12. Heaps - Data Structures using C++ by Varsha Patil

1Oxford University Press © 2012

Data Structures Using C++ by Dr Varsha Patil

12. HEAPS



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



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



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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



Types of Heap Min-heap

Max-heap



Min-Heap Data

all >= Data

all >= Data

Fig 12.5 :Structure of min-heap



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



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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Fig 12.7 An example of a max-heap



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



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



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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)/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



68 46 22 35 02 13 09

0913235

2246

68

A Heap Tree

Representation of heap by using array



Heaps as Abstract Data Type

Declare create() : Heap

Insert(Heap,Data) : Heap

Deletemaxval(Heap) : Heap

ReheapDown(Heap, Child) : Heap

ReheapUp(Heap, Root) : Heap

End



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



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



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



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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, 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



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



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



Heap Applications Heaps are used commonly in the following

operations:

Selection algorithm

Scheduling and prioritizing (priority queue)

Sorting



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



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



Applications of priority queues where heaps are implemented are as follows:

CPU scheduling

I/O scheduling

Process scheduling.



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



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



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



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



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



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



Fig. Representation of a node of binomial

heap

Parent

KeyDegree

Child Sibling



Fig. Representation of binomial heap of Fig. “A”

using five-tuple nodeNull111

Null

Null22

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Null

Null

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Null

Null



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



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



Fibonacci Heap An example of the Fibonacci heap consisting of

5 min-heap-ordered trees and 15 nodes

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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



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



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



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



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



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Chapter 12….!

12. Heaps - Data Structures using C++ by Varsha Patil

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