Let's see how basic operations work with binary max heaps. What is particularly easy for binary max heaps is finding the maximum value without extracting it. I mean, it is easy to implement GetMax operation. Well, recall that the main property of that binary max heap tree is the following. For each edge its top value is greater or equals than its bottom value. this means that if we go from bottom to top now in our trees, the values can only increase. This in particular means that the maximum value is stored at the root of our tree. So just to implement GetMax, we just return the value at the root of our tree. And this takes us just a constant time of course. Now let see how inserting a new element into to the max binary heap works. So first of all a new element should be attached somewhere to our tree. We cannot attach it to the root in this case for example, because the root already has two children. Therefore, we just attach it to some leaf. Let's select for example, the leaf seven and attach a new node to it. The new node in this case has value 32. Well, it is still a binary tree. Right? Because seven, before attaching seven, had zero children, now it has just one child. So it is still a binary tree. However, the heap property might potentially be violated. And it is violated actually in this case, right? Which is shown by this red edge. So for this red edge the value of it parent which is seven, is less than the value of its child which is 32. So we need to fix it somehow. So to fix it we just allow that the new element to sift up. So this new element has value 32, which is relatively large with respect to all other elements in this tree, so we need to move it somewhere closer to the root. So the process of moving it closer to the roof is called sifting up. So the first thing to do is we need to fix this problematic edge. To fix it, we perform the following simple operation. We just swap the corresponding two elements. In this case, we'll swap seven and 32. After they swap, there is no problem on this edge. However, it might be the case that the new element 32 is still smaller. Is still greater than its parent and this is the case, in our toy example. So the parent of 32 is now 29, which is smaller than 32, so we still need to fix this red problem. And we just repeat this process, we again swap the new element with its parent, right? So we swap it and now we see that the property is satisfied for all edges in this binary tree. So what we've just done is that we let the new element to sift up. And what is important to note here is that we maintained the following invariant, that the heap property at any point of time of sifting the new element up, the heap property is violated on at most one edge of our binary tree. So and if we see that there is a problematic edge, we just swap its two elements, right? And each time during this process the problematic node gets closer to the root. This in particular implies that the number of swaps required is at most the height of this tree. Which in turn means that the running time of insertion procedure, as well as the running time of the sifting up procedure, in this case is big O of the tree height. Now let's see how the extract max procedure works for binary max heaps. First of all, recall that we already know that the maximum value is stored at the root of the tree. However, we cannot just take and detach the root node because it will leave two sub trees, right? So we need to somehow preserve the structure of the tree. What is easy to detach from a binary tree is any leaf. So let's do the following, let's select any leaf of our tree and let's replace the root with this leaf. So in this case this produces the following tree. This potentially might violate the heap property. And in this case, this does violate the property. So the new root 12, is less than both its children. So the property is violated on two edges. So 12 is a relatively small number in this case. So we need to move it down to the leaves. Great, so for this we will implement a new procedure, which is called SiftDown, okay? So, similarly to SiftUp, we are going to replace, to replace the new element with one of its children. In this case we have a choice actually, we can replace it either with its left child or with its right child. By thinking a little bit we realize that it will make more sense to replace it with the left child in this case. Because the left child is larger than the right child, because after this, after we replace 12 with 29, the right problematic edge will be fixed automatically, right? So this is how we are going to perform the SiftDown procedure. Once again, we select the largest of two child and we replace. the problematic node with this larger child. As you can see, the right problematic edge is fixed automatically. The left edge is also fixed, just because we swapped two elements. However, the new problematic node might introduce new problems, right closer to the bottom of the tree. Now we see that there is still a problematic edge, so in this case, we have just one edge so 12 is smaller than 14, but it is greater than seven, so we are safe in the right tree. In this case we swap 14 with 12 and after that we just get a tree where the property is satisfied on all edges. So once again we maintain the following invariant. At each point of time we have just one problematic node, and we always solve the problematic node. With the larger one of its children, so that to fix both problematic edges. Right? And the problematic node always gets closer to the leaf, which means that the total running time of the extract max as well as the sift down procedures is proportional to the tree height. Now, when we have implemented both procedures, sifting up and sifting down, it's not so difficult to implement also the ChangePriority procedure. So assume that we have an element for which we would like to change its priority. This means that we are going either to decrease its priority or increase its priority. Well, to fix the potential problems that might be introduced by changing its priority, we are going to call either sifting up or sifting down. Well, let me illustrate this again on the toy example. Assume that we are going to change the priority of this leaf 12. So we've just changed it. We just increased the priority of this element to 35. In this case, we potentially introduced some problems and we need to fix some. Well we see that 35 is a relatively large number which means that we need to sift it up. So we need to move it closer to the root. So to do this we just call SiftUp procedure. Which repeatedly swaps the problematic node with its parent, so in this case this will produce the following sequence of swaps. First will swap 35 with 18 this gives us the following picture, we see there is still a problem 35 is still larger than its parent so we swap it again. Now we see that 35 is smaller than its parent. And actually, the heap property is satisfied for all edges. Once again, what is important in this case is that at each point of time, the heap property is violated on at most one edge of our tree. So since our problematic node always gets closer to the root at each step, I mean, after each swap. We conclude that the running time of change priority procedure is also at most Big O of the tree height. There is an elegant way of removing an element from the binary max heap. Namely it can be done just by calling two procedures that we already have. So I assume that we have a particular element that we're going to remove. So the first step to do is we just change its priority to plus infinity, that is, to a number which is definitely larger than all the elements in our binary MaxHeap. When we call it, the change priority procedure will sift this element to the top of our tree, namely to the root of our tree. Then to remove this element it is enough to call the extract max procedure. So in this particular example it will work as follows. So assume that we're going to remove the element 18, which is highlighted here on this slide. So we first change it's priority to infinity. Then the ChangePriority procedure calls the SiftUp procedure. This procedure realizes that there is, that the property is violated on this edge. And swaps these two elements. Then it swaps the next two elements and each at this point well this, this node that we're going to remove is at the root. Well, to remove this node, we just call the ExtractMax procedure. So recall that the first step of ExtractMax is to replace the root node with any leaf. So let's select, for example, 11. So we replace, we replace the root with 11. Then we need to call sift down, just to let this new root go closer to the leaves. Well, in this case, 11 will be replaced first by 42, then there is still a problem on the edge from 11 to, to 18. So we swap 11 with 18 and finally we swap 11 with 12. Well, once again since everything boils down just to two procedures. First is change priority. And the second one is extracting the max. And they all, they both work in time proportional to the tree height. So we conclude that the running time of the remove procedure is also, at most, Big O of the tree height. So to summarize, we were able to implement all max binary heap operations in time proportional to the tree height, and the GetMax procedure even works in constant time in our current implementation. So we definitely would like to keep our trees shallow. And this will be the subject of our next video.
