heap_sort jkghjkhef

7/28/2019 HEAP_SORT jkghjkhef

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Session: Jan May 2013

EEC 012 Data Structures 1Prof. (Dr) N. Guruprasad

HEAP TREE & HEAP SORT


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EEC 012 Data Structures 2

Heap tree : defined as binary tree with keysassigned to nodes provided following two conditions

are met :

1) ( tree shape requirement )

all levels are full exceptpossibly the last level; only some right most leaves may

be missing.

2) ( parental dominance requirement )key at each

node is greater than or equal to keys at its children.

Prof. (Dr) N. Guruprasad


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HEAP TREE

10

5 7

4 2 1



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Not HEAP - violates shape requirement

10

5 7

2 1



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Not HEAP - violates dominance requirement

10

5 7

6 2 1



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Applications :Interval scheduling we may have a list of tasks with

certain start and finish times and we want to do as many of

these tasks as possible in a given period of time.

Priority queue scheduling in OS

Event-driven simulators to maintain the list of events to be

simulated in order of their time of occurrence.



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Two variations :

1) Max heapnode is greater than or equal to its

children. ( descending heap )

2) Min heapnode is smaller than or equal to its

children. ( ascending heap )

Construction of Heap :

1) Bottom up heap construction2) Top down heap construction



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Bottom up heap construction :

Initializes a binary tree and heapifies as follows :

1) Starting with last parental node, algorithm checks

whether parental dominance holds for key at this

node

2) If it does not; algorithm exchanges nodes key K

with larger key K of its children and checks whether

parental dominance holds for K in its new position.3) This process continues until parental dominance

requirement for K is satisfied.



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4) After completing the Heapification ofsubtree rooted

at current parental node, algorithm proceeds to do the

same for nodes immediate predecessor.

5) Algorithm stops after this is done for trees root.



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Example : Construct a heap for the followingelements :

2 9 7 6 5 8

using bottom up technique.



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Algorithm : HeapBottomUp(H[1..n])

// Purpose : Construct heap from elements of given array by

bottom up algorithm

// Input : Array H[1..n] of orderable items

// Output : Heap H[1..n]

for i [ n/2 ] downto 1 do

k Iv H[k]

heap false


S


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while not heap and 2 * k n do

j 2 * kif j < n

if H[ j ] < H [ j+1 ]

j j + 1

if v H [ j ]

heap true

else

H[k] H[j];k j

H[ k ] v


S i J M 2013


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Top down heap construction :

Constructs a heap by successive iterations.1) Attach a new node K after last leaf in existing heap.

2) Shift K up to its appropriate place in new heap as

follows :1) Compare K with parents key; if parent K; STOP tree

is heap

2) Otherwise swap these two keys and compare K with

new parent.

3) Swapping continues until K is not greater than its

last parent or it reaches root.


S i J M 2013


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Example : Construct a heap for the following

elements :

50 25 30 75 100 45 80

using top down approach.


Session Jan Ma 2013


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Heap sort :

Interesting sorting algorithm by J.W.J. WILLIAMS.

Basically a two stage algorithm that works as follows :

Stage 1 ) ( Heap construction ) construct heap for given

array.Stage 2 ) ( Maximum deletion ) apply root deletion

operation n-1 times to remaining heap.

NOTE : under array implementation of HEAP; element

being deleted is placed last in an array.




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EEC 012 D t St t 16

Example : Sort the following elements using HeapSort.

13 86 43 38 54 23 08 63

Solution : Two stage process :

1) Using top down approach we build heap tree

2) By successively deleting the root we get the elements

sorted

P f (D ) N G d

heap_sort jkghjkhef

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