11.4. Heaps as a Sorting Algorithm

So, before we leave heaps, let us see how to use heaps to sort. So, we want to sort a list of values. So, what we can do is, we can first build a heap, right. So, we now start with some arbitrary sequence of values, x 1 to x n, then we build a heap and we possibly get reordered in some way of x i 1, x i 2, x i n, but this is not in sorted way, this is in heap structure, that is, we know, that the maximum is at the left and so on.

This chunk introduces the concept of using heaps for sorting. When you take a collection of numbers and organize them into a heap, you can rearrange them so that the largest number is always at the root of the heap. This means that while the elements may not be in sorted order yet, they are structured in a way that makes extracting the largest (or smallest) values efficient.

Think of this as organizing a line of people based on their heights. If you build a human pyramid where the tallest person is always at the top, while everyone else may not be perfectly lined up, you can easily find out who is the tallest (the root of the heap) and remove them from the top whenever needed.

Now, I do delete max and I put this guy out, right. So, I know that this is the maximum. Then, after this, this something will come to the front. At the next point, that will come out, and then I will get the second max and so on, right. So, if I do delete max n times, clearly, at each point I get the next maximum. So, I am extracting elements from the tree in descending order, and so I get a sorted output.

In this chunk, we learn about the process of extracting the maximum elements from the heap. By performing the 'delete max' operation, we can repeatedly remove the largest value (which will be at the root of the heap) and subsequently fill the gap with a new value from the heap, restoring the heap properties. Each extraction provides the next largest number, allowing us to create a sorted list in descending order.

Imagine a game where players are ranked based on their scores. As you call out the highest score, that player leaves the game, and the next highest score automatically rises to the top. By continuing this process until everyone has been called out, you end up with a list of players in descending order of their scores.

I can reverse set, I can keep in ascending order, it does not matter, but I am just extracting the elements in a particular order. And so this is a trivial sorting algorithm. Now, each extraction takes log n time because it is a delete time delete max, right, or a delete max or a delete min depending on what type of heap for using. So, you do n such extraction. So, in log n time you can get the elements out in sorted order and to put them in, you took only order n time. So, overall we sorted n log n time.

This chunk discusses the efficiency of the heap sort process and how it leads to a sorted array. Each 'delete max' operation, which takes logarithmic time, is performed n times for each element in the heap, leading to an overall time complexity of O(n log n) for the sorting process. This makes heaps particularly efficient for sorting large lists.

Consider organizing a music playlist. If you repeatedly select and remove the top track (which is your favorite right now) and put it in a separate 'played list', you can end up with a list of your favorites sorted by the order they were played. Even if you remove tracks in logarithmic time, the process of organizing the playlist remains efficient.

There is a small subtle thing that you can ask about this. The question is, where these values go, right. So, initially, in the first iteration, if I have my heap looking like this, this is my maximum. So, it looks like I have to put in a new list, but what happens after this step is, that this value gets inserted here and then it percolates down by the heap if I fix operation, right. So, when I delete the value at the root, I take the last leaf, put its value in the root and then I put it down so that all the heap property is restored. And after this, now my heap only looks like this because this value is gone.

This chunk emphasizes the mechanics of heap sort, specifically concerning where the values go post-extraction. Instead of moving the extracted values to another list, the procedure places them at the end of the array that holds the heap structure. As you repeatedly extract values, the remaining heap reduces in size, and values are effectively sorted in their correct positions.

Think of it like packing books on a shelf. Instead of taking books out and placing them in a different box, you take the largest book from the top shelf, replace it with the last book on the shelf, and then rearrange the remaining books. This way, as you continue to remove books, the shelf empties from the top to the bottom while still ensuring that the books left are arranged correctly.

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