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COMP 250

Lecture 13

mergesort, quicksort

Oct. 6, 2017

Time complexity

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𝑂 𝑛• List operations:

findMax, remove

• grade school addition or subtraction

• …..

𝑂 𝑛2

• insertion/selection/bubble sort

• grade school multiplication

• ……

𝑂 𝑙𝑜𝑔2 𝑛• convert to binary

• binary search

• ……

210 ≈ 103

220 ≈ 106

230 ≈ 109

Computers perform ~109 operations per second.

𝑛

210 ≈ 103

220 ≈ 106

230 ≈ 109

𝑛2

106

1012

1018

𝑙𝑜𝑔2 𝑛

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20

30

Computers perform ~109 operations per second.

centuriessecond

minutes…hours

Better sorting algorithms ?

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𝑂 𝑛 < ? < 𝑂( 𝑛2 )

Mergesort

Given a list, partition it into two halves (1st & 2nd).

Sort each half (recursively).

Merge the two halves.

This turns out to be much faster than the other list sorting algorithms we have seen.

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mergesort

mergesort

mergepartition

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

return listelse{

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

}}

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

return listelse{

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

}}

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

return listelse{

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

}}

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mergesort

mergesort

mergepartition

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merge

head1

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merge

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merge

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merge

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1234

merge

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12345

merge

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123456

merge

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1234567merge

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123456781011

merge

head2

….and so on until one list is empty.

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merge

Then, copy the remaining elements.

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merge( list1, list2){initialize list to be emptywhile (list1 is not empty) & (list2 is not empty){

if (list1.first < list2.first)list.addlast( list1.removeFirst(list1) )

elselist.addlast( list2.removeFirst(list2) )

}while list1 is not empty

list.addlast( list1.removeFirst(list1) )while list2 is not empty

list.addlast( list2.removeFirst(list2) )

return list}

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merge( list1, list2){initialize list to be emptywhile (list1 is not empty) & (list2 is not empty){

if (list1.first < list2.first)list.addlast( list1.removeFirst(list1) )

elselist.addlast( list2.removeFirst(list2) )

}while list1 is not empty

list.addlast( list1.removeFirst(list1) )while list2 is not empty

list.addlast( list2.removeFirst(list2) )

return list}

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

return listelse{

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

}}

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list1

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𝑙𝑜𝑔2𝑛

𝑛

Q: How many operations are required to mergesorta list of size 𝑛 ?

A: 𝑂( 𝑛 𝑙𝑜𝑔2 𝑛 )

This will become more clear a few lectures from now when we discuss recurrences.

𝑙𝑜𝑔2𝑛

𝑛

𝑛

210 ≈ 103

220 ≈ 106

230 ≈ 109

𝑛2

106

1012

1018

𝑙𝑜𝑔2 𝑛

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20

30

𝒏 𝒍𝒐𝒈𝟐 𝒏

𝟏𝟎𝟒

~𝟏𝟎𝟕

~𝟏𝟎𝟏𝟎

𝑛 𝑙𝑜𝑔2 𝑛 is much closer to 𝑛than to 𝑛2

𝑛

210 ≈ 103

220 ≈ 106

230 ≈ 109

𝑛2

106

1012

1018

𝑙𝑜𝑔2 𝑛

10

20

30

Computers perform ~109 operations per second.

milliseconds minutes hours centuries

𝒏 𝒍𝒐𝒈𝟐 𝒏

𝟏𝟎𝟒

~𝟏𝟎𝟕

~𝟏𝟎𝟏𝟎

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mergesortquicksort

𝑂 𝑛 < 𝑂( 𝑛 𝑙𝑜𝑔2 𝑛) ≪ 𝑂( 𝑛2 )

bubble sortselection sortinsertion sort

Quicksort

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quicksort(list){if list.length <= 1

return listelse{

pivot = list.removeFirst() // or some other element list1 = list.getElementsLessThan(pivot)list2 = list.getElementsGreaterOrEqual(pivot)list1 = quicksort(list1)list2 = quicksort(list2)

}return concatenate( list1, e, list2 )

}

Quicksort

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quicksort(list){if list.length <= 1

return listelse{

pivot = list.removeFirst() // or some other element list1 = list.getElementsLessThan(pivot)list2 = list.getElementsGreaterOrEqual(pivot)list1 = quicksort(list1)list2 = quicksort(list2)

}return concatenate( list1, e, list2 )

}

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quicksort

quicksort

partition

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concatenate

pivot

Quicksort can be done “in place”

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quicksort( low, high ){if low > high

returnelse{

pivot = ___; // select index in {low, …, high} partitionIndex = makePartition (low, high, pivot)quicksort(low, partitionIndex - 1)quicksort(partionIndex + 1, high)

}}

Quicksort partitioning can be done ‘in place’ using a clever swapping and scanning technique. (See web for details, if interested.)

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pivot

Mergesort vs. Quicksort

• Mergesort typically uses an extra list. More space can hurt performance for big lists.

• We will discuss worst case performance of quicksort later in the course.

• See stackoverflow if you want opinions on which is better.

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