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Before we introduce heap sort, we need to introduce the concept of a heap. In a heap, the elements are organized in a tree similar to a family tree. Each element has exactly two children, except at the bottom right, where missing elements are allowed. This tree also satisfies what is called the “heap property”: each ball is brighter than its children. This mean the brightest ball is at the top. Let’s take it out, and put it aside. Now the tree is in an illegal state: an element is missing at the top. We’ll replace it with one from the bottom, where missing elements are allowed. Now the heap property is broken. We’ll fix this by executing a procedure called sift-down. We’ll gradually move the top element downwards, until it sinks down to where it belongs. At each step we’ll swap it with the brighter of its two children. Now let’s take out the top element again. Repeating this over and over produces the elements in sorted order. Now our short sighted robot will perform the same thing. It may not be obvious, but these elements are already organized in a heap: The top element is first… then its children… then their children… and so on. Let’s extract the top element… and put it last. It’s now out of heap and in its final position. We’ll place the last element at the top, and execute sift-down. First, we’ll find the brighter of the two children… Then make sure that a swap is needed… swap… And continue downwards… And so on… We’ll repeat this until the entire list is sorted. All we have left to explain is how we built the heap in the first place. So now let’s start with a random permutation. To make a heap out of it, we’ll execute a procedure called heapify. We’ll execute sift down on every element starting from the bottom up. We’ll start by sifting down the lowest element that has children. This already fixed a small portion of our tree, and now we’ll continue sifting down the rest of the elements. And here’s how it looks like when our robot does it. When the heap is ready, it will start removing the top elements one by one, as shown before. Just to complete the basic set of heap operations, here’s how new elements are added. We’ll place the new element at the bottom, and execute a procedure called sift-up. As long as this ball is brighter than its parent, we’ll swap. In heap sort, we can create the initial heap by repeatedly adding all the elements, but this is less efficient than the heapify procedure shown before.