Question

Recall the insertion sort algorithm (contiguous version) as discussed in this chapter. Assume the following list of keys: 30,20,35,27,96,82,56,60,48,75,5,80 Exactly how many key comparisons are executed to sort this list using the insertion sort algorithm?

Short answer

33 key comparisons are executed.

Step-by-step solution

Understanding Insertion Sort

The insertion sort algorithm works by dividing the array into a sorted and an unsorted part. It iteratively takes one element from the unsorted part and inserts it into the correct position in the sorted part. This continues until all elements are sorted.

Initialize Variables

Start with the first element (30) as sorted. The rest of the list is unsorted. Initialize a variable to count the key comparisons, starting at zero.

Process Each Element

For each element in the unsorted portion (from the second element to the last), compare it to elements in the sorted portion, moving from right to left, until you find the right position or reach the beginning of the array. Insert the element and count each comparison until its correct position is found.

Execute Insertion Sort and Count Comparisons

1. Start with the second element, 20: Compare with 30. (1 comparison) 2. Third element, 35: Compare with 30. (1 comparison) 3. Fourth element, 27: Compare with 35, 30, then 20. (3 comparisons) 4. Fifth element, 96: Compare with 35. (1 comparison) 5. Sixth element, 82: Compare with 96, then 35. (2 comparisons) 6. Seventh element, 56: Compare with 82, 35, then 30. (3 comparisons) 7. Eighth element, 60: Compare with 82, 56, and 35. (3 comparisons) 8. Ninth element, 48: Compare with 60, 56, 35, and 30. (4 comparisons) 9. Tenth element, 75: Compare with 82, then 60. (2 comparisons) 10. Eleventh element, 5: Compare with all elements 75 to 20. (11 comparisons) 11. Last element, 80: Compare with 96, then 82. (2 comparisons) Total Comparisons = 1 + 1 + 3 + 1 + 2 + 3 + 3 + 4 + 2 + 11 + 2 = 33.

Conclusion

By following the insertion sort algorithm and counting each comparison as we sorted the example list, we determined there were a total of 33 key comparisons required to sort the given list.

Key concepts

Key Comparisons

Key comparisons are an essential part of many sorting algorithms, including insertion sort. They are the operations where two elements are checked against each other to decide their order. During insertion sort:

• We compare elements from the unsorted section with elements in the sorted section.
• We count each of these comparisons to understand the algorithm's effort to organize the list.

For example, if you have a list starting with [30, 20, 35], a key comparison takes place when 20 is compared with 30 to decide if any swapping is needed.
In our given example, we executed a total of 33 key comparisons to sort the list of numbers:

• Single comparisons occur often when elements are already somewhat ordered.
• Multiple comparisons happen more frequently when inserting a smaller element into the middle of a sorted list.

Understanding the number of key comparisons required provides insight into the insertion sort's efficiency.

Sorting Algorithms

Sorting algorithms are methods used to order elements in a particular data structure according to a defined sequence. Insertion sort is one of these methods characterized by its simplicity and ease of implementation.
Insertion sort belongs to the class of comparison-based sorting algorithms. Here's how it differs from others in its class:

• While quicksort breaks down the list into smaller parts, insertion sort builds a sorted array one element at a time.
• Unlike mergesort, which divides and conquers, insertion sort works by taking elements from one side of the list and carefully inserting each into its proper place on the other side.

Insertion sort is particularly efficient for small datasets or lists that are already partially sorted, as it minimizes unnecessary operations compared to other more complex algorithms.

Algorithm Analysis

Algorithm analysis helps us understand the efficiency and performance of an algorithm. For sorting algorithms like insertion sort, it's crucial to analyze based on time complexity.

• The best-case time complexity of insertion sort is \( O(n) \), occurring when the list is already sorted.
• The average case is \( O(n^2) \), which reflects inserting each element into a nearly random spot in the sorted portion.
• The worst-case scenario also yields \( O(n^2) \), such as when the list is sorted in reverse order.

Algorithm analysis not only considers the time complexity but also appreciates the following aspects:

• The number of key comparisons and swaps, which impacts CPU resource usage.
• The algorithm's adaptability to different types of data inputs.

By analyzing algorithms, students and professionals can choose the most appropriate sorting technique based on the data characteristics they confront.