Question

Assume the following list of keys: 12,38,45,50,55,5,30 The first five keys are in order. To move 5 to its proper position using the insertion sort algorithm as described in this chapter, exactly how many key comparisons are executed?

Short answer

5 key comparisons are needed.

Step-by-step solution

Initial List and Positioning

Start with the list of keys: 12, 38, 45, 50, 55, 5, 30. The first five elements (12, 38, 45, 50, and 55) are already in sorted order. The task is to move the key '5' to its proper position in the sorted part of the list.

Position '5' for Insertion

Identify the position where ‘5’ should be inserted. Starting from the end of the sorted portion (55), compare ‘5’ with each key in reverse order to find the correct position.

First Comparison

Compare '5' with '55'. Since '5' is less than '55', a swap or shift is necessary. Continue to the next comparison.

Second Comparison

Compare '5' with '50'. Again, '5' is less than '50', so another swap or shift is needed. Move to the next element.

Third Comparison

Compare '5' with '45'. '5' is still less than '45'. The process is repeated for the next element.

Fourth Comparison

Compare '5' with '38'. '5' is less than '38', so continue comparing.

Fifth and Final Comparison

Compare '5' with '12'. Here, '5' is less than '12', confirming that '5' should be inserted before '12', making this the last comparison.

Key concepts

Sorting Algorithm

Insertion sort is a simple yet effective sorting algorithm used to organize data in a list. The idea behind insertion sort is straightforward. You loop through the list, one element at a time, and insert each item into its correct position relative to the already sorted portion of the list.

Let's break it down! Think of sorting playing cards. You pick each card one by one and place it in the right spot among the cards you've already sorted. This is exactly how insertion sort works! It builds a sorted list in place, and it gradually increases the size of this sorted portion of the list.

Why is insertion sort popular? Due to its simplicity, it is easy to code and perfect for lists that are mostly sorted already. However, be aware that insertion sort is less efficient on large lists than more advanced algorithms like quicksort or mergesort.

Key Comparisons

Key comparisons are a crucial part of any sorting algorithm, including insertion sort. They determine where an element should be placed in the sorted list.

In our insertion sort example, key comparisons are made to determine the correct placement for the number '5'. Specifically, '5' is compared to each of the preceding numbers in the list (55, 50, 45, 38, 12) until we find the appropriate spot. Each comparison tells the algorithm whether to shift the current key further down the list.

In terms of insertion sort, key comparisons are synonymous with search operations that help find the perfect spot for each new element from the unsorted portion of the list into the sorted part.

Data Structures

Data Structures are the foundation of storing and organizing data efficiently. In the context of sorting algorithms, the type of data structure can greatly impact the performance.

With insertion sort, lists are the common data structures used. Lists are simple, essentially providing a sequential collection of elements upon which insertion sort performs its operations efficiently.

The decision of which data structure to use can depend on the context of the data. However, in insertion sort, the traditional array (or a similar contiguous memory structure) works the best, given how each element shifts directly during the sorting phase. This choice offers simplicity, which is ideal if wrapped with insertion sort.

Algorithm Efficiency

Algorithm efficiency is a vital consideration in implementing any sorting algorithm. It refers to how quickly and effectively an algorithm solves a problem, typically in terms of time and space complexity.

For insertion sort, efficiency depends heavily on the initial order of the elements in the list. If a list is almost sorted, insertion sort performs fairly well because fewer shifts are needed. However, in the worst-case scenario, where the list is sorted in reverse, every new element needs comparisons and shifts through the entire sorted section, giving insertion sort a time complexity of \(O(n^2)\).

So, while insertion sort might not be the fastest for large datasets, it's a great choice for smaller ones or lists that are already close to being sorted. Its simplicity and possibility for implementation with less overhead also make it an attractive choice in specific scenarios.