Question

Consider the following list: 5,18,21,10,55,20 The first three keys are in order. To move 10 to its proper position using the insertion sort as described in this chapter, exactly how many key comparisons are executed?

Short answer

3 comparisons are executed.

Step-by-step solution

Understanding the Problem

We have a list of numbers: 5, 18, 21, 10, 55, 20. The task is to determine how many comparisons are needed to insert the number 10 into its proper position using insertion sort, given that the first three numbers (5, 18, 21) are already ordered.

Setup for Insertion Sort

Insertion sort works by taking an element from the unsorted portion of the list and inserting it into its correct position within the sorted portion. Here, 10 is the next element to be inserted into the already sorted list: 5, 18, 21.

First Comparison: 10 and 21

The number 10 is compared to the largest element in the sorted portion, which is 21. Since 10 is less than 21, a swap is necessary, and another comparison will be needed. This counts as the first comparison.

Second Comparison: 10 and 18

Now 10 is compared with the next largest element, 18. Since 10 is less than 18, we need to swap again and conduct another comparison. This counts as the second comparison.

Third Comparison: 10 and 5

Finally, 10 is compared with 5. Since 10 is greater than 5, 10 will be placed right after 5. No further comparisons are needed, and thus this counts as the third comparison.

Conclusion of Comparisons

In total, three comparisons were made to insert 10 into its proper position in the sorted section of the list, completing the process.

Key concepts

Key Comparisons

When learning how to sort a list using the insertion sort algorithm, one of the critical components is understanding key comparisons. Key comparisons are instances where two elements of a list are checked against each other to determine their correct order. In the context of an insertion sort, these comparisons allow us to insert an element from the unsorted part of the list into its rightful place within the sorted section.

In our example problem, there are three key comparisons needed to correctly position the number 10 among the previously ordered numbers 5, 18, and 21. Here's a breakdown of how each comparison works in this scenario:

• First Comparison: 10 is compared to 21. Since 10 is smaller, this initiates the process of finding the correct spot for 10 in the sorted list.
• Second Comparison: With 10 less than 18 as well, we perform another comparison to continue finding its position.
• Third Comparison: Finally, 10 is compared with 5. Since 10 is greater, this solidifies its position directly after 5 in the sorted sequence.

These comparisons not only help in understanding the insertion sort but also demonstrate the importance of determining the number of operations involved in sorting a list.

Sorting Algorithm

The insertion sort is a fundamental sorting algorithm that is often introduced in introductory computer science courses. It is appreciated for its simplicity and effectiveness when dealing with small or partially sorted arrays. The idea is to build a sorted portion of the list one element at a time, by inserting each new element into its proper place among the already sorted numbers.

Here's how it typically works:

• Start with the first element already sorted: Consider it a single-item list which is by default sorted.
• Pick the next element: Take an element from the unsorted section and determine its correct position in the sorted section with key comparisons.
• Shift the sorted elements: Move the elements that are larger than the picked element to make space for it.
• Insert the element: Place the picked element into its correct position.
• Repeat: Continue this process until all elements are sorted.

The insertion sort algorithm is particularly useful for small datasets or when the list is already partially sorted, as the number of comparisons and shifts required are minimized in such scenarios.

Data Structures

Understanding data structures is essential when working with sorting algorithms like insertion sort. A data structure is a specific way of organizing data so that it can be accessed and manipulated efficiently. In the context of sorting, we often work with arrays or lists, which are linear data structures that store elements in a specific order.

For insertion sort, the data structure is typically a linear array or list, which provides efficient access and modification capabilities needed for sorting:

• Array/ List Access: Direct access to each element in constant time, which is crucial for comparing elements during sorting.
• Shifting Elements: Insertion sort requires the ability to shift elements to make space for new ones. Arrays and lists allow this operation, though it may sometimes require additional computational effort, especially for large datasets.
• Dynamic vs. Static: Lists (dynamic) are more flexible than arrays (static) as they can grow in size, while arrays require explicit definition of their size.

Selecting the right data structure can influence the efficiency of the insertion sort, and understanding these structures will aid in grasping challenges and solutions in sorting algorithms.