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PREPARED BYyR.MADHUMITHA

yG.KARTHIKAyV.LAKSHMI PRIYA

yA.MOHANA PRIYAyU.GNANAMBIGAI

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Why we do sorting?y Commonly encountered programming task in

computing.

y

Examples of sorting:y List containing exam scores sorted from Lowest to

Highest or from Highest to Lowest

y List containing words that were misspelled and belisted in alphabetical order.

y List of student records and sorted by studentnumber or alphabetically by first or last name.

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SORTINGy Sorting is any process of arranging items in

some sequence and/or in different sets, and

accordingly, it has two common, yet distinctmeanings:

y ordering: arranging items of the same kind,class, nature, etc. in some ordered sequence,

y categorizing: grouping and labeling items withsimilar properties together (by sorts).

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Sortingy Card players all know how to sort

y First card is already sortedy With all the rest,

Scan back from the end until you find the first card larger thanthe new one,

Move all the lower ones up one slot

insert it

Q

2

9

A

K

10

J

2

2

9

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Insertion sorty Complexity

y For each card

y Scan O(n)

y Shift up O(n)

y Insert O(1)

y Total O(n)y First card requires O(1), second O(2),

y Forn cards operations O(n2)7 i

i=1

n

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Sorting - Insertion sorty Complexity

y For each card

y Scan O(n)

y Shift up O(n)

y Insert O(1)y Total O(n)

y First card requires O(1), second O(2),

y For n cards operations O(n2)

7 i

i=1

n

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Algorithm :Insertion Sorty void Insertionsort (ElementType A[],int N)y {y

intj,p;y Element Type Tmp;y for(p=1;p

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HEAP SORTy At each level in the binary tree created for

Merge Sort, there are n elements, with O(1)

time spent at each elementy O(n) running time for processing one level

y The height of the tree is O(log n)

y Therefore, the time complexity is O(nlog n)

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What is aheap ?y Definitions of heap:

1. A large area of memory from which the

programmer can allocate blocks as needed,and deallocate them (or allow them to begarbage collected) when no longer needed

2. A balanced, left-justified binary tree inwhich no node has a value greater than thevalue in its parent

y Heapsort uses the second definition

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HEAP PROPERTYy A node has the heap property if the value in the node

is as large as or larger than the values in its children

12

8 3

Blue node has

heap property

12

8 12

Blue node has

heap property

12

8 14

Blue node does not have

heap property

All leaf nodes automaticallyhave the heap propertyA binary tree is a heap if all nodes in it have theheap property

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SHIFTING UPy Given a node that does not have the heap

property, you can give it the heap property by

exchanging its value with the value of thelarger child12

8 14

Blue node does not haveheap property

14

8 12

Blue node hasheap property

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A SAMPLE HEAPyHeres a sample binary tree after it has been

heapified

19

1418

22

321

14

119

15

25

1722

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yNotice that heapified does not meansorted

yHeapifying does not change the shape ofthe binary tree; this binary tree isbalanced and left-justified because it

started out that way

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QUICK SORT1) Divide : If the sequence S has 2 or more elements,

select an element x from S to be your pivot. Anyarbitrary element, like the last, will do. Remove all the

elements of S and divide them into 3 sequences:L, holds Ss elements less than xE, holds Ss elements equal to xG, holds Ss elements greater than x

2) Recurse: Recursively sort L and G3) Conquer: Finally, to put elements back into S in order,

first inserts the elements of L, then those of E, andthose of G.

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1)Select: pick an element

2) Divide: rearrange elements

so that x goes to its final position E

3)Recurse and Conquer: recursively

sort

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QUICK SORT RUNNING TIMEy Worst case: when the pivot does not divide the sequence in twoy At each step, the length of the sequence is only reduced by 1y Total running time

y General case:y Time spent at level i in the tree is O(n)

y Running time: O(n) * O(height)

y Average case:y O(n log n)

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SHELL SORTy Founded by Donald Shell and named the sorting

algorithm after himself in 1959.y 1st algorithm to break the quadratic time barrier but

few years later, a sub quadratic time bound wasproven

y Shellsort works by comparing elements that aredistant rather than adjacent elements in an array orlist where adjacent elements are compared.

y Shellsort uses a sequence h1, h2, , ht called theincrement sequence. Any increment sequence is fineas long as h1 = 1 and some other choices are betterthan others.

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y Shellsort makes multiple passes through a listand sorts a number of equally sized sets using

the insertion sort.y Shellsort improves on the efficiency of

insertion sort byquickly shifting values totheir destination.

y Shellsort is also known as diminishingincrement sort.

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Empirical Analysis of Shellsort

(Advantage)y Advantage of Shellsort is that its only

efficient for medium size lists. For bigger

lists, the algorithm is not the best choice.Fastest of all O(N^2) sorting algorithms.

y 5 times faster than the bubble sort and alittle over twice as fast as the insertion sort,

its closest competitor.

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Empirical Analysis of Shellsort

(Disadvantage)y Disadvantage of Shellsort is that it is a

complex algorithm and its not nearly as

efficient as the merge, heap, and quick sorts.y The shell sort is still significantly slower thanthe merge, heap, and quick sorts, but itsrelatively simple algorithm makes it a goodchoice for sorting lists of less than 5000items unless speed important. It's also anexcellent choice for repetitive sorting ofsmaller lists.

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THANK YOU