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8/8/2019 09. Merge Sort
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DATASTRUCTURE
S
MAHESH GOYANI
MAHATMA GANDHI INSTITUE OF TECHNICAL EDUCATION & RESEARCH CENTER
mgoyani@rediffmail.com
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IVIDE & CONQUER
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SortSort
SortSort SortSort
SortSort SortSort SortSort SortSort
IVIDE & CONQUER
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IVIDE & CONQUER
CombineCombine
CombineCombine CombineCombine
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MergeSort Cost (real order)MergeSort Cost (real order)
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Minimal effort splitting
Lots of effort recombining
Lots of effort splitting
Minimal effort recombining
Four Recursive Sorts
Merge Sort Insertion Sort
Quick Sort Selection Sort
Size of Sublists
n/2,n/2 n-1,1
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In plain English: if the size of the array > 1, split thearray into two halves, and recursively sort both halves;
when the sorts return, merge the two halves
In pseudocode:
if size equals 1
return
copy first half into leftArray
copy second half into rightArray
sort (leftArray)
sort (rightArray)
merge leftArray with rightArray
TERMINOLOGY
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MERGE SORT
Mergesort is a divide and conquer algorithm that does exactlythat.
It splits the list in half
It sorts the two halves
Then merges the two sorted halves together Mergesort can be implemented recursively
The merge sort algorithm involves three steps:
If the number of items to sort is 0 or 1, return
Recursively sort the first and second halves separately
Merge the two sorted halves into a sorted group
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Merge SortMerge Sort
8814
982562
52
79
3023
31
Divide and Conquer
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Merge SortMerge Sort
8814
982562
52
79
3023
31Split Set into Two
(no real work)
25,31,52,88,98
sort the first half.
14,23,30,62,79
sort the second half.
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Merge SortMerge Sort
Merge two sorted lists into one
25,31,52,88,98
14,23,30,62,79
14,23,25,30,31,52,62,79,88,98
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MERGE SORT
3 5 15 28 30 6 10 14 22 43 50a:a: b: b:
aSize: 5aSize: 5 bSize: 6 bSize: 6
tmp:tmp:
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MERGE SORT
5 15 28 30 10 14 22 43 50a:a: b: b:
tmp:tmp:
i=0i=0
k=0k=0
j=0j=0
3 6
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MERGE SORT
5 15 28 30 10 14 22 43 50a:a: b: b:
tmp:tmp:
i=0i=0
k=0k=0
3
j=0j=0
3 6
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MERGE SORT
3 15 28 30 10 14 22 43 50a:a: b: b:
tmp:tmp:
i=1i=1 j=0j=0
k=1k=1
3 5
5 6
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MERGE SORT
3 5 28 30 10 14 22 43 50a:a: b: b:
3 5tmp:tmp:
i=2i=2 j=0j=0
k=2k=2
6
15 6
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MERGE SORT
3 5 28 30 6 14 22 43 50a:a: b: b:
3 5 6tmp:tmp:
i=2i=2 j=1j=1
k=3k=3
15 10
10
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MERGE SORT
10
3 5 28 30 6 22 43 50a:a: b: b:
3 5 6tmp:tmp:
i=2i=2 j=2j=2
k=4k=4
15 10 14
14
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MERGE SORT
1410
3 5 28 30 6 14 43 50a:a: b: b:
3 5 6tmp:tmp:
i=2i=2 j=3j=3
k=5k=5
15 10 22
15
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MERGE SORT
1410
3 5 30 6 14 43 50a:a: b: b:
3 5 6tmp:tmp:
i=3i=3 j=3j=3
k=6k=6
15 10 22
2215
28
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MERGE SORT
1410
3 5 30 6 14 50a:a: b: b:
3 5 6tmp:tmp:
i=3i=3 j=4j=4
k=7k=7
15 10 22
2815
28 43
22
G SO
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MERGE SORT
1410
3 5 6 14 50a:a: b: b:
3 5 6tmp:tmp:
i=4i=4 j=4j=4
k=8k=8
15 10 22
3015
28 43
22
30
28
MERGE SORT
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MERGE SORT
1410
3 5 6 14 50a:a: b: b:
3 5 6 30tmp:tmp:
i=5i=5 j=4j=4
k=9k=9
15 10 22
15
28 43
22
30
28 43 50
Done.
MERGE SORT
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MERGE SORT
Sort Sort
Merge
COMPLEXITY
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At each level of recursion, all of the elements are copied; thus thedecomposition phase takes n log n time
At each level of recursion, all of the elements are copied in themerge operation; thus the merge phase takes n log n time
T = Td + Tm = n log n + n log n = 2 (n log n )
Merge sort is O(n log n
)
COMPLEXITY
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Merging two sorted arraysMerging two sorted arrays
20
13
7
2
12
11
9
1
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Merging two sorted arraysMerging two sorted arrays
20
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9
1
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Merging two sorted arraysMerging two sorted arrays
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Merging two sorted arraysMerging two sorted arrays
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Merging two sorted arraysMerging two sorted arrays
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Merging two sorted arraysMerging two sorted arrays
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Merging two sorted arraysMerging two sorted arrays
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Merging two sorted arraysMerging two sorted arrays
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Time = Θ (n) to merge a
total of n elements (linear
time).
Recursion tree
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Recursion tree
Solve T (n) = 2T (n/2) + cn, where c > 0 is constant.
T (n/2) T (n/2)
cn
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cn
T (n/4) T (n/4) T (n/4) T (n/4)
cn/2 cn/2
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
h = lg n
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
h = lg n
cn
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
h = lg n
cn
cn
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
h = lg n
cn
cn
cn
…
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cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ (1) …
h = lg n
cn
cn
cn
#leaves = n Θ (n) …