Now we're going to look at binary heaps, which is an ingenious and very simple data structure that's going to help us implement all the priority queue operations quickly. So the idea of a binary heap is based on the idea of a complete binary tree. Well, first of all, a binary tree is either empty or it's a node with links to left and right binary trees. So that's an example of a binary tree. A complete binary tree is one that's perfectly balanced, except possibly for the bottom level. So there might be a few nodes on the bottom level and one level lower than the bottom level, but otherwise all the levels are full. And we'll see how that looks in just a second. The property of a complete tree is at the height of a complete tree with N nodes is the biggest integer less than log base 2 of N. And that's really easy to convince yourself that that's true because the height if we add nodes one at a time going from left to right on the bottom level say, the height only increases when N is the power of 2. Complete binary trees actually happen in nature. Here's an example of one that goes one, two, three, four levels at least. So 16 bushes at the end there. All right, now the way we're going to use a complete binary trees to implement priority queues is to first of all associate information with each node. We'll put our keys in the nodes. And also we're going to represent it with an array. So when we start putting the keys in the nodes, we're going to impose one more condition that's called heap ordering. And the idea is that the parent's key is going to be no smaller than its children's key, and that's true for every node in the tree. The array representation, all we do is we put, we start with indices at 1. It's a little less calculation. That way, we leave a of zero empty. And then we just take the nodes in level order. So first we put the root, then we put the two nodes on the first level going left from right, and then all the nodes on the third level going from left to right and so forth. This is interesting because we can draw the tree to get more intuition about what's happening, but in the actual data structure representation, we don't need any links at all, it's just an array. The way that we move around the tree is by doing arithmetic on the indices. So let's look at a few properties of binary heaps. So that's complete binary trees represented in array with keys in the nodes, satisfying heap order property. Well, the first thing is that a of 1 is the largest key. It's larger than the keys in this two children and they're larger than theirs and so forth, so it's the largest key in the data structure. The other thing is as I just mentioned, you can use the array indices to move through the tree. For example, if the node is at position k, index k in the array then its parent is at k over 2 and that's the integer divide. So the parents of say H and G are both N, H is at 10 G is at 11, N is at 5. So both of those are 10 over 2, 11 over 2 integer divide is 5. And going the other way, it's easy to see that the children of the node at k are 2k and 2k plus 1. So we don't need explicit links at all to represent these data structures, we can just use array indices. So that's the basic setup or the invariant that we want to maintain in this data structure. And now, what we are going to do is take a look at just a couple of different scenarios that we violate that invariant temporarily, and then fix it. And that's going to give us the flexibility that we need to implement priority queue operations. So one scenario shown here is, if for whatever reason a child's key becomes larger than its parents key. So in this case, we have an example where T, the node T here its value changes and it becomes larger than its parent key, P. So the heap order condition is satisfied everywhere except at this node. Well, it's easy to fix this one. All we do is exchange the key in the child with the key in the parent. After that exchange, then that would have T up here and P down here. Then the heap order condition is satisfied at that node because the parent was smaller, so that one's smaller. The southern is still smaller, so T after it's exchanged up here will be bigger than both its children. But the heap conditional will be violated because T is still smaller than S. So we have to do it again, exchange it with S. So we move up the tree exchanging the larger key with its smaller parent until we get to a point where it's larger than both its children. That's restoring the heap border along a path from the place where it's violated to the root. You can think of that as kind of like the well-known Peter Principle where a node gets promoted to a level where it finally can't be better than its boss, the level of its maximum incompetence. And implementing that in code is really easy. We call that the swim operation. Swims up to the top, and if we have a node at index k and we know the heap condition is violated there. As long as we're not at the root and k's parent, k over 2 is less than a of k then we just exchange it with its parent and move up. That's the swim operation to eliminate violation when a key value increases. So for example, this gives us a way to insert a new element into a heap. What we do is we add a new node at the end of the heap, so that's one position over. The thing is remember represented in array one two three and so forth. So the next empty position in the array, there's a place to put a new node. And then we just declare that that's part of a heap and that node, well if it's less than its parent, we're fine. But in general, we have to check whether the heap condition is violated and exchange it with its parent as long as it's smaller. And that's just perform the swim operation. So if N is the number of items in the heap, defined to be in the heap, we're going to increment it, store our new key there and then perform the swim operation. So that's a quick implementation of the insert operation. And notice since it's just going from bottom to top in the heap, it takes at most 1 plus log base 2 of N compares. Now, there's another scenario where a key becomes smaller. For whatever reason a parent becomes the key and decreases, it might become smaller than one or both of its children's. In this case, the value apposition two has changed to H for whatever reason, in that smaller, in this case then both its children. So how do we fix that violation? Well, that one's also easy. We figure out which of the children is larger. In this case it's the S. And we exchange that one with the one that's violating the condition. So that's moving the smaller key down. After that exchange, then S is in position two and it's bigger than both P and H. And the heap condition is only violated again where H is sitting. And again, we keep going until getting to the bottom, or getting to a place where both children are smaller. And that's maybe a little bit what happens when a new boss is hired from the outside and then the two subordinates struggle to take over that position and then the boss would get demoted to it's level of competence. And again, that level of flexibility, here's the implementation of it. And again, it's quite straightforward using the index arithmetic to move around in the heap, and that's called the sink operation because we're going down in the heap. Now for a position K, then what we need to worry about is the nodes at 2k and 2k plus one. The first thing to check is find out which one is bigger, it's either 2k or 2k plus one and so set J accordingly. So that's J now is after this statement, is the larger of the two children. And don't forget to check that we're going off the end of the heap. And then if the K is not less than either one of those, then we're done. If it is, then we exchange with the larger of the two children and move down the heap. Again just a few lines of code to eliminate the violation when a key value in a heap decreases. And that one we're going to use to implement delete the maximum in a heap. So, delete the maximum we have to do two things. One thing is the size of the heap is got to go down by one. The other thing is return the maximum. While we know the one that we want to return is the one at the root, so we'll save that value away to return to the client. And then since it has to go down by one, the place to remove the element from the heap is at the end of the array because it's now going to have to not occupy that position. So we take that element and replace the root with it. So move the H up, and actually put the root value there, just exchange them but it's no longer in the heap. Now, that element which went from the bottom to the top is most likely going to violate the heap order, it's going to be smaller than one or its both of its children, so we do a sink. Now, in this case, to implement delete max we save away that value at the root in max, and we eliminate loitering by nulling out that vacated position, then return the max value. So that's an implementation of the delete max operation for heap using a sink, where a key value decreases, goes down in the heap. So, let's just take a look at what happens with a real heap with the demo when we do these things. And you'll have a good feeling for how this data structure works. So, we're starting at some point where we have these 10 keys in a heap and its heap order. So we've drawn the data structure with the links so we have an intuition for what's going on, but all the program sees is the array in gray at the bottom where T is in position one, P and R in position two and three and so forth. So, now suppose we were supposed to add S. So to add it to the heap, that's got to go in the position at the end. And now we've added it to the heap by just incrementing in and putting it in there. But now we have to bring the heap order back into condition and so now that key is larger than its parents. So we're going to swim it up, exchange it with its parent as long as it's smaller than its parent. So, first thing it goes up the exchange with the S, it's still bigger than P so we exchange it with the P and now we're done because S is not bigger than T and the heap order condition is now satisfied everywhere in the heap. So, with just two exchanges we insert that new element into the heap in this case. Now, suppose the next operation is remove the maximum. So, we're going to take T and we're going to exchange it with the last element, and then declare that to be no longer part of the heap. So now, we have to bring the heap order back because it might be violated at the root. So now we invoke the sink where we exchange that node with the larger of its two children until we find a place where it's larger than both of those children. So, S is the larger of the two children R and S, and now H is still smaller than both its children, so we promote the larger which is P. And now H has no right child, just a left child and it's larger than that one so now we're finished with that operation. We've removed the maximum and we still have our data structure heap order and our N keys stored in the first N positions in the array. Let's remove the maximum again. Again, we take it out by exchanging this time G with a root, and then decrease the size of the heap by one and just take that out. Now the node at the root violates the heap orders, so we have to exchange it with the largest of its two children, in this case that's R. Again G is not larger than its two children so we have to exchange it with the larger of the two, that's O, and now we're done, we've removed the largest again. Now suppose we insert S back into the heap. So, that's adding it at the end, violates the heap order, exchange it with the parent through smaller and keep doing until we get to a place where it's larger than it's two children, in this case S goes all the way up to the root, and then the I is all heap ordered again. So that's a little survey of some operations on a heap and you can see how every operation is done with just a few exchanges along the path from the bottom to the top, or the top to the bottom. Okay, here's the complete Java implementation of a priority queue, we're using the binary heap data structure. It's actually not so different from the elementary implementations that we looked at in the last section. Our representation is an array of keys and a size, that's the number of items in the heap. For simplicity, we'll show the code where the client gives the capacity of the heap. We can use resizing array in industrial strength implementation, the same that we did for stacks and other data structures where we use arrays. So, we'll build a new array of keys and we have to use an ugly cast because of generic arrays in Java, and that's where it's comparable and we need one more than the capacity to handle this thing where we don't use position zero. So, the priority queue operations is the insert in delMax that we just showed in the previous slides. Is empty, it is just checking whether YN is equal to zero. We have the swim and sink functions that we showed earlier. And then we have helper functions less and exchange that access the array directly so that the code doesn't have to access it directly. And that's a complete implementation of priority queues in Java. This implementation by itself is extremely significant because it's really very simple. Optimal representation of the data and only a little arithmetic with array indices. But as you can see by looking at this table, it gives us a way to implement priority queues where both operations are guaranteed to happen in log N time. Now, experts have worked to come up with improvements on this and there are slight improvements possible. We can make the heap d way rather than just two way. And depending on the frequency of execution of the uncertain delMax operations, that might work out better. There's an advanced data structure called a Fibonacci heap, where inserts are done in constant time and delete max done in log N time, on average over all the operations. That one's generally too complicated to use in practice. But still again, using theory as a guide maybe there's a way to decrease costs a little bit from binary heaps. Of course, we cannot get down to having constant time for all operations. Why? Well, we can start with a heap, by inserting all the elements and then deleting the maximum and getting a sort done and that would be linear time if we had this kind of variation, If we had Constantine's operations for both uncertain delMax. But for certain applications we can get close to constant time for one or the other operations and that will be useful in different implementations. Now, there's an important consideration that we have to bring up related to the programming language, and this is a more general consideration and usually we bring into focus in algorithms but it's worthwhile mentioning. We're assuming that the client doesn't get to change the keys while they're on the priority queue. It's better not to assume that, it's better to arrange for that or implementations by using keys that are immutable, whose values don't change. There's many reason that immutable keys are that programming languages provide the capability to build immutable keys and this is a fine example of one. So, we'll talk more about that in a minute. The other things that we didn't talk about, the implementation should throw an exception if the client tries to delete from an empty priority queue, and we should have a no-argument constructor and use a resizing array to account for a gradual growth and shrinkage in a industrial strength implementation. Usually, we provide two implementations: one that's max oriented, one that's min oriented so that nobody gets confused and they're the same except less and greater switch. And we'll see later on, there's times when we want to expand the API and provide other operations like removing an arbitrary item from the priority queue, or give the client in the API the capability of changing the priority of an item. Or sink and swim methods are good for making this happen, but we'll delay these implementations until we need them in a more complicated algorithm. So what about immutability? So, everything in Java is implemented as a data type, a set of values and operations on those values. And the idea of immutable data type is you can't change the value once it's created. So that's kind of like when you create a literal value to be assigned to an integer, it has that value. So, here's an example, say using the data type for vectors, might be a way to implement vectors. So we put the word final to means that instance methods can't be overridden, and not only that, instance variables are private, they can't be seen from the outside and they don't change. And so a constructor for an immutable vector data type, it might take an array as its argument. And that array, it's got value stored in it, say it doubles and those can change but what immutable implementation would do would be to copy those values into the local data array instance variable and then those values are not going to change. In the instance, methods won't change them and the client can't change them. So, that value stays the same. Lots of implementations, data type implementations in Java are immutable. String is immutable. When you create a string that value doesn't change. If you want a new string, you have to create a new string using concatenation or some other operation. And the same with the wrapper types like integer and double or color and lots of things. Whereas on the other hand, sometimes the whole purpose of a data type is to maintain a changing value. A good example as a counter or a stack. So you wouldn't put those things on a priority queue because the value is changing, but the other ones you would. So the advantages of immutability and again, maybe this isn't the place to really solve those advantages, it's more for a programming language course, is that it really simplifies debugging. We can have more confidence that our priority queue operations are going to work correctly, if we know that the type of data that's on the property queue is immutable. If the client could change the values, how do we know that the heap order operation is preserved? If we want the client to be able to change the values, we're going to provide methods for that purpose as I just mentioned. There's many other reasons that people use immutable data types. There is a disadvantage that you have to create a new object for every data type value. But for a lot of applications that disadvantage is not viewed to be significant compared to the advantages. Here's a quote from one of Java׳s architect Josh Block, "Classes should be immutable unless there's a very good reason to make the mutable. If a class cannot be made immutable, you should still limit its mutability as much as possible." And many programmers live by that kind of preset. So that's a basic implementation of priority queues using the heap data structure represented as an array.