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Today, we are going to talk about priority queues. Can anyone tell me what they think a priority queue is?
Isn't it like a regular queue but with jobs that have priorities?
Exactly! In a regular queue, we use a first-in-first-out method, while in a priority queue, we execute jobs based on their assigned priorities. Can someone give me an example of where we might use a priority queue?
Like in a hospital, where patients are treated based on the severity of their condition.
Great example! Priority queues are widely used in systems like job scheduling. A challenge arises when we need to manage these jobs effectively, and this is where heaps come into play.
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Heaps are designed to maintain a balanced binary tree structure. Can anyone describe what we mean by a balanced tree?
I think it means that both sides of the tree should be almost the same size, right?
Precisely! This balance ensures efficient operations. Can anyone tell me what properties define a heap?
There are structural properties, like filling from top to bottom, and value properties, where the parent node must be larger than its children.
Exactly! Remember the mantra: 'structure first, values follow.' This leads to enhancements in efficiency for managing jobs with varying priorities.
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Now, letβs look at what happens when we insert a value into a heap. How do we ensure the heap property is maintained?
We could just add the new element at the end and then compare it to its parent?
Exactly! This process, known as 'bubbling up,' allows us to maintain the max-heap property. What happens if our new value is smaller than its parent?
Then, we stop bubbling up since the property is satisfied!
Correct! Now, if we delete the maximum value, what procedure do we need to follow?
We replace it with the last element and then bubble down to restore properties.
Perfect! This back-and-forth swapping keeps the heap properties intact, leading to efficient job scheduling.
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The section elaborates on the implementation of heaps, comparing them to traditional data structures, highlighting their structural and value properties, and explaining how they allow efficient insertion and extraction of elements based on priorities.
In this section, we explore the concept of heaps, a specialized type of binary tree, used to maintain a collection of elements with priorities. Heaps facilitate the operation of a priority queue efficiently, contrasting with traditional lists or queues that handle elements based solely on their order of arrival. The key operations associated with heaps are inserting new elements and deleting the maximum element, both of which are executed in logarithmic time when using a balanced tree.
Heaps have two critical properties: 1) Structural Property: A heap must fill every level from top to bottom and left to right; this ensures that the heap maintains a balanced structure. 2) Value Property: For max heaps, the value of each parent node must be greater than or equal to that of its children. If these properties are violated during insertion or deletion operations, adjustments, known as 'heapifying,' may need to be made to achieve a valid heap again.
The use of heaps significantly improves the performance for applications that require frequent priority adjustments, reducing the overall time complexity from O(n^2) with sequential lists to O(n log n), making it a critical structure in algorithms dealing with scheduling and resource management.
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A heap is a binary tree with two properties, the first property is structural: which are the values which are stored in the tree, remember that leaves, nodes in a binary tree may have 0 children, 1 child or 2 children. So, we could have in general a very uneven structure. Heaps have a very regular structure when we have a heap we have a binary tree in which we fill each level from top to bottom, left to right.
In other words, at any point a heap consists of a number of filled levels and then in the bottom level we have nodes filled from left to right and then possibly some unfilled nodes. The other property with the heap, the first property is structural, it just tells us how the values look in the heap.
A heap is a specific kind of binary tree, which is organized in a particular way. The first property, known as the structural property, dictates how the tree is organized. In a heap, every level of the tree is filled from top to bottom and left to right. This means that nodes have a very regular structure, ensuring no gaps are left until the bottom level is reached. The nodes may have 0, 1, or 2 children, but always follow the filling rule, which helps in maintaining balance within the tree.
Think of a heap like a pyramid of blocks. Each level can only be filled completely before moving on to the next, and the blocks (or nodes) should be placed together snugly so that they form a solid structure without leaving empty spaces at the top levels.
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The second property about the heap is the values themselves. So, the heap property in this case what we call the max heap property because we are interested in maximum values says that every value is bigger than the values of its 2 children. So, at every node if you look at the value and we look at the value in the left child and the right child then the parent will have a larger value than the both the children.
The second important aspect of heaps is known as the max heap property. This property states that for any given node in the heap, the value of that node must be greater than or equal to the values of its children. This means that the maximum element is always at the root of the tree, and every parent node will always be greater than its child nodes. This property is crucial for efficiently prioritizing elements in a priority queue.
Consider a small family where parents are always taller (in height) than their children. In this analogy, the parents represent the nodes of the heap while the children represent the nodes below them. Just as the parents are taller than their kids, in a heap, each parent node should have a value greater than its child nodes.
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Let us see some examples. Here is a four node heap, because it has four nodes we fill the root in the first level and finally, in the second level we have only one node which is a leftmost and notice that the values are correctly ordered, 24 is bigger than 11 and 7, 11 is bigger than its only child 10, 7 has no children. So, there are no constraints.
Here is another example where the bottom level is actually full, here we have 7 nodes and once again at every point we see that 11 is bigger than 10 and 5, 7 is bigger than 6 and 5. So, we have the value property - the max heap property along with the structural property.
The provided examples illustrate what a typical heap looks like and demonstrate both the structural and max heap properties. In the first example, with four nodes, the top-level node (the root) has the highest value, and as we move down the tree, the values of the child nodes are less than or equal to their parent. The second example has a complete bottom level, confirming that all nodes are filled from left to right at every level, adhering to both properties.
You can think of this like a family tree where the grandparents' status (wealth, age) is higher than that of their children or grandchildren. Just like the tree where the grandparents are at the top and better off than any of their descendants.
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Here is an example of something which is structurally not a heap because it is a heap we should have it filled from top to bottom, left to right. So, we should have a node here to the left of 7 before we add a right child therefore, this is not a heap.
Similarly, here the node 8 could not have been added here before we filled in the right child of 7. So, once again this has not been filled correctly left to right, top to bottom and therefore, this is not a heap.
The examples of incorrect structures show what happens when the filling rules of a heap are violated. A node must have its left child filled before adding a right child, and gaps from top to bottom are not allowed, which confirms that some of these structures do not qualify as heaps. These common mistakes help emphasize the importance of maintaining the heap structure correctly.
Imagine building a multi-story parking garage; you cannot start filling the upper levels before the lower levels are filled upβif you donβt, it may weaken the structure and create problems, similar to what happens in heaps when their filling order is not respected.
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Key Concepts
Heap Structure: Heaps are binary trees filled level by level, from top to bottom and left to right.
Max-Heap Property: In a max-heap, each parent node's value is greater than or equal to its child nodes' values.
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An example of a max-heap: A tree where the root has a value higher than both its left and right child nodes.
Inserting a new value into a max-heap and bubbling it up to maintain the correct order.
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In a heap, values grow from deep, parent strong, child weak. Keep it balance, donβt be meek!
Imagine an orchard where trees (parents) grow larger (greater value) than the small plants (children) under them. To ensure the orchard stays organized and all grow uniformly, we pick the tallest tree first!
H.E.A.P. - Hierarchical Elements Arrange Priority.
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Review the Definitions for terms.
Term: Heap
Definition:
A special tree-based data structure that satisfies the heap property, where each parent node is greater than its children in a max-heap.
Term: Priority Queue
Definition:
An abstract data type where each element has a priority and the element with the highest priority is served before others.
Term: Binary Tree
Definition:
A tree data structure in which each node has at most two children, referred to as the left and right child.