulate data elements. Accessing elements in an array is efficient, typically with a time complexity of O(1), which means you can reach any element instantly using its index. This feature is beneficial during the sorting process, as elements can be compared and repositioned quickly.

When using insertion sort on array-based lists, like in the original exercise, elements are shifted within the array to their correct position, maintaining the array's efficiency and compactness. This iterative repositioning is key to forming a sorted list.

Iterative Sorting

Iterative sorting techniques, such as insertion sort, involve the repeated execution of a set of operations until a certain condition is met. These methods gradually refine the dataset into its sorted form. Unlike recursive sorting techniques, iterative sorting relies on loops and repeated steps rather than function calls.

Insertion sort embodies iterative sorting perfectly. It involves multiple passes through the array, during which elements are repositioned within the growing sorted portion of the list. Each pass is a step closer to achieving a completely sorted list. The number of iterations, or passes, directly affects the performance of the sort.

For example, in the original problem, insertion sort performs seven passes to organize the list up to a certain point. It iteratively compares each element with those preceding it and places it at its correct position in the sorted section. This method is effective for lists that are already partially sorted, making it a preferred choice in such scenarios.

Computational Thinking

Computational thinking is an essential skill for solving problems systematically and efficiently using computer science principles. At its core, it involves understanding a problem deeply, breaking it down into manageable parts, and developing a clear and logical sequence of steps to solve it.

In the context of the original exercise, computational thinking is applied by analyzing how the insertion sort algorithm can be implemented on an array-based list. Students learn to decompose the task of sorting a list into simpler operations, such as comparing elements and shifting them.

The practice of identifying patterns and abstractions, and applying these to solve complex problems like sorting, embodies computational thinking. Students also develop the ability to evaluate the efficiency of their solutions, refining them to improve performance and accuracy. Through exercises like these, learners enhance their critical thinking and problem-solving abilities, important not only in computing but also in other domains.