Question

Assume the following list of keys: 8,28,31,20,55,46 The first three keys are in order. To move 20 to its proper position using the insertion sort algorithm as described in this chapter, exactly how many key comparisons are executed?

Short answer

2 key comparisons are required.

Step-by-step solution

Understanding the Problem

We are given a list of keys: 8, 28, 31, 20, 55, 46. The task is to find how many key comparisons are needed to insert the key '20' into its correct position using the insertion sort algorithm.

Locate Key Position for Insertion

The list of keys is: 8, 28, 31, 20, 55, 46. Among these keys, the first three (8, 28, 31) are in order. To find the correct position for '20', we need to compare it with these sorted keys.

Compare 20 with 31

First, compare the key '20' with the key '31', which is currently in the third position. Since 20 < 31, we will move 31 one position to the right.

Compare 20 with 28

Next, compare '20' with the previous key in the list, which is '28'. Since 20 < 28, move 28 one position to the right.

Compare 20 with 8

Then, compare '20' with key '8'. Since 20 > 8, no further move is required, and 20 is placed in this current position, which is after 8.

Count Key Comparisons

Two key comparisons were performed: 20 vs 31 and 20 vs 28. Therefore, the number of key comparisons required for inserting 20 in its correct position is 2.

Key concepts

Key Comparisons

In the context of the insertion sort algorithm, key comparisons are the backbone of determining the correct position of an element in a list. Each comparison involves looking at the element (or "key") being sorted and comparing it with the elements already sorted to decide its rightful place.

Here's how it works:

• When a key is selected for placement, it needs to be compared with those already sorted.
• Start from the end of the sorted subsection and compare moving backward.
• If the current key is less than the compared key, shift the compared key to the right and continue.
• Stop the process when the current key is no longer less than the compared key.

In the given exercise with the keys 8, 28, 31, 20, the key '20' is compared with '31' and then '28'. After these comparisons, 20 finds its place between 8 and 28. Thus, the number of key comparisons to correctly position '20' is crucial to understanding and optimizing sorting algorithms.

Algorithm Analysis

Algorithm analysis is a critical aspect of understanding how efficiently an algorithm performs, particularly in sorting tasks. When analyzing insertion sort, one focuses on:

• Time Complexity: Refers to how the execution time scales with the number of elements. In the worst case, insertion sort has a time complexity of \(O(n^2)\) due to comparing each element with all previous sorted ones.
• Space Complexity: Insertion sort is an in-place sort, which means it requires a constant amount of additional memory or \(O(1)\).
• Operation Count: Counting operations like key comparisons and swaps help in practical performance assessment.

For insertion sort, understanding the number of key comparisons and operations gives insight into how well it performs given a specific dataset. In real-world scenarios, insertion sort is mainly efficient for smaller datasets or nearly sorted lists, offering a simplified yet effective method in such cases.

Sorting Algorithms

Sorting algorithms are designed to arrange data in a particular order, typically ascending or descending. Insertion sort is one of the fundamental approaches used in sorting:

• Insertion Sort: Orders lists by comparing each element with those already sorted and placing it in the correct position.
• Characteristics: It's simple, intuitive, and works well for small datasets or lists nearly in order.

The benefit of insertion sort lies in its straightforward method, making it a good choice for educational purposes and understanding basic sorting mechanics. Although it's not as efficient as quicksort or mergesort, its ease of implementation is why it often serves as an introductory sorting algorithm in computer science courses.

Data Structures

Understanding data structures is key when applying sorting algorithms like insertion sort. A data structure determines how data is stored, organized, and manipulated. With insertion sort, the data structure used is typically a list or an array.

• Lists and Arrays: These structures allow elements to be accessed by index, providing the ability to compare and swap elements with ease.
• Mutable: Lists are mutable, meaning elements can be changed, moved, and inserted, which is essential for sorting algorithms like insertion sort.

Insertion sort leverages the characteristics of lists and arrays to efficiently place elements in order by making comparisons and shifting elements as needed. This direct manipulation within a list makes insertion sort intuitive and easy to implement, especially in educational settings where students are learning the basic principles of sorting and data organization.