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COMP 250

Lecture 15

recurrences 2:mergesort & quicksort

Oct. 13, 2017

Recall: Mergesort

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810

311

617

16254

13

810

311

617

16254

13

1368

1011

2457

1316

12345678

10111316

mergesort

mergesort

mergepartition

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mergesort(list){if list.length == 1

return listelse{

mid = (list.size - 1) / 2list1 = list.getElements(0,mid)list2 = list.getElements(mid+1, list.size-1)list1 = mergesort(list1)list2 = mergesort (list2)return merge( list1, list2 )

}} = + 2 2

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= + +

What if is not even ?

e.g. 13 = ∗ 13 + 6 + 7In general, one should write the recurrence as:

round down round up

In COMP 250, one typically assumes = 2 for recurrences that involve ( ).The more general recurrence has roughly the same solution.

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6

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Recall this slide fromlecture 13:

Q: How manyoperations arerequired to mergesorta list of size ?A: ( )

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mergesort(list){if list.length <= 4 // or some other base case

return slowSort(list) // see lecture 7else{

mid = (list.size - 1) / 2list1 = list.getElements(0,mid)list2 = list.getElements(mid+1, list.size-1)list1 = mergesort(list1)list2 = mergesort (list2)return merge( list1, list2 )

}} = + 2 2 , ≥ 5

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if list.length <= 4return slowSort(list)

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if list.length <= 4return slowSort(list)

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if list.length <= 4return slowSort(list)

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if list.length <= 4return slowSort(list)

For large , this is thedominant term.

For large , this is notthe dominant term.

Quicksort

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810

311

617

16254

13

3617254

10111613

12345678

10111316

quicksort

quicksort

partition

1234567

10111316

concatenate

pivot

Quicksort worst case example

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110

811

637

16254

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108

11637

16254

13

12345678

10111316

quicksort

quicksort

partition

concatenate

pivot

2345678

10111316

Quicksort worst case

= + − 1From last lecture….

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Quicksort worst case

= + − 1= ( + 1)2

From last lecture….

which is ( )21

How to reduce the chances of worst case ?(unbalanced partition)

“Median of three” technique:

Let the pivot be themedian of first,middle, and last elements in list.

e.g. pivot = median( 1, 13, 4) = 4

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(size-1)/2

size-1

110

811

613

716

2534

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23

110

811

613

716

2534

0

(size-1)/2

size-1

123

108

116

137

165

4partition

Works well inpractice, especiallyif the list is alreadynearly sorted.

(Details omitted.)

pivot

Exercise

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Seelecturenotes

Exercise

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Seelecturenotes