data structures using c++ by dr varsha patil oxford...


A specialized tree-based data structure known as heaps

Usage of heaps efficiently for applications such as priorityqueues

How heaps are implemented using arrays

A few more applications such as selection problem andevent simulation


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 iscompletely 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 valueof each node is greater than or equal to the key value of itschildren, or the key value of each node is lesser than orequal to the key value of its children


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5065

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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


Min-heap

Max-heap


Data

all >= Dataall >= Data

Fig 12.5 :Structure of min-heap


In min-heap, the key value of each node is lesser than orequal to the key value of its children

In addition, every path from root to leaf should be sortedin ascending order

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Fig 12.6 An example of a min heap


A max-heap is where the key value of a node is greaterthan the key values in all of its sub trees

In general, whenever the term ‘heap’ is used by itself

Data

all <= Dataall <= Data


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Fig 12.7 An example of a max-heap


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 rightsub tree is completely full of height h − 2, or

its left sub tree is completely full of height h − 1 and its rightsub 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


A binary tree has the heap property if :

it is empty or

the key in the root is larger than either children andboth sub trees have the heap property


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Fig 12.8 Sample Heap

Implementation of Heap


In this array,

1. parent of index ith node is at index (i − 1)/2

2. left child of index ith node is at index 2 X i + 1

3. right child of index ith node is at index 2 X i + 2

Data 9 8 4 6 2 3

Index 0 1 2 3 4 5 6 7


68 46 22 35 02 13 09

0913235

2246

68

A Heap Tree

Representation of heap by using array


Declare create() : Heap

Insert(Heap,Data) : Heap

Deletemaxval(Heap) : Heap

ReheapDown(Heap, Child) : Heap

ReheapUp(Heap, Root) : Heap

End


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


The Reheap Up operations repair the structure so that it isa heap by lifting the last element up the tree until thatelement reaches a proper position in the tree

ReheapUp

ReheapUp Operation


To restore the heap, we need an operation that will sinkthe root down until the heap ordering property is satisfiedand thus the operation ReheapDown comes into action

ReheapUDown Operation


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Original Tree, not a Heap

Example

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Root 21 moved down (right)

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Moved down again yielding a heap

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Insert into heap—A new key can be inserted into a heap. Initially, anew key is inserted by locating the first empty leaf location in anarray, and the ReheapUp operation places it in a proper location inthe heap

Delete into heap—The key can be deleted from heap and it is theroot value. After deletion, the heap without root is repaired byReheapDown operation

The last node key is placed at root and then ReheapDown operationplaces it at proper location


Organize the entire collection of data elements as a binary treestored in an array indexed from 1 to n, where for any node atindex i, its two children, if exist, will be stored at index 2 X i + 1and 2 X i + 2

the top part in which the data elements are in they Divide thebinary tree into two parts: r original order and the bottom partin which the data elements are in their heap order, where eachnode is in higher order than its children, if any

Start the bottom part with the half of the array, which containsonly leaf nodes. Of course, it is in heap order, because the leafnodes have no child


Move the last node from the top part to the bottom part,compare its order with its children, and swap its locationwith its highest order child if its order is lower than anychild

Repeat the comparison and swapping to ensurethe bottom part is in heap order again with this new nodeadded

Repeat step 4 until the top part is empty. At this time, thebottom part becomes a complete heap tree


Heaps are used commonly in the following operations:

Selection algorithm

Scheduling and prioritizing (priority queue)

Sorting


For the solution to the problem of determining the kthelement, we can create the heap and delete k − 1 elementsfrom it, leaving the desired element at the root.

So the selection of the kth element will be very easy as it isthe root of the heap

For this, we can easily implement the algorithm of theselection problem using heap creation and heap deletionoperations

This problem can also be solved in O(nlogn) time usingpriority queues


The heap is usually defined so that only the largestelement (that is, the root) is removed at a time.

This makes the heap useful for scheduling andprioritizing

In fact, one of the two main uses of the heap is as a prioriyqueue, which helps systems decide what to do next


Applications of priority queues where heaps areimplemented are as follows:

CPU scheduling

I/O scheduling

Process scheduling.


Other than as a priority queue, the heap has one otherimportant usage: heap sort

Heap sort is one of the fastest sorting algorithms,achieving speed as that of the quicksort and merge sortalgorithms

The advantages of heap sort are that it does not userecursion, and it is efficient for any data order

There is no worse-case scenario in case of heap sort


The steps for building heap sort are as follows:

Build the heap tree

Start Delete Heap operations, storing each deletedelement at the end of the heap array

After performing step 2, the order of the elements will beopposite to the order in the heap tree

Hence, if we want the elements to be sorted in ascendingorder, we need to build the heap tree in

Descending order—the greatest element will have thehighest priority


Note that we use only one array, treating its partsdifferently

When building the heap tree, a part of the array will beconsidered as the heap, and the rest part will be theoriginal array

When sorting, a part of the array will be the heap, and therest part will be the sorted array


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 anyother node; moreover, if the children of the root arenumbered from left to right by k − 1, k − 2, …, 0, the child iis the root of a subtree


A binomial heap H is a set of binomial trees that satisfiesthe 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 onebinomial tree in H whose root has degree k


The node of a binomial heap can be represented by five tuples as shown in fig

1. Parent—Pointing to parent node

2. Key—Key value, that is, data

3. Degree—Degree of each node, that is, the number of children it has

4. 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


Parent

Key

Degree

Child Sibling


Null

11

1

Null

Null

2

2

13

1

26

0

Null Null

19

0

Null Null


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


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


An example of the Fibonacci heap consisting of 5 min-heap-ordered trees and 15 nodes

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19


Fibonacci heap can be represented using the Fibonacci heap nodes

The representation of such a node is shown in Figure

Parent

Key

Degree

Mark

Left Child Right


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


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


A complete or nearly complete binary tree where each node is greateror equal to its decedents with each subtree satisfying this property iscalled 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 binarytree. It is easy to maintain fixed relationship between a node and itschildren

Among many applications of heap, the key ones are the following:priority queue, sorting, and selection


Priority queue is implemented using heap by maintaining itsrelationship 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, thelargest element is to be picked up for processing known as selectionalgorithm

data structures using c++ by dr varsha patil oxford...

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