Question

Consider the following list: 7,28,31,40,5,20 The first four keys are in order. To move 5 to its proper position using the insertion sort as described in this chapter, exactly how many key comparisons are executed?

Short answer

4 comparisons.

Step-by-step solution

Understand Insertion Sort

Insertion sort works by repeatedly taking an element from the unsorted part of the list and inserting it into its correct position within the sorted part. This involves shifting elements in the sorted section until the right spot is found.

Identify Sorted and Unsorted Sections

The given list is: 7, 28, 31, 40, 5, 20. The sorted section is the first four elements (7, 28, 31, 40), and the unsorted section starts from the fifth element, which is 5.

Move 5 to Correct Position

Begin with the unsorted element 5. Start comparing it with the last element of the sorted section, which is 40. Continue comparing 5 with each element of the sorted section: - Compare 5 with 40 and find 5 < 40, so shift 40 to the right. - Compare 5 with 31 and find 5 < 31, so shift 31 to the right. - Compare 5 with 28 and find 5 < 28, so shift 28 to the right. - Compare 5 with 7 and find 5 < 7, so shift 7 to the right. After these comparisons, the position for 5 is found at the beginning of the list.

Count the Comparisons

As we compare 5 against each element of the sorted section (40, 31, 28, and 7) and shift them, we count one comparison per shift. That's a total of 4 comparisons.

Key concepts

Sorting Algorithms

Sorting algorithms are fundamental tools in computer science, used to arrange data into a specific order. In everyday life, sorting is applied when arranging items, such as numbers or letters, in ascending or descending order. In the context of programming, sorting helps with organization, efficiency, and data management.

There are several types of sorting algorithms, each with its own strengths and weaknesses. Some common sorting algorithms include:

• Bubble Sort: Repeatedly swaps adjacent elements if they are in the wrong order.
• Merge Sort: Divides the list into halves, sorts them, and merges them back together.
• Quick Sort: Selects a 'pivot' element and partitions the other elements into two sub-arrays.
• Insertion Sort: Builds the final sorted list one item at a time, much like sorting playing cards by hand.

Each of these algorithms has unique characteristics and performance metrics, making them suitable for different applications.

Algorithm Analysis

Algorithm analysis is about understanding the efficiency of an algorithm. It's essential to determine how an algorithm performs in terms of time (time complexity) and space (space complexity).

Analyzing algorithms involves assessing:

• Time Complexity: How the execution time increases with the size of the input data.
• Space Complexity: How much additional memory the algorithm requires as the input size grows.

For insertion sort, the time complexity in the worst and average cases is \(O(n^2)\), where \(n\) is the number of elements. This is because it may need to compare and move each element of the list individually.

Understanding these complexities helps in choosing the most appropriate algorithm for a specific task, balancing speed and resources.

Key Comparisons

Key comparisons are a crucial element of sorting algorithms. They allow the algorithm to determine the correct order of elements by comparing their values.

In the context of insertion sort, a key comparison occurs every time the active, unsorted element is compared with an element of the sorted section. Let's dive into the process:

With the initial list \(7, 28, 31, 40, 5, 20\), the number 5 must find its proper position amongst the sorted portion. Insertions are made through key comparisons:

• Compare 5 with 40, and since 5 is smaller, 40 is shifted.
• Next, 5 is compared with 31, and 31 is shifted as well because 5 is smaller.
• This process repeats for 28 and 7, each resulting in a shift until 5 finds its proper place.

The exercise demonstrates four key comparisons. Each comparison informs whether a shift is necessary, progressively sorting the list by inserting the new element. Key comparisons are vital in maintaining the efficiency and correctness of sorting algorithms.