Lecture -16-merge sort (slides).pptx

Merge sort Analysis and complexity
Design and Analysis of Algorithms
1

Divide and Conquer Approach
Step 1:
• If the problem size is small, solve this problem
directly
• Otherwise, split the original problem into 2 or
more sub-problems with almost equal sizes.
Step 2:
• Recursively solve these sub-problems by applying
this algorithm.
Step 3:
• Merge the solutions of the sub- problems into a
solution of the original problem.
2

• Time complexity:
– where S(n) is time for splitting
– M(n) is time for merging
– b and c are constants
Example
• Binary search
• Quick sort
• Merge sort
T(n)=



2T(n/2) + S(n) + M(n)
b
, n  c
, n < c
Advanced Algorithms Analysis and Design
Time Complexity of General Algorithms

Merge-Sort
• This is the first divide-and-conquer algorithm:
we solve the problem by dividing the problem
into smaller sub-problems, we recursively call
the algorithm on the sub-problems, and we
then recombine the solutions to the sub-
problems to create a solution to the overall
problem.
4

Merge-Sort
• We will recursively define merge sort as follows:
1. If the list is of size 1,
2. Otherwise, a. Divide the unsorted list into two
sub-lists,
b. Recursively call merge sort on each sub-list, and
c. Merge the two sorted sub-lists together into a
single sorted list.
5

Merge-sort is based on divide-and-conquer approach
and can be described by the following three steps:
Divide Step:
• If given array A has zero or one element, return S.
• Otherwise, divide A into two arrays, A1 and A2,
• Each containing about half of the elements of A.
Recursion Step:
• Recursively sort array A1, A2
Conquer Step:
• Combine the elements back in A by merging the sorted
arrays A1 and A2 into a sorted sequence.
Merge-sort

Visualization of Merge-sort as Binary Tree
• We can visualize Merge-sort by means of binary tree
where each node of the tree represents a recursive call
• Each external node represents individual elements of
given array A.
• Such a tree is called Merge-sort tree.
• The heart of the Merge-sort algorithm is conquer step,
which merge two sorted sequences into a single sorted
sequence
• The merge algorithm is explained in the next

• Sort the array [14, 10, 4, 8, 2, 12, 6, 0] in the ascending
order
Solution:
• Divide 14, 10, 4, 8, 2, 12, 6,0
14, 10, 4, 8 2, 12, 6,0
14, 10 4, 8 2, 12 6, 0
14 10 4 8 2 12 6 0
Merge-sort
Merge-sort Merge-sort
Sorting Example: Divide and Conquer Rule

• Recursion and Conquer
0, 2, 4, 6, 8, 10, 12, 14
4, 8, 10, 14 0, 2, 6, 12
10, 14 4, 8 2, 12 0, 6
14 10 4 8 2 12 6 0
Merge
Merge Merge
Merge Merge
Merge Merge
Contd..

Merge-sort(A, f, l)
1. if f < l
2. then m = (f + l)/2
3. Merge-sort(A, f, m)
4. Merge-sort(A, m + 1, l)
5. Merge(A, f, m, l)
Merge-sort Algorithm

Merge(A, f, m, l)
1. T[f..l] declare temporary array of same size
2. i  f; k  f; j  m + 1 initialize integers i, j, and k
3. while (i  m) and (j  l)
4. do if (A[i]  A[j]) comparison of elements
5. then T[k++]  A[i++]
6. else T[k++]  A[j++]
7. while (i  m)
8. do T[k++]  A[i++] copy from A to T
9. while (j  l)
10. do T[k++]  A[j++] copy from A to T
11. for i  p to r
12. do A[i]  T[i] copy from T to A
Merge-sort Algorithm

• Let T(n) be the time taken by this algorithm to sort an
array of n elements dividing A into sub-arrays A1 and A2.
• It is easy to see that the Merge (A1, A2, A) takes the
linear time. Consequently,
T(n) = T(n/2) + T(n/2) + θ(n)
T(n) = 2T (n/2) + θ(n)
• The above recurrence relation is non-homogenous and
can be solved by any of the methods
– Defining characteristics polynomial
– Substitution
– recursion tree or
– master method
Analysis of Merge-sort Algorithm

n
n
T
n
T 
 )
2
(
.
2
)
(
2
)
2
(
.
2
)
2
( 2
n
n
T
n
T 

2
3
2
2
)
2
(
.
2
)
2
(
n
n
T
n
T 

.
.
.
2
)
2
(
.
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)
2
( 3
4
3
n
n
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n
T 

1
1
2
)
2
(
.
2
)
2
( 


 k
k
k
n
n
T
n
T
Analysis: Substitution Method

n
n
n
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n
n
T
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T 



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 )
2
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.
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)
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( 2
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n
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.


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
times
k
k
k
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

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

 n
.
.
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)
2
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(

n
k
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k
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)
2
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( 

)
log
.
(
)
( 2 n
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T 


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


times
k
k
k
n
n
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T
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


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 n
.
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)
2
(
.
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(
k
n
n k


 2
log
2
:
that
suppose
us
Let
n
n
n
n
n
T
n
n
T 2
2 log
.
log
.
)
1
(
.
)
(
Hence, 




Lecture -16-merge sort (slides).pptx

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