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All right, so I want to introduce a new data structure that I really like, and this is called a heap.

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So we're going to talk about heaps now and we'll have a few lectures on them.

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They're really cool, pretty simple data structure that can do some really interesting things.

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So let's get into it.

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So what is the heat?

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Well, it's similar to looking to a binary search tree.

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If we're just talking about the things that we've seen so far in the course, so it's the same looking

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in theory, but not necessarily implemented in the same way.

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And it's also not necessarily used for the same type of operations as a binary search tree.

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So let's talk about the defining characteristics of a heat.

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So structurally, a heap has to be a perfect binary tree up to at least, but not including the last

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level.

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What that means is like if you look at this picture right here, let's start up here and kind of encompass

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this entire area, not including the last level, right?

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So let's go here and all the way over here, I'm trying to draw in a circle around the part that would

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be a perfect binary tree.

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So let's take a look at that.

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Each level actually has double the notes of what the parent level has, so we start at the root and

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look at the level below the we noticed that the route has one node, right?

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The level below it has two nodes.

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The level below that node has four nodes.

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The level below that node has eight right one two three four five six seven eight.

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So if you think about it, it's kind of like if you start the level zero.

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Each level that you go down is basically two to the power of the level, so it's consecutive powers

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of two.

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So two to the zero is one.

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The next level two to the one is to the next level, two to the two is for two to the three is eight.

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And if we filled out this whole level, it would be 16, right?

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The last level, we do not have to have completely filled out.

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But another important property of a heap like structurally is that the heat must be built from left

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to right.

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So when we build the heap, we actually have to like, let's say we're starting at the root.

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We have to fill a left node first before the right node and so on.

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So we kind of go down into the right, starting from the left right.

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So if we just had one node and we wanted to add another node, we'd have to add this one first and then

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we'd have to add that one.

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And then we have to have this one and this one and then that one and that one and then kind of fell

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into levels like this.

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And you notice that's kind of what's happening on the bottom level.

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So it's fine that there's like not nodes here like this missing nodes here on this level.

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But we could not, for example, let's say we didn't have these two nodes here and we had these nodes

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here.

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That's not OK.

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We would have to fill these two nodes before we could do these two and before we can do these two and

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so on and so forth.

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So another thing is if the heap is a men heap, every parent node, but must be less than or equal to

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each of its children if the parent isn't, if he was a max heaped in the opposite has to be true.

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So a heap is really only concerned with whether the parent is less than or greater than the children.

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Right?

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And in binary search tree, we kind of had like, let's take an example here.

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So if we had like five year at the root node, if we got the number three, we would want to put it

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on the left right.

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If we got the number seven, we would want to put it on the right, right.

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So like everything to the right of the root node we're considering would be greater than its value,

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right?

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Everything in the left would be less then.

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So the heap isn't really about that.

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The heap is about just having the stuff below these be less than it.

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So.

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And we're always going to build it from left to right, no matter what.

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So what would happen is basically.

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You know, we just need to make sure that when you're.

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When you have a heap like, for example, this route right here has to be greater than the left child

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and greater than the right child if it's a max heap.

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If it's a men heave, this route right here has to be less than the left child and less than the right

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child.

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So that's what we're talking about.

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So let's take a look at a mini example.

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So you notice here that five is less than 25 and less than 21.

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In fact, five is less than everything, and it's sub trees here, right, because at each level, the

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things below it have to be greater than right.

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So 25, everything below 25 is greater than greater than 25, right?

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So you notice like twenty four to forty one, all these numbers are greater.

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Twenty seven.

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So that property has to be maintained.

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So let's go ahead and watch this part of this -- get built so you can better understand how it works

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as far as building a house structurally, what we were talking about, you know, coming from left and

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going to the right.

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So we're just going to start at this point where it's partially completed.

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And let's just say that we insert 48.

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We're going to put 48 right here.

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Forty two would go right here.

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Forty one would go right here.

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Fifty two right there, 69 right there.

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Seventy seven seventy one.

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You notice this just doesn't really matter.

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We're just it doesn't really matter what the values are in this situation.

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We're just building it out from left to right.

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So that's the purpose of showing this, you know, is that you have to build it out like this.

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Right.

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So unfortunately, though, incoming data is not always going to arrive in a perfect order like that,

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you notice that this just like if we're going back, you can see us just building perfectly.

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We just we just so happen to be getting all these values that are greater than the parent.

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We care about it being a men heap, right?

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So they're supposed to be greater than the parent, but we just keep getting like perfect values that

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are always like 52 is greater than that.

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Oh, in 69 next.

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That's great because it's bigger than that.

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You know, we come over here.

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These are these are all greater.

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All of these children that we're putting in are all greater than the parent know that we're attaching

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them to your.

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So very rarely will the data actually be like that and arrive in that perfect order, so let's see what

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a real heap insertion will look like.

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So let's say we want to insert tape.

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So this is kind of a situation where we're going to do what we did before.

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Twenty eight is greater than 25.

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We're going to want to put it here, no matter what, because the structural property.

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Says that we need to build the heap this way, building it from the left to the right.

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So we go ahead and we put twenty eight here what we would do next as we would compare it.

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So we would say is 20 less than 25.

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No, it's it's not less than 25.

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So we don't really have to like worry about doing anything.

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The heat property is still satisfied, right?

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The mini property here.

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So let's say we want to insert three now we know where it's going to go, right?

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We just put one right here.

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So now we need to put one right here.

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That's the next spot.

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We go ahead and we put three right there.

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Then we say is three less than 25.

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Three is less than 25, so we actually need to do something about that.

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And what are we going to do?

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Well, we're going to swap.

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We're going to swap three with twenty five and bring twenty five down here.

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So we swap those.

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And 25 comes down here.

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We're not done yet, though we have to compare it and tell it cannot be basically pushed up anymore.

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This is what we would call bubbling up.

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So we're taking the value and bubbling it up to the position it belongs in.

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So we look at the next parent, right?

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Three came up to here.

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Twenty five went down.

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Now we say, OK, let's check to see if three is less than five.

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The parent, right?

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Yes, it is.

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So we have another situation where it's still listed, so we're going to have to do another swap.

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So we swapped three and five here.

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Now three is at the top.

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So now the heat property is fulfilled, right?

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If we look at everything in the heap, we notice that, OK, three children, are they less?

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Yeah, five is less.

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21 is less.

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What about five or 10?

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At least it's greater, right?

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So three is less than its children, right?

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So yeah, five is greater than three.

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Twenty one is greater than three.

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Five.

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Let's look at it's children.

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Yeah, twenty eight is greater than five.

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Twenty five is carrying the five, so the heat property is satisfied.

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The structure is fine.

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Everything has been built from the left, so we're looking good.

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Let's say that we insert 21.

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So this is the next spot.

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And just regardless of what the value is, we have to put something here because we're building from

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the left to the right.

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21 happens to be the same value as the parent.

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We go ahead and we check is 21 less than 21.

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No, it's not as the same, right?

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It's equal.

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So we actually don't do anything and we leave it there.

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And the heat property is still satisfied because it's a less than or equal to thing, right?

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As far as looking at the children, we want the children to be greater than or equal to.

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We want the parent to be less than or equal to four men heap, of course, because that's the example

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we're covering, right?

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So how do we implement a heap?

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Do we make nodes and use pointers?

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Remember, the binary search tree had a node class and then we kind of had a binary search tree class

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where we could have like a left child and a right child right there, the sub trees and stuff like that.

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So we were just traversing pointers and creating a data structure on our own.

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Like that, right?

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So we actually don't have to do that there as hip is so simple that we can just use an array.

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And so you might be thinking, well, OK, we have all this stuff in the array, but how do we know

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where to put the stuff in there?

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Like how does this tree?

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Thing translate into an array.

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Well, the array is essentially like a flattened version of the tree.

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And since the heat property is so simple, right, it's like, you know, we build from left to right

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and we also just need to have the children be both less than the parent or if it's a, you know, men

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heave, we want the children to both be greater than the parent, right?

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So we can actually just mention three kind of properties about the positioning of the elements in the

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array to make it heat.

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So.

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If we say that the index of the left child and so let's say we're talking about some node at position

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I so for example, let's take a look at the root node, right?

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We noticed that three is the root of this tree right here.

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And let's take a look at the array.

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Three Is it the first position, right?

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So the rule will always be at the first position.

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You can always just remember that.

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So what is the index of the Route three?

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Well, it's zero.

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So let's take a look at this property right here.

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Index of left child of the node at two eye plus one.

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So let's look at the tree.

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What is the left child of the three?

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It's five, right?

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This says two is plus one.

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Well, what's eye in this case?

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I is the index and zero two times zero plus one is one right?

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Zero plus one.

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So we get to index one, right?

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That's the answer of this right here.

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So if you go to index one was there, it's five.

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Oh, well, yes.

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So it corresponds with the left one.

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Let's take a look at this index of right child at two plus two.

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So two times zero plus two is two.

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What is the right child of three?

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It's 21, two times zero plus two is two, we said.

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So if we go to index two, it's twenty one.

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Well, let's check.

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Let's check some other stuff.

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So we'll just check the whole tree.

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Five.

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Right, so left child.

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Twenty eight two times.

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Five say the it was the index of five.

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It's one right.

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Two times one plus one is three.

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So let's go to index three.

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It's twenty eight right.

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So there's the left child appropriately stored.

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And then what about the right child, so the right child is to two times II plus two eyes, one right

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for five, so we have two plus two, that's four.

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So we go to index for us, there's 25 that corresponds with the tree as well, and then 21.

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Let's just go ahead and look at that.

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So 21.

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If we look at where that is stored, it's that well, this is this this position right here, we can

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look at this, this 21 right here.

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We have two ones in here.

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But.

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If we look at this twenty one right here, if we go to a times two plus one, right?

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That's five if you look right here.

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Twenty one is the child right?

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And we look at index five.

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Twenty one is here.

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So I just want to walk you through and show you really what it all breaks down to as in putting it in

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the array.

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So when we flatten this out, the reason why we can use an array is our ability to use these formulas

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to be able to access the children.

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And it works, vice versa.

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You know, the opposite way as well.

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The index of the parent can be said to be at the floor, so we'll kind of like round down, right?

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Of I minus one, so index minus one divided by two.

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So, for example, let's just take a look at like twenty one.

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At the very bottom here, so.

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Twenty one is the index five, right?

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So if we say five minus one, that's four and we do four divided by two.

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Well, that divide evenly, right?

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So we say four divided by two is two and that gives us twenty one.

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Let's take something over here, for example.

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So what about this?

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Like twenty five, right?

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Twenty five, is it for four, minus one is three.

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Three divided by two.

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We're just going to round that down and say that it goes in one, right?

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So if we go to here, if we say that the answer's one, we see that the parent is stored there.

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Five.

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So pretty cool, right?

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So what is all this even useful for?

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Like, why would we use a heap over some other data structure?

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Well, heaps are really useful for keeping track of the minimum or maximum values in your data collection.

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There also is a cool, efficient sorting algorithm that we will look at in a lecture or two in the future,

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which is heap sort and that uses a heap.

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So also, just in general, most operations on Heap are quite efficient.

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We will be looking at the time complexity of certain operations, and you'll notice that most of them

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are like, you know, logarithmic or you might have like in log in if you're building an entire heap

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or you might just have linear time and or, you know, even constant time as far as finding the maximum

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or minimum thing.

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So definitely a pretty efficient data structure.

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All right.

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So that hopefully serves as a decent introduction to heaps.

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In the next few lectures, we'll be getting into more specifics.

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So we'll be talking about not only how to insert things in the heat, but how to delete things and how

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to make sure that the heap stays a heap when you are deleting items.

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We'll also talk about heaps sort and also really cool lecture.

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Hopefully, we can talk about a quicker, more efficient way to build a heap as well.

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All right.

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So with that, I'll see you in the next lecture.