12.1.4. Example of Insertion Sort

So, let us continue with a discussion of sorting, and look at another simple sorting algorithm. So, as we said before that are many more motivation for sorting; starting from searching, to removing duplicates, to computing some statistical properties, such as frequency tables, and so on.

This introduces the topic of sorting algorithms, specifically focusing on Insertion Sort. The motivation for sorting includes various applications such as making searching easier, eliminating duplicates from lists, and calculating statistical details about data sets like frequency tables.

Think of sorting like organizing books on a shelf. You might sort them alphabetically by the author's name or by genre. This makes it easier to find a particular book, just like sorting algorithms organize data for easier retrieval.

So, the example that we have in hand, is one way we are ask to sort a bunch of exam papers in descending order of marks. The second strategy to sort this bunch of a exam papers would be the following. So, you take the top most paper in this stack that you have, and create a new stack that. Now you take the second paper and compare it to the first paper. If the mark is bigger, you put it about, because you wanted it in descending order. If the mark is smaller, you put it below.

Insertion Sort starts by taking the first element (the top-most paper, in this analogy) from an unsorted collection and placing it into a new sorted collection. Each subsequent element is then compared to the sorted elements and placed appropriately to maintain the order. This method continues until all elements are sorted.

Imagine you have a stack of exam papers that you want to sort by scores. You start with one paper, then take another and decide where it fits. If it's higher than what you have, it sits on top. If it’s lower, it goes underneath. You continue this process with each new paper until all are sorted in a nice stack.

So, we start by building a sorted sequence with one element. So, this is called insertion sort. And we insert each unsorted element into the correct place in the already unsorted sequence. So it is because of the insertion step, that every element is inserted into its correct place.

The key aspect of Insertion Sort is the actual insertion process. A sorted sequence starts with one element. Each new element is then compared to the already sorted sequence—inserting it into the correct position. This step is repeated until all elements are sorted.

Picture organizing a deck of cards. You begin with one card facing up, then take each new card and slide it into the right position among the previously sorted cards. This ensures that at any point, the cards leading up to that moment are in the correct order.

So, this is a very straight forward iterative implementation. So, what will do is, we have an initial array a, and position 0 to n minus 1. So, what will do; of course, is at the beginning we have to nothing. So, we cannot assume that we actually start with position 1. So, we start with position 1, and then we look backwards.

Insertion Sort can be implemented iteratively. You start at the second element in the array (index 1) and look backwards to find where it fits in relation to previous elements. If the current element is smaller than the one before it, they are swapped, and this continues until the current element is larger than the previously sorted elements or until the beginning of the array is reached.

Imagine you're organizing a line of people by height. You start with the person at the second position, compare their height to the one before them, and move them backward in the line until they are in the right place. This iterating backward continues until everyone is lined up correctly.

So, this brings the value that we started with, is in correct position. So, in general, if we had; say done up to sum i and then I start walk in backward. Then so long as i is smaller than i minus 1, I will exchange. I will keep exchanging, until I reach a position, where a find that the values on the left, are smaller than this value, and I will stop.

The time complexity of Insertion Sort is generally O(n^2) in the worst case because each element needs to be compared to potentially all already sorted elements. However, if the list is already sorted or nearly sorted, the algorithm performs significantly quicker—closer to linear time (O(n)).

Think about how much easier it would be to find a person's height in a queue if everyone is already in order. If you have to place a new person in a queue of already sorted heights, you can quickly find the right spot by comparing just a few heights instead of all of them, making the process much faster.

So, once again as we saw for selection sort. There is a natural way to think of insertion sort as a recursive thing. We sort part of the array, and then we insert an element into that to grow it.

Insertion Sort can also be implemented recursively. In this approach, you recursively sort the first part of the array, then insert the next element into that sorted section. The recursion continues until the entire array is sorted, functioning much like the iterative version.

Imagine teaching someone to organize cards. First, you teach them to organize a small portion of the cards, and then you show them how to add another card into that already organized pile. This continues until all cards are organized, demonstrating how recursive sorting works.

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