Download Heap1

Heap
Objectives
Upon completion you will be able to:
• Define and implement heap structures
• Understand the operation and use of the heap ADT
• Design and implement selection applications using a heap
• Design and implement priority queues using a heap
Data Structures: A Pseudocode Approach with C, Second Edition
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Basic Concepts
A heap is a binary tree whose left and right subtrees have values
less than their parents. We begin with a discussion of the basic
heap structure and its two primary operations, reheap up and
reheap down.
• Definition
• Maintenance Operations
Data Structures: A Pseudocode Approach with C, Second Edition
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Basic Concepts
Definition
A heap is a binary tree structure with the following
properties:
1. The tree is complete or nearly complete.
2. The key value of each node is greater than or equal
to the key value in each of its descendents.
•
•
The root of a heap is guaranteed to hold the largest node in
the tree; its subtrees contain data that have lesser values.
Heaps are often implemented in an array rather than a
linked list.
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Sometimes this structure is called a max-heap.
A min-heap is to create a heap which the key value in a node is
less than the key values in all of its subtrees.
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The reheap up operation reorders a “broken” heap by floating
the last element up the tree until it is in its correct location in
the heap.
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Reheap Down Operation reorders a “broken” heap by
pushing the root down the tree until it is in its correct position
in the heap.
Data Structures: A Pseudocode Approach with C, Second Edition
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Heap Implementation
Heaps are usually implemented in an array structure. In this
section we discuss and develop five heap algorithms.
• Reheap Up
• Reheap Down
• Build a Heap
• Insert a Node into a Heap
• Delete a Node from a Heap
Data Structures: A Pseudocode Approach with C, Second Edition
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Heap Implementation
The relationship between a node and its children is fixed and
can be calculated as shown below:
1. For a node located at index i, its children are found at:
a. Left child: 2i + 1
b. Right child: 2i + 2
2. The parent of a node located at index i is located at (i -1)/2
3. Given the index for a left child, j, its right sibling, if any, is
found at j + 1. Conversely, given the index for a right child, k,
its left sibling, which must exist, is found at k - 1.
4. Given the size, n, of a completed heap, the location of the
first leaf is (n/2)
5. Given the location of the first leaf element, the location of
the last nonleaf element is one less.
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