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1Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
12. HEAPS
2Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Objectives A specialized tree-based data structure known
as heaps
Usage of heaps efficiently for applications such as priority queues
How heaps are implemented using arrays
A few more applications such as selection problem and event simulation
3Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Basic Concepts Definition of heaps: A heap is a binary tree having the following
properties:
It is a complete binary tree, that is, each level of the tree is completely filled, except at the bottom level
At this level, it is filled from left to right
It satisfies the heap-order property, that is, the key value of each node is greater than or equal to the key value of its children, or the key value of each node is lesser than or equal to the key value of its children
4Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
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Fig 12.1 Heap with height three
Fig 12.1 Heap with height two
Fig 12.1 Heap with height one
Fig 12.1 Fig 12.2 Fig 12.3
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Fig 12.4 Binary Trees with no Heap
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Data Structures Using C++ by Dr Varsha Patil
Types of Heap Min-heap
Max-heap
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Data Structures Using C++ by Dr Varsha Patil
Min-Heap Data
all >= Data
all >= Data
Fig 12.5 :Structure of min-heap
8Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Min-Heap In min-heap, the key value of each node is
lesser than or equal to the key value of its children
In addition, every path from root to leaf should be sorted in ascending order
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Fig 12.6 An example of a min heap
9Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Max-Heap A max-heap is where the key value of a node is
greater than the key values in all of its sub trees
In general, whenever the term ‘heap’ is used by itself
Data
all <= Data
all <= Data
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Data Structures Using C++ by Dr Varsha Patil
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Fig 12.7 An example of a max-heap
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Data Structures Using C++ by Dr Varsha Patil
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Fig 12.7 An example of a max-heap
12Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Structure Property A binary tree is complete if it is of height h and
has 2h+1 − 1 nodes. A binary tree of height h is complete if
it is empty, or its left sub tree is complete of height h − 1 and
its right sub tree is completely full of height h − 2, or
its left sub tree is completely full of height h − 1 and its right sub tree is complete of height h − 1.
A complete tree is filled from the left when all the leaves are on
the same level or two adjacent ones
all nodes at the lowest level are as far to the left as possible
13Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Heap-order Property A binary tree has the heap property if :
it is empty or
the key in the root is larger than either children and both sub trees have the heap property
14Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
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Fig 12.8 Sample Heap
Implementation of Heap
15Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
In this array,1. parent of index ith node is at index (i − 1)/22. left child of index ith node is at index 2 X i + 13. right child of index ith node is at index 2 X i + 2
Data 9 8 4 6 2 3Index
0 1 2 3 4 5 6 7
16Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
68 46 22 35 02 13 09
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A Heap Tree
Representation of heap by using array
17Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Heaps as Abstract Data Type
Declare create() : Heap
Insert(Heap,Data) : Heap
Deletemaxval(Heap) : Heap
ReheapDown(Heap, Child) : Heap
ReheapUp(Heap, Root) : Heap
End
18Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Operations on Heaps Create—To create an empty heap to which ‘root’
points
Insert—To insert an element into the heap
Delete—To delete max (or min) element from the heap
ReheapUp—To rebuild heap when we use the insert() function
ReheapDown—To build heap when we use the delete() function
19Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
ReheapUp The Reheap Up operations repair the structure
so that it is a heap by lifting the last element up the tree until that element reaches a proper position in the tree
ReheapUp
ReheapUp Operation
20Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
ReheapDown To restore the heap, we need an operation that
will sink the root down until the heap ordering property is satisfied and thus the operation ReheapDown comes into action
ReheapUDown Operation
21Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
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Original Tree, not a Heap
Example
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Data Structures Using C++ by Dr Varsha Patil
Root 21 moved down (right)
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Data Structures Using C++ by Dr Varsha Patil
Moved down again yielding a heap
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24Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Insert into heap—A new key can be inserted into a heap. Initially, a new key is inserted by locating the first empty leaf location in an array, and the ReheapUp operation places it in a proper location in the heap
Delete into heap—The key can be deleted from heap and it is the root value. After deletion, the heap without root is repaired by ReheapDown operation
The last node key is placed at root and then ReheapDown operation places it at proper location
25Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
The steps for creating heaps are as follows:
Organize the entire collection of data elements as a binary tree stored in an array indexed from 1 to n, where for any node at index i, its two children, if exist, will be stored at index 2 X i + 1 and 2 X i + 2
the top part in which the data elements are in they Divide the binary tree into two parts: r original order and the bottom part in which the data elements are in their heap order, where each node is in higher order than its children, if any
Start the bottom part with the half of the array, which contains only leaf nodes. Of course, it is in heap order, because the leaf nodes have no child
26Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
The steps for creating heaps are as follows:
Move the last node from the top part to the bottom part, compare its order with its children, and swap its location with its highest order child if its order is lower than any child
Repeat the comparison and swapping to ensure the bottom part is in heap order again with this new node added
Repeat step 4 until the top part is empty. At this time, the bottom part becomes a complete heap tree
27Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Heap Applications Heaps are used commonly in the following
operations:
Selection algorithm
Scheduling and prioritizing (priority queue)
Sorting
28Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Selection Problem For the solution to the problem of determining
the kth element, we can create the heap and delete k − 1 elements from it, leaving the desired element at the root.
So the selection of the kth element will be very easy as it is the root of the heap
For this, we can easily implement the algorithm of the selection problem using heap creation and heap deletion operations
This problem can also be solved in O(nlogn) time using priority queues
29Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Scheduling and prioritizing (priority
queue) The heap is usually defined so that only the
largest element (that is, the root) is removed at a time.
This makes the heap useful for scheduling and prioritizing
In fact, one of the two main uses of the heap is as a prioriy queue, which helps systems decide what to do next
30Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Applications of priority queues where heaps are implemented are as follows:
CPU scheduling
I/O scheduling
Process scheduling.
31Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Sorting Other than as a priority queue, the heap has one
other important usage: heap sort
Heap sort is one of the fastest sorting algorithms, achieving speed as that of the quicksort and merge sort algorithms
The advantages of heap sort are that it does not use recursion, and it is efficient for any data order
There is no worse-case scenario in case of heap sort
32Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Heap Sort The steps for building heap sort are as follows:
Build the heap tree Start Delete Heap operations, storing
each deleted element at the end of the heap array
After performing step 2, the order of the elements will be opposite to the order in the heap tree
Hence, if we want the elements to be sorted in ascending order, we need to build the heap tree in
Descending order—the greatest element will have the highest priority
33Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Note that we use only one array, treating its parts differently
When building the heap tree, a part of the array will be considered as the heap, and the rest part will be the original array
When sorting, a part of the array will be the heap, and the rest part will be the sorted array
34Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Binomial Trees and Heap
Binomial tree is an ordered tree defined recursively
For the binomial tree Bk, There are 2k nodes The height of the tree is k There are exactly (ki) nodes at depth i for i = 0,
1, …, k The root has degree k, which is greater than
that of any other node; moreover, if the children of the root are numbered from left to right by k − 1, k − 2, …, 0, the child i is the root of a subtree
35Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Binomial Heap A binomial heap H is a set of binomial trees that
satisfies the following binomial heap properties
Each binomial tree in H follows the min-heap property. We say that each such tree is minheap- ordered
For any non-negative integer k, there is utmost one binomial tree in H whose root has degree k
36Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Representation of Binomial Heap
The node of a binomial heap can be represented by five tuples as shown in fig
1. Parent—Pointing to parent node2. Key—Key value, that is, data3. Degree—Degree of each node, that is, the number of children it has4. Child—Pointing to any of its child node (mostly pointing to its leftmost child)5. Siblings—Pointing to sibling node, that is, used to maintain the singly-circular lists of siblings
37Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Fig. Representation of a node of binomial
heap
Parent
KeyDegree
Child Sibling
38Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Fig. Representation of binomial heap of Fig. “A”
using five-tuple nodeNull111
Null
Null22
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260
Null
Null
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Null
Null
39Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Operations on Binomial Heaps
There are various operations of binomial heaps CreateBHeap—Creates an empty binomial heap, that
is, simply allocates and returns an object H, where head[H] = null
FindMinimumKey—Returns a pointer to the node with the minimum key in an n-node binomial heap H
UnitingTwoBHeap—Takes the union of the two binomial heaps.
InsertNode—Inserts node into binomial heap H ExtractMinimumKeyNode—Extracts the node with
minimum key from binomial heap H and returns the pointer to the extracted node
DecreaseKey—Decreases the key of a node in a binomial heap H to a new value k
DeleteKey—Deletes the specified key from binomial heap H
40Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Fibonacci Heap Like binomial heap, Fibonacci heap is a
collection of min-heap-ordered trees
The trees in a Fibonacci are not constrained to be binomial trees
The roots of all the trees in Fibonacci heap are linked together using left and right pointers into circular doubly-linked list called root list of the Fibonacci heap
41Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Fibonacci Heap An example of the Fibonacci heap consisting of
5 min-heap-ordered trees and 15 nodes
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42Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Representation of Fibonacci Heap
Fibonacci heap can be represented using the Fibonacci heap nodes
The representation of such a node is shown in Figure
ParentKey
DegreeMark
Left Child Right
43Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
The node of Fibonacci heap can be represented
by seven tuples Parent—Pointing to parent node Key—Key value, that is, data Degree—Degree of each node, that is, the number of
children it has Child—Pointing to any of its child node (mostly
pointing to its leftmost child) Mark—The Boolean-valued field indicates whether
the node has lost a child since the last time the node was made the child of another node. The newly created nodes are,unmarked (i.e., default value is false)
Left—Pointing to the left sibling node, that is, used to maintain the doubly circular lists of siblings
Right—Pointing to the right sibling node, that is, used to maintain the doubly circular lists of siblings
44Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Operations on Fibonacci Heaps
CreateHeap—Creates an empty Fibonacci Heap, that is, simply allocates and returns an object H, where min[H]=null
FindMinimumKey—Returns a min[H], that is, pointer to the node with the minimum key in an n-node Fibonacci heap H
UnitingTwoFHeap—Takes the union of the two Fibonacci heaps
InsertNode—Inserts node into Fibonacci heap H ExtractMinimumKeyNode—Extracts the node with
minimum key from Fibonacci heap H and returns the pointer to the extracted node
DecreaseKey—Decreases the key of a node in a Fibonacci heap H to a new value k
DeleteKey—Deletes the specified key from Fibonacci heap H
45Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Summary A complete or nearly complete binary tree where each
node is greater or equal to its decedents with each subtree satisfying this property is called as heap
The basic operations on heap are as follows: insert, delete, ReheapUp, and ReheapDown
Heap can be implemented using an array as it is a complete binary tree. It is easy to maintain fixed relationship between a node and its children
Among many applications of heap, the key ones are the following: priority queue, sorting, and selection
46Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
Priority queue is implemented using heap by maintaining its relationship of element with other members in a list
One of the popular sorting techniques is heap sort that uses heap
The popularly heap is used in application where at each stage, the largest element is to be picked up for processing known as selection algorithm
Summary
47Oxford University Press © 2012
Data Structures Using C++ by Dr Varsha Patil
END Of
Chapter 12….!