12.1.7. Comparative Performance of Sorting Algorithms

If you look at the analysis, it is quite similar to selection sort that we sort before. Inserting a value means we have to walk down to that segment till the very end. Now of course, one may argue, that to find the position to insert, you can use binary search. We need not go one element at a time.

The performance of different sorting algorithms can be compared based on their operations. For insertion sort, inserting an element requires finding the correct position in a sorted portion of the list. While finding this position can potentially be made more efficient using binary search, shifting elements to make room for the new entry would still require linear time to execute.

Think of it like organizing a bookshelf. If you have each book sorted and want to insert a new book, you could search for the right spot quickly if you were to use specific indicators (like a binary search). However, you still need to physically move the books next to that spot, and that movement takes time, just like in insertion sort.

So, insertion sort is again an order n square search. So, once again as we saw for selection sort. There is a natural way to think of insertion sort as a recursive thing.

The worst-case time complexity for insertion sort is O(n^2), which occurs when elements are arranged in the opposite order to that of the desired sorting. For analyzing the insertion sort algorithm, we can see that it behaves similarly to selection sort from a performance perspective. This algorithm also lends itself to a recursive understanding, where you can view sorting as continuously growing an already sorted section by inserting one new element at a time.

Imagine you are assembling a jigsaw puzzle. Each time you add a new piece to the right spot (just like inserting an element in insertion sort), you might need to check every piece already in place to find the right fit. If the pieces are all jumbled up, you'd have to go through many pairs before finding a spot, thus taking longer.

So, insertion sort usually behaves better than selection sort, and both of them are better than bubble sort. In insertion sort, we will find the element is already in the correct position.

Insertion sort is particularly efficient when dealing with nearly-sorted data because it minimizes the number of moves needed to place each element in the right place. In fact, if you are working with a list that is already sorted, the insertion sort can perform in linear time, O(n), making it a very quick sorting method under certain circumstances.

Consider a scenario where you're preparing a list of numbers for an exercise and you find that they are already arranged properly. In this case, you simply check each one against the new addition, and if everything is in order, you hardly need to make any changes—just like how insertion sort would behave in a nearly-sorted list.

But this is not to say that these algorithms are all equally bad. In particular it turns out that if actually look at experimental evidence, then insertion sort usually behaves better than selection sort, and both of them are better than bubble sort.

The comparative performance of sorting algorithms shows that while all three algorithms (insertion sort, selection sort, bubble sort) have quadratic time complexities, their practical efficiencies vary. Experimental evidence suggests that insertion sort is often the fastest in practice, even compared to selection sort and bubble sort, particularly with smaller datasets or those that are nearly sorted.

Imagine you are tasked to clean a cluttered room. If most of the items are already in their right places, a little tidying (like insertion sort) takes much less time than sorting through all items completely (like selection or bubble sort). In this way, insertion sort tends to be more efficient in scenarios where work has already been partially done.

--