shell sort--sorting techniques

7/30/2019 Shell Sort--Sorting techniques

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Shell Sort

CSET 3150


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Sorting and Searching

Shell Sortmore efficient than Bubble, Insertion and

Selection sortscompares distant items first and works waydown to nearby items

interval is called the gap

in a simple implementation, gap begins at one-half the length of the list and is successivelyhalved until each item is compared withneighbor


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Shell Sort

A variation of the insertion sortBut faster than O(n2)

Done by sorting subarrays of equallyspaced indices

Instead of moving to an adjacent locationan element moves several locations away

Results in an almost sorted array

This array sorted efficiently with ordinaryinsertion sort


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Shell SortGiven an array of n items

Start with the first item, and then take every n/2items to form a sub- arrayTake the second item and every n/2 item fromthere to form your second sub-array

Continue to form sub-arrays until every item isincluded in a sub-arrayWhen you have all of your sub-arrays sort them byan insertion sortThen take the first elements of the sorted

subarrays as the first elements in a new big array;the second elements as the next group; etc.Form new subarrays by selecting every (n/2)/2items and then process as before

Continue until you select 1 item


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Shell Sort

An array and the subarrays formed by grouping

elements whose indices are 6 apart.


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Shell Sort

The subarrays of the previous figure after they are

sorted, and the array that contains them.


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Shell Sort

The subarrays of the array in previous figure formed by

grouping elements whose indices are 3 apart


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Shell Sort

The subarrays of previous figure after they are sorted,

and the array that contains them.


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Another Shell Sort ExampleConsider sorting the following list byShell sort:

45

23

4

9

6

7

91

8

12

24

9

24

45

91

6

8

23

4

7

12

9

6

4

24

8

7

45

23

12

91

4

8

9

12

45

6

7

23

24

91

4

6

8

7

9

23

12

24

45

91

4

6

7

8

9

12

23

24

45

91

GAP=3 GAP=2 GAP=1


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Choosing the gap size

The idea of the decreasing gap size is that the listbecomes more and more sorted each time the gapsize is reduced, therefore you dont (for example)

want to have a gap size of 4 followed by a gap size of2 because youll be sorting half the numbers a secondtime.

There is no formal proof of a good initial gap size,

but about a 10th the size of N is a reasonable start.Try to use prime numbers as your gap size, or oddnumbers if you can not readily get a list of primes(though note gaps of 9, 7, 5, 3, 1 will be doing less

work when gap=3).


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Efficiency of Shell Sort

Efficiency is O(n2) for worst case

Ifn

is a power of 2Average-case behavior is O(n1.5)

Any time the variable space is even,add 1

This also results in O(n1.5)


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Comparing the Algorithms

Best Average WorstCase Case Case

Selection sort O(n2

) O(n2

) O(n2

)Insertion sort O(n) O(n2) O(n2)

Shell sort O(n) O(n1.5) O(n1.5)

The time efficiencies of three sorting algorithms


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Comparing O(N2), O(N1.25) & O(N)

O(N) O(N^1.25) O(N^2)

1 1 1

2 2.378414 4

3 3.948222 9

4 5.656854 16

5 7.476744 25

6 9.390507 36

7 11.38604 49

8 13.45434 64

9 15.58846 81

10 17.78279 100

11 20.03276 121

12 22.33452 144

13 24.68478 169

14 27.08071 196

15 29.51985 225

16 32 256

17 34.51923 289

18 37.07581 324

19 39.66815 361

20 42.29485 400

21 44.9546 441

22 47.64621 484

23 50.36859 529

24 53.12073 576

25 55.9017 625

0

100

200

300

400

500

600

700

1 3 5 7 911

13

15

17

19

21

23

25

Size of input (N)

Time

(T)

O(N)

O(N^1.25)

O(N^2)


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The family of sorting methods

Main sorting themes

Comparison-basedsorting

Transpositionsorting

BubbleSort

Insert andkeep sorted

Insertionsort

Treesort

Divide andconquer

QuickSort MergeSort

ProxmapSort

RadixSort

ShellSort

Diminishingincrement

sorting

Address--basedsorting

Selectionsort

Heapsort

Priority queuesorting

shell sort--sorting techniques

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