p p chapter 10 has several programming projects, including a project that uses heaps. p p this...
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Chapter 10 has several programming projects, including a project that uses heaps.
This presentation shows you what a heap is, and demonstrates two of the important heap algorithms.
HeapsHeaps
Data StructuresData Structuresand Other Objectsand Other ObjectsUsing JavaUsing Java
HeapsHeaps
A heap is a certain kind of complete binary tree.
HeapsHeaps
A heap is a certain kind of complete binary tree.
When a completebinary tree is built,
its first node must bethe root.
When a completebinary tree is built,
its first node must bethe root.
Root
HeapsHeaps
Complete binary tree.
Left childof theroot
The second node isalways the left child
of the root.
The second node isalways the left child
of the root.
HeapsHeaps
Complete binary tree.
Right childof the
root
The third node isalways the right child
of the root.
The third node isalways the right child
of the root.
HeapsHeaps
Complete binary tree.
The next nodesalways fill the next
level from left-to-right..
The next nodesalways fill the next
level from left-to-right..
HeapsHeaps
Complete binary tree.
The next nodesalways fill the next
level from left-to-right.
The next nodesalways fill the next
level from left-to-right.
HeapsHeaps
Complete binary tree.
The next nodesalways fill the next
level from left-to-right.
The next nodesalways fill the next
level from left-to-right.
HeapsHeaps
Complete binary tree.
The next nodesalways fill the next
level from left-to-right.
The next nodesalways fill the next
level from left-to-right.
HeapsHeaps
Complete binary tree.
HeapsHeaps
A heap is a certain kind of complete binary tree.
Each node in a heapcontains a key thatcan be compared toother nodes' keys.
Each node in a heapcontains a key thatcan be compared toother nodes' keys.
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HeapsHeaps
A heap is a certain kind of complete binary tree.
The "heap property"requires that each
node's key is >= thekeys of its children
The "heap property"requires that each
node's key is >= thekeys of its children
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Adding a Node to a HeapAdding a Node to a Heap
Put the new node in the next available spot.
Push the new node upward, swapping with its parent until the new node reaches an acceptable location. 19
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Adding a Node to a HeapAdding a Node to a Heap
Put the new node in the next available spot.
Push the new node upward, swapping with its parent until the new node reaches an acceptable location. 19
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Adding a Node to a HeapAdding a Node to a Heap
Put the new node in the next available spot.
Push the new node upward, swapping with its parent until the new node reaches an acceptable location. 19
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Adding a Node to a HeapAdding a Node to a Heap
The parent has a key that is >= new node, or
The node reaches the root. The process of pushing the
new node upward is called reheapificationreheapification upward. 19
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Removing the Top of a HeapRemoving the Top of a Heap
Move the last node onto the root.
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Removing the Top of a HeapRemoving the Top of a Heap
Move the last node onto the root.
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Removing the Top of a HeapRemoving the Top of a Heap
Move the last node onto the root.
Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.
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Removing the Top of a HeapRemoving the Top of a Heap
Move the last node onto the root.
Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.
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Removing the Top of a HeapRemoving the Top of a Heap
Move the last node onto the root.
Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.
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Removing the Top of a HeapRemoving the Top of a Heap
The children all have keys <= the out-of-place node, or
The node reaches the leaf.
The process of pushing the new node downward is called reheapificationreheapification downward.
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Implementing a HeapImplementing a Heap
We will store the We will store the data from the data from the nodes in a nodes in a partially-filled partially-filled array.array.
An array of dataAn array of data
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Implementing a HeapImplementing a Heap
Data from the root Data from the root goes in thegoes in the first first location location of the of the array.array.
An array of dataAn array of data
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Implementing a HeapImplementing a Heap
Data from the next Data from the next row goesrow goes in the in the next two array locations.
An array of dataAn array of data
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Implementing a HeapImplementing a Heap
Data from the next Data from the next row goesrow goes in the in the next two array locations.
An array of dataAn array of data
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Implementing a HeapImplementing a Heap
Data from the next Data from the next row goesrow goes in the in the next two array locations.
An array of dataAn array of data
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We don't care what's inWe don't care what's inthis part of the array.this part of the array.
Important Points about the ImplementationImportant Points about the Implementation
The links between the tree's The links between the tree's nodes are not actually stored as nodes are not actually stored as pointers, or in any other way.pointers, or in any other way.
The only way we "know" that The only way we "know" that "the array is a tree" is from the "the array is a tree" is from the way we manipulate the data.way we manipulate the data.
An array of dataAn array of data
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Important Points about the ImplementationImportant Points about the Implementation
If you know the index of a If you know the index of a node, then it is easy to figure node, then it is easy to figure out the indexes of that node's out the indexes of that node's parent and children. Formulas parent and children. Formulas are given in the book.are given in the book.
[0][0] [1] [2] [3] [4]
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A heap is a complete binary tree, where the entry at each node is greater than or equal to the entries in its children.
To add an entry to a heap, place the new entry at the next available spot, and perform a reheapification upward.
To remove the biggest entry, move the last node onto the root, and perform a reheapification downward.
Summary Summary
THE ENDTHE END
Presentation copyright 1999 Addison Wesley Longman,For use with Data Structures and Other Objects Using Javaby Michael Main.
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Students and instructors who use Data Structures and Other Objects Using Java are welcometo use this presentation however they see fit, so long as this copyright notice remainsintact.