Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two...

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Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy Divide the list into two smaller lists of about equal sizes Sort each smaller list recursively Merge the two sorted lists to get one sorted list How do we divide the list? How much time needed? How do we merge the two sorted lists? How much time needed?

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Transcript of Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two...

Page 1: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 1

Mergesort

Based on divide-and-conquer strategy

Divide the list into two smaller lists of about equal sizes

Sort each smaller list recursively Merge the two sorted lists to get one sorted list

How do we divide the list? How much time needed?

How do we merge the two sorted lists? How much time needed?

Page 2: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 2

Dividing

If the input list is a linked list, dividing takes (N) time We scan the linked list, stop at the N/2 th entry

and cut the link

If the input list is an array A[0..N-1]: dividing takes O(1) time we can represent a sublist by two integers left

and right: to divide A[left..right], we compute center=(left+right)/2 and obtain A[left..center] and A[center+1..right]

Try left=0, right = 50, center=?

Page 3: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 3

Mergesort

Divide-and-conquer strategy recursively mergesort the first half and the second

half merge the two sorted halves together

Page 4: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 4

http://www.cosc.canterbury.ac.nz/people/mukundan/dsal/MSort.html

Page 5: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 5

How to merge? Input: two sorted array A and B Output: an output sorted array C Three counters: Actr, Bctr, and Cctr

initially set to the beginning of their respective arrays

(1)   The smaller of A[Actr] and B[Bctr] is copied to the next entry in C, and the appropriate counters are advanced

(2)   When either input list is exhausted, the remainder of the other list is copied to C

Page 6: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 6

Example: Merge

Page 7: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 7

Example: Merge...

Running time analysis: Clearly, merge takes O(m1 + m2) where m1 and m2 are

the sizes of the two sublists.

Space requirement:merging two sorted lists requires linear extra memoryadditional work to copy to the temporary array and back

Page 8: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 8

Page 9: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 9

Analysis of mergesort Let T(N) denote the worst-case running time of

mergesort to sort N numbers.

Assume that N is a power of 2.

Divide step: O(1) time Conquer step: 2 T(N/2) time Combine step: O(N) time Recurrence equation:

T(1) = 1 T(N) = 2T(N/2) + N

Page 10: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 10

Analysis: solving recurrence

kNN

T

NN

T

NNN

T

NN

T

NNN

T

NN

TNT

kk

)2

(2

3)8

(8

2)4

)8

(2(4

2)4

(4

)2

)4

(2(2

)2

(2)(

Since N=2k, we have k=log2 n

)log(

log

)2

(2)(

NNO

NNN

kNN

TNTk

k

Page 11: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 11

Page 12: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 12

An experiment

Code from textbook Unix time utility

Page 13: Sorting I / Slide 1 Mergesort Based on divide-and-conquer strategy * Divide the list into two smaller lists of about equal sizes * Sort each smaller list.

Sorting I / Slide 13

Question:


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