quadratic sorting - university of colorado bouldermain/supplements/pdf/notes13.pdf · quadratic...

1

The presentation illustrates two quadratic sorting algorithms:Selectionsort and Insertionsort. Before this lecture, students shouldknow about arrays, and should have seen some motivation for sorting(such as binary search of a sorted array).

❐ Chapter 13 presents severalcommon algorithms forsorting an array of integers.

❐ Two slow but simplealgorithms areSelectionsort andInsertionsor t.

❐ This presentationdemonstrates how the twoalgorithms work.

Quadratic Sorting

Data Structuresand Other ObjectsUsing C++

2

The picture shows a graphical representation of an array which we willsort so that the smallest element ends up at the front, and the otherelements increase to the largest at the end. The bar graph indicates thevalues which are in the array before sorting--for example the firstelement of the array contains the integer 45.

Sorting an Array of Integers

❐ The pictureshows anarray of sixintegers thatwe want tosort fromsmallest tolargest

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][0] [1] [2] [3] [4] [5]

3

The first sorting algorithm that we'll examine is called Selectionsort. Itbegins by going through the entire array and finding the smallestelement. In this example, the smallest element is the number 8 atlocation [4] of the array.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

The Selectionsort Algorithm

❐ Start byfinding thesmallestentry.

[5]AAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

[5][0] [1] [2] [3] [4] [5]

4

Once we have found the smallest element, that element is swappedwith the first element of the array...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]



❐ Swap thesmallestentry withthe firstentry.



AAAAAA

[5][0] [1] [2] [3] [4] [5]

5

...like this.

The smallest element is now at the front of the array, and we havetaken one small step toward producing a sorted array.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]



❐ Swap thesmallestentry withthe firstentry.



AAAAAA

0

10

20

30

40

50

60

70

[1][0] [1] [2] [3] [4] [5]

6

At this point, we can view the array as being split into two sides: To theleft of the dotted line is the "sorted side", and to the right of the dottedline is the "unsorted side". Our goal is to push the dotted line forward,increasing the number of elements in the sorted side, until the entirearray is sorted.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ Part of thearray is nowsorted.



AAAAAA

0

10

20

30

40

50

60

70

[1]

AAAAAAAAAAAAAAAAAAAAAAAA





AAAAAAAAAAAA

Sorted side Unsorted side

[0] [1] [2] [3] [4] [5]

7

Each step of the Selectionsort works by finding the smallest element inthe unsorted side. At this point, we would find the number 15 at location[5] in the unsorted side.

[1] [2] [3] [4] [5] [6]AAAAAAAAAAAAAAAAAAAAA


AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][2] [3] [4] [5][2] [3] [4] [5]


❐ Find thesmallestelement inthe unsortedside.






AAAAAAAAAAAA


[0] [1] [2] [3] [4] [5]

8

This small element is swapped with the number at the front of theunsorted side, as shown here...



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][3] [4] [5] [6][3] [4] [5] [6]


❐ Find thesmallestelement inthe unsortedside.

❐ Swap withthe front ofthe unsortedside.






AAAAAAAAAAAA


[0] [1] [2] [3] [4] [5]

9

...and the effect is to increase the size of the sorted side by oneelement.

As you can see, the sorted side always contains the smallest numbers,and those numbers are sorted from small to large. The unsorted sidecontains the rest of the numbers, and those numbers are in noparticular order.



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][3] [4] [5] [6][3] [4] [5] [6]


❐ We haveincreased thesize of thesorted sideby oneelement.






AAAAAAAAAAAA


[0] [1] [2] [3] [4] [5]

10

Again, we find the smallest entry in the unsorted side...



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][3] [4] [5][3] [4] [5]


❐ The processcontinues...






AAAAAAAAAAAA


Smallestfrom

unsorted

[0] [1] [2] [3] [4] [5]

11

...and swap this element with the front of the unsorted side.



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][4] [5] [6][4] [5] [6]








AAAAAAAAAAAA


[0] [1] [2] [3] [4] [5]

Swap

with

front

12

The sorted side now contains the three smallest elements of the array.



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][4] [5] [6][4] [5] [6]








AAAAAAAAAAAAAAAAAA

Sorted side Unsorted sideSorted side

is bigger

[0] [1] [2] [3] [4] [5]

13

Here is the array after increasing the sorted side to four elements.



AAAAAA



AAAAAA



AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][5] [6][5] [6]


❐ The processkeeps addingone morenumber to thesorted side.

❐ The sorted sidehas the smallestnumbers,arranged fromsmall to large.






AAAAAAAAAAAAAAAAAA


[0] [1] [2] [3] [4] [5]

14

And now the sorted side has five elements.

In fact, once the unsorted side is down to a single element, the sort iscompleted. At this point the 5 smallest elements are in the sorted side,and so the the one largest element is left in the unsorted side.

We are done...



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][6][6]


❐ We can stopwhen theunsorted sidehas just onenumber, sincethat numbermust be thelargest number.

[0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA

Sorted side Unsorted sid

15

...The array is sorted.

The basic algorithm is easy to state and also easy to program.



AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ The array isnow sorted.

❐ We repeatedlyselected thesmallestelement, andmoved thiselement to thefront of theunsorted side. [0] [1] [2] [3] [4] [5]

16

Now we'll look at another sorting method called Insertionsort. The endresult will be the same: The array will be sorted from smallest to largest.But the sorting method is different.

However, there are some common features. As with the Selectionsort,the Insertionsort algorithm also views the array as having a sorted sideand an unsorted side, ...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

The Insertionsort Algorithm

❐ TheInsertionsortalgorithmalso viewsthe array ashaving asorted sideand anunsortedside. [0] [1] [2] [3] [4] [5]

17

...like this.

However, in the Selectionsort, the sorted side always contained thesmallest elements of the array. In the Insertionsort, the sorted side willbe sorted from small to large, but the elements in the sorted side willnot necessarily be the smallest entries of the array.

Because the sorted side does not need to have the smallest entries, wecan start by placing one element in the sorted side--we don't need toworry about sorting just one element. But we do need to worry abouthow to increase the number of elements that are in the sorted side.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ The sortedside startswith just thefirstelement,which is notnecessarilythe smallestelement. [1]



AAAAAA

[1][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


18

The basic approach is to take the front element from the unsortedside...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ The sortedside growsby taking thefrontelementfrom theunsortedside...

[1] [2]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA

[1] [2][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


19

...and insert this element at the correct spot of the sorted side.

In this example, the front element of the unsorted side is 20. So the 20must be inserted before the number 45 which is already in the sortedside.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ ...andinserting itin the placethat keepsthe sortedsidearrangedfrom smallto large. [1] [2]



AAAAAA



AAAAAA

[1] [2][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


20

After the insertion, the sorted side contains two elements. These twoelements are in order from small to large, although they are not thesmallest elements in the array.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ In thisexample, thenew elementgoes in frontof theelement thatwas alreadyin the sortedside. [1] [2]



AAAAAA



AAAAAA

[1] [2][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


21

Sometimes we are lucky and the newly inserted element is already inthe right spot. This happens if the new element is larger than anythingthat's already in the array.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ Sometimeswe are luckyand the newinserted itemdoesn't needto move atall.

[1] [2] [3]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA

[1] [2] [3][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


22

Sometimes we are lucky twice in a row.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❐ Sometimeswe are luckytwice in arow.

[1] [2] [3] [4]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA




AAAAAAAA

AAAAAAAAA

[1] [2] [3] [4][0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


23

The actual insertion process requires a bit of work that is shown here.The first step of the insertion is to make a copy of the new element.Usually this copy is stored in a local variable. It just sits off to the side,ready for us to use whenever we need it.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

How to Insert One Element

❶ Copy thenew elementto a separatelocation.

[1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA

AAAAAAAAAAAAAAAAAAAAA


AAAAAA

[1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAA

[0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


24

After we have safely made a copy of the new element, we start shiftingelements from the end of the sorted side. These elements are shiftedrightward, to create an "empty spot" for our new element to be placed.

In this example we take the last element of the sorted side and shift itrightward one spot...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]


❷ Shiftelements inthe sortedside,creating anopen spacefor the newelement.

[1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAA


AAAAAA

[0] [1] [2] [3] [4] [5]

25

...like this.

Is this the correct spot for the new element? No, because the newelement is smaller than the next element in the sorted section. So wecontinue shifting elements rightward...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA

AAAAAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]


❷ Shiftelements inthe sortedside,creating anopen spacefor the newelement.



AAAAAA

[4][4][0] [1] [2] [3] [4] [5]

26

This is still not the correct spot for our new element, so we shift again...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70



AAAAAA



AAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]


❷ Continueshiftingelements...



AAAAAA

[3][3][0] [1] [2] [3] [4] [5]

27

...and shift one more time...

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70



AAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]


❷ Continueshiftingelements...



AAAAAA

[2][2][0] [1] [2] [3] [4] [5]

28

Finally, this is the correct location for the new element. In general thereare two situations that indicate the "correct location" has been found:

1. We reach the front of the array (as happened here), or

2. We reached an element that is less than or equal to the newelement.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][1] [2] [3] [4] [5]AAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]


❷ ...until youreach thelocation forthe newelement.



AAAAAA

[1][1][0] [1] [2] [3] [4] [5]

29

Once the correct spot is found, we copy the new element back into thearray. The number of elements in the sorted side has increased byone.

[1] [2] [3] [4] [5] [6]0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6][1] [2] [3] [4] [5]AAAAAAAAAAAAAAAAAAAAA


AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA

[1] [2] [3] [4] [5]


❸ Copy thenew elementback into thearray, at thecorrectlocation.



AAAAAA

[0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


30

The last element of the array also needs to be inserted. Start bycopying it to a safe location.



AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]




AAAAAA

❐ The lastelementmust also beinserted.Start bycopying it...

[0] [1] [2] [3] [4] [5]






AAAAAAAAAAAAAAAAAA


31

Now we will shift elements rightward. How many shifts will be done?



AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

A Quiz

[6][6]

How manyshifts willoccur before wecopy thiselement backinto the array?



AAAAAA

[0] [1] [2] [3] [4] [5]

32

These four elements are shifted. But the 7 at location [0] is not shiftedsince it is smaller than the new element.



AAAAAA

AAAAAAAAA

AAAAAAAAAAAAAAAA

AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

A Quiz

[2][2]AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA


AAAAAA

❐ Four itemsare shifted.

[0] [1] [2] [3] [4] [5]

33

The new element is inserted into the array.



AAAAAA



AAAAAA



AAAAAA



AAAAAAAAA



AAAAAA



AAAAAA

0

10

20

30

40

50

60

70

[1] [2] [3] [4] [5] [6]

A Quiz



AAAAAA

❐ Four itemsare shifted.

❐And thenthe element iscopied backinto the array.

[0] [1] [2] [3] [4] [5]

34

A quick summary of these algorithms' timing properties.

❐ Both Selectionsort and Insertionsort have a worst-case time of O(n2), making them impractical forlarge arrays.

❐ But they are easy to program, easy to debug.

❐ Insertionsort also has good performance when thearray is nearly sorted to begin with.

❐ But more sophisticated sorting algorithms areneeded when good performance is needed in allcases for large arrays.

Timing and Other Issues

35

Feel free to send your ideas to:

Michael Main

[email protected]

THE END

Presentation copyright 1997 Addison Wesley Longman,For use with Data Structures and Other Objects Using C++by Michael Main and Walter Savitch.

Some artwork in the presentation is used with permission from Presentation Task Force(copyright New Vision Technologies Inc) and Corel Gallery Clipart Catalog (copyrightCorel Corporation, 3G Graphics Inc, Archive Arts, Cartesia Software, Image ClubGraphics Inc, One Mile Up Inc, TechPool Studios, Totem Graphics Inc).

Students and instructors who use Data Structures and Other Objects Using C++ are welcometo use this presentation however they see fit, so long as this copyright notice remainsintact.

quadratic sorting - university of colorado bouldermain/supplements/pdf/notes13.pdf · quadratic...

Documents