mergesort, quicksort Oct. 6, 2017 1 Time complexity 2 2 - - PowerPoint PPT Presentation

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mergesort, quicksort Oct. 6, 2017 1 Time complexity 2 2 - - PowerPoint PPT Presentation

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COMP 250 Lecture 13 mergesort, quicksort Oct. 6, 2017 1 Time complexity 2 2 convert to binary List operations: insertion/selection/ findMax, remove bubble sort grade school

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SLIDE 1

1

COMP 250

Lecture 13

mergesort, quicksort

  • Oct. 6, 2017
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SLIDE 2

Time complexity

2

𝑃 π‘œ

  • List operations:

findMax, remove

  • grade school addition
  • r subtraction
  • …..

𝑃 π‘œ2

  • insertion/selection/

bubble sort

  • grade school

multiplication

  • ……

𝑃 π‘šπ‘π‘•2 π‘œ

  • convert to binary
  • binary search
  • ……
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SLIDE 3

210 β‰ˆ 103 220 β‰ˆ 106 230 β‰ˆ 109

Computers perform ~109 operations per second.

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π‘œ

210 β‰ˆ 103 220 β‰ˆ 106 230 β‰ˆ 109

π‘œ2

106 1012 1018 π‘šπ‘π‘•2 π‘œ 10 20 30

Computers perform ~109 operations per second.

centuries second minutes…hours

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SLIDE 5

Better sorting algorithms ?

5

𝑃 π‘œ < ? < 𝑃( π‘œ2 )

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SLIDE 6

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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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7 8 10 11 13 16

mergesort mergesort merge partition

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mergesort(list){ if list.length == 1 return list else{ 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 list else{ 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 list else{ 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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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7 8 10 11 13 16

mergesort mergesort merge partition

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1 3 6 8 10 11 2 4 5 7 13 16

merge

head1 head2

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13

1 3 6 8 10 11 2 4 5 7 13 16 1

merge

head1 head2

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1 3 6 8 10 11 2 4 5 7 13 16 1 2

merge

head1 head2

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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3

merge

head1 head2

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SLIDE 16

16

1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4

merge

head1 head2

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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5

merge

head1 head2

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SLIDE 18

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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6

merge

head1 head2

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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7

merge

head1 head2

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SLIDE 20

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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7 8 10 11

merge

head2

….and so on until

  • ne list is empty.
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1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7 8 10 11 13 16

merge

Then, copy the remaining elements.

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merge( list1, list2){ initialize list to be empty while (list1 is not empty) & (list2 is not empty){ if (list1.first < list2.first) list.addlast( list1.removeFirst(list1) ) else list.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 empty while (list1 is not empty) & (list2 is not empty){ if (list1.first < list2.first) list.addlast( list1.removeFirst(list1) ) else list.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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8 10 3 11 6 1 7 16 2 5 4 13

mergesort(list){ if list.length == 1 return list else{ 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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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 8 10 3 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 8 10 3 8 10 8 10 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 8 10 3 8 10 8 10 3 8 10 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 8 10 3 11 6 1 8 11 10 6 6 11 8 10 1 6 11 3 8 10 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 8 10 3 11 6 1 8 11 10 6 6 11 8 10 1 6 11 3 8 10 1 3 6 8 10 11 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 5 4 7 16 2 8 10 3 11 6 1 7 16 2 5 4 13 13 5 8 11 7 10 6 16 4 4 5 7 16 6 11 8 10 1 6 11 3 8 10 2 7 16 4 5 13 1 3 6 8 10 11 2 4 5 7 13 16 list1 list2

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8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 7 16 2 5 4 13 8 10 3 11 6 1 5 4 7 16 2 8 10 3 11 6 1 7 16 2 5 4 13 13 5 8 11 7 10 6 16 4 4 5 7 16 6 11 8 10 1 6 11 3 8 10 2 7 16 4 5 13 1 3 6 8 10 11 2 4 5 7 13 16 1 2 3 4 5 6 7 8 10 11 13 16

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π‘šπ‘π‘•2π‘œ

π‘œ

Q: How many

  • perations are

required to mergesort a list of size π‘œ ? A: 𝑃( π‘œ π‘šπ‘π‘•2 π‘œ ) This will become more clear a few lectures from now when we discuss recurrences. π‘šπ‘π‘•2π‘œ

π‘œ

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π‘œ

210 β‰ˆ 103 220 β‰ˆ 106 230 β‰ˆ 109

π‘œ2

106 1012 1018 π‘šπ‘π‘•2 π‘œ 10 20 30 𝒐 π’Žπ’‘π’‰πŸ‘ 𝒐 πŸπŸπŸ“ ~πŸπŸπŸ– ~𝟐𝟏𝟐𝟏

π‘œ π‘šπ‘π‘•2 π‘œ is much closer to π‘œ than to π‘œ2

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SLIDE 36

π‘œ

210 β‰ˆ 103 220 β‰ˆ 106 230 β‰ˆ 109

π‘œ2

106 1012 1018 π‘šπ‘π‘•2 π‘œ 10 20 30

Computers perform ~109 operations per second.

milliseconds minutes hours centuries

𝒐 π’Žπ’‘π’‰πŸ‘ 𝒐 πŸπŸπŸ“ ~πŸπŸπŸ– ~𝟐𝟏𝟐𝟏

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37

mergesort quicksort

𝑃 π‘œ < 𝑃( π‘œ π‘šπ‘π‘•2 π‘œ) β‰ͺ 𝑃( π‘œ2 )

bubble sort selection sort insertion sort

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SLIDE 38

Quicksort

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

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quicksort(list){ if list.length <= 1 return list else{ 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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8 10 3 11 6 1 7 16 2 5 4 13 3 6 1 7 2 5 4 10 11 16 13 1 2 3 4 5 6 7 8 10 11 13 16

quicksort quicksort partition

1 2 3 4 5 6 7 10 11 13 16

concatenate pivot

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Quicksort can be done β€œin place”

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

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

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8 10 3 11 6 1 7 16 2 5 4 13 3 6 1 7 2 5 4 8 10 11 16 13

quicksort quicksort partition

1 2 3 4 5 6 7 8 10 11 13 16

pivot

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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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