DESIGN AND ANALYSIS OF ALGORITHM (DAA)

PREPERED BY
R.VENGATESH(13BCS2095)
S.SURYA(13BCS2087)
S.VIGNESH(13BCS2096)
V.VIGNESH
BALAKRISHNAN(13BCS2097)
M.VADIVEL(13BCS2089)
GUIDED BY
S.SARAVANAN
(Asst.professor)
Dept. of CSE

 Definition of Sorting
 Brute force
◦ Selection sort
◦ Bubble sort
◦ Radix sort
 Divide and Conquer
◦ Merge sort
◦ Quick sort
 Conclusion
17/07/2014 2

Sorting: an operation that segregates items into
groups according to specified criterion.
A = { 3 1 6 2 1 3 4 5 9 0 }
A = { 0 1 1 2 3 3 4 5 6 9 }

1. We have two group of items:
◦ sorted group, and
◦ unsorted group
1. Initially, all items are in the unsorted group. The
sorted group is empty.
◦ We assume that items in the unsorted group unsorted.
◦ We have to keep items in the sorted group sorted.
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1. Select the “best” (eg. smallest) item from the
unsorted group, then put the “best” item at the
end of the sorted group.
2. Repeat the process until the unsorted group
becomes empty.
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5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

5 1 3 4 6 2
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Comparison
Data Movement
Sorted

Largest

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

5 1 3 4 2 6
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Comparison
Data Movement
Sorted

Largest

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

Largest

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

Largest

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
17/07/2014 36
Comparison
Data Movement
Sorted

2 1 3 4 5 6
17/07/2014 37
Comparison
Data Movement
Sorted

2 1 3 4 5 6
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Comparison
Data Movement
Sorted

2 1 3 4 5 6
17/07/2014 39
Comparison
Data Movement
Sorted

Largest

1 2 3 4 5 6
17/07/2014 40
Comparison
Data Movement
Sorted

1 2 3 4 5 6
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Comparison
Data Movement
Sorted
DONE!

 Running time:
◦ Worst case: O(N2
)
◦ Best case: O(N2
)
17/07/2014 42

Adjacent elements are compared and exchanged.
It is repeatedly done
At the end the largest element is bubbled to the last
position in the list
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17/07/2014 44
9, 6, 2, 12, 11, 9, 3, 7
6, 9, 2, 12, 11, 9, 3, 7
6, 2, 9, 12, 11, 9, 3, 7
6, 2, 9, 12, 11, 9, 3, 7
6, 2, 9, 11, 12, 9, 3, 7
6, 2, 9, 11, 9, 12, 3, 7
6, 2, 9, 11, 9, 3, 12, 7
6, 2, 9, 11, 9, 3, 7, 12The 12 is greater than the 7 so they are exchanged.The 12 is greater than the 7 so they are exchanged.
The 12 is greater than the 3 so they are exchanged.The 12 is greater than the 3 so they are exchanged.
The twelve is greater than the 9 so they are exchangedThe twelve is greater than the 9 so they are exchanged
The 12 is larger than the 11 so they are exchanged.The 12 is larger than the 11 so they are exchanged.
In the third comparison, the 9 is not larger than the 12 so no
exchange is made. We move on to compare the next pair without
any change to the list.
In the third comparison, the 9 is not larger than the 12 so no
exchange is made. We move on to compare the next pair without
any change to the list.
Now the next pair of numbers are compared. Again the 9 is the
larger and so this pair is also exchanged.
Now the next pair of numbers are compared. Again the 9 is the
larger and so this pair is also exchanged.
Bubblesort compares the numbers in pairs from left to right
exchanging when necessary. Here the first number is compared
to the second and as it is larger they are exchanged.
Bubblesort compares the numbers in pairs from left to right
exchanging when necessary. Here the first number is compared
to the second and as it is larger they are exchanged.
The end of the list has been reached so this is the end of the first pass. The
twelve at the end of the list must be largest number in the list and so is now in
the correct position. We now start a new pass from left to right.
The end of the list has been reached so this is the end of the first pass. The
twelve at the end of the list must be largest number in the list and so is now in
the correct position. We now start a new pass from left to right.

17/07/2014 45
6, 2, 9, 11, 9, 3, 7, 122, 6, 9, 11, 9, 3, 7, 122, 6, 9, 9, 11, 3, 7, 122, 6, 9, 9, 3, 11, 7, 122, 6, 9, 9, 3, 7, 11, 12
6, 2, 9, 11, 9, 3, 7, 12
Notice that this time we do not have to compare the last two
numbers as we know the 12 is in position. This pass therefore only
requires 6 comparisons.
Notice that this time we do not have to compare the last two
numbers as we know the 12 is in position. This pass therefore only
First Pass
Second Pass

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2, 6, 9, 9, 3, 7, 11, 122, 6, 9, 3, 9, 7, 11, 122, 6, 9, 3, 7, 9, 11, 12
6, 2, 9, 11, 9, 3, 7, 12
2, 6, 9, 9, 3, 7, 11, 12
Second Pass
First Pass
Third Pass
This time the 11 and 12 are in position. This pass therefore only
This time the 11 and 12 are in position. This pass therefore only

17/07/2014 47
2, 6, 9, 3, 7, 9, 11, 122, 6, 3, 9, 7, 9, 11, 122, 6, 3, 7, 9, 9, 11, 12
6, 2, 9, 11, 9, 3, 7, 12
2, 6, 9, 9, 3, 7, 11, 12
Second Pass
First Pass
Third Pass
Each pass requires fewer comparisons. This time only 4 are needed.Each pass requires fewer comparisons. This time only 4 are needed.
2, 6, 9, 3, 7, 9, 11, 12Fourth Pass

17/07/2014 48
2, 6, 3, 7, 9, 9, 11, 122, 3, 6, 7, 9, 9, 11, 12
6, 2, 9, 11, 9, 3, 7, 12
2, 6, 9, 9, 3, 7, 11, 12
Second Pass
First Pass
Third Pass
The list is now sorted but the algorithm does not know this until it
completes a pass with no exchanges.
The list is now sorted but the algorithm does not know this until it
completes a pass with no exchanges.
2, 6, 9, 3, 7, 9, 11, 12Fourth Pass
2, 6, 3, 7, 9, 9, 11, 12Fifth Pass

17/07/2014 49
2, 3, 6, 7, 9, 9, 11, 12
6, 2, 9, 11, 9, 3, 7, 12
2, 6, 9, 9, 3, 7, 11, 12
Second Pass
First Pass
Third Pass
2, 6, 9, 3, 7, 9, 11, 12Fourth Pass
2, 6, 3, 7, 9, 9, 11, 12Fifth Pass
Sixth Pass
2, 3, 6, 7, 9, 9, 11, 12
This pass no exchanges are made so the algorithm knows the list is
sorted. It can therefore save time by not doing the final pass. With
other lists this check could save much more work.
This pass no exchanges are made so the algorithm knows the list is
sorted. It can therefore save time by not doing the final pass. With
other lists this check could save much more work.

 Running time:
◦ Worst case: O(N2
)
◦ Best case: O(N)
 Variant:
◦ bi-directional bubble sort
 original bubble sort: only works to one direction
 bi-directional bubble sort: works back and forth.
17/07/2014 50

 This sort is unusual because it does not directly
compare any of the elements
 We instead create a set of buckets and repeatedly
separate the elements into the buckets
 On each pass, we look at a different part of the
elements
17/07/2014 51

 Assuming decimal elements and 10 buckets, we
would put the elements into the bucket associated
with its units digit
 The buckets are actually queues so the elements are
added at the end of the bucket
 At the end of the pass, the buckets are combined in
increasing order
17/07/2014 52

 On the second pass, we separate the elements based
on the “tens” digit, and on the third pass we separate
them based on the “hundreds” digit
 Each pass must make sure to process the elements
in order and to put the buckets back together in the
correct order
17/07/2014 53

 Each element is examined once for each of the
digits it contains, so if the elements have at most M
digits and there are N elements this algorithm has
order O(M*N)
 This means that sorting is linear based on the
number of elements
 Why then isn’t this the only sorting algorithm used?
17/07/2014 54

17/07/2014 55
The unit digit is 0
The unit digit is 1
 The unit digit is 2
 The unit digit is 3

17/07/2014 56
The unit digits are already in order
Now start sorting the tens digit

Values in the buckets are now in order
17/07/2014 57
The unit and tens digits are already in order
Now start sorting the hundreds digit

The most well known algorithm design strategy:
1. Divide instance of problem into two or more
smaller instances
2. Solve smaller instances recursively
3. Obtain solution to original (larger) instance by
combining these solutions
17/07/2014 58

17/07/2014 59
subproblem 2
of size n/2
subproblem 1
of size n/2
a solution to
subproblem 1
a solution to
the original problem
a solution to
subproblem 2
a problem of
size n

 Sorting: mergesort and quicksort
 Tree traversals
 Binary search
 Matrix multiplication-Strassen’s algorithm
 Convex hull-QuickHull algorithm
17/07/2014 60

Merge sort
 It stores by dividing array into two
halves
 Sort each them and merge them
into single sorted one
17/07/2014 61
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
3 8 2 9 1 7 4 5
2 3 8 9 1 4 5 7
1 2 3 4 5 7 8 9
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
3 8 2 9 1 7 4 5
2 3 8 9 1 4 5 7
1 2 3 4 5 7 8 9

Merge-Sort Example
17/07/2014 62

674523 14 6 3398 42
674523 14 6 3398 42
2398
2398 2398 2398 2398
Merge
23
23
98
98
45 14
45 1445 14
Merge
14
14
45
45
17/07/2014 63

Merge
98 451423
674523 14 6 3398 42
674523 14 6 3398 42
2398
2398 2398 2398 2398 2398
45 14
45 1445 141445
14
14
4523
23
98 45
45
98
98
676 33 42
676 676
Merge
6
6
67
67
33 4233 42
Merge
33
33
42
42
17/07/2014 64

98 451423
674523 14 6 3398 42
674523 14 6 3398 42
2398
2398 2398 2398 2398 2398
45 14
45 1445 141445
14
14
4523
23
98 45
45
98
98
676 33 42
676 6766
6
67
67
33 4233 4233
33
42
42
Merge
676 42336
6
67 33
33
4242
42
67
67
17/07/2014 65

98 451423
674523 14 6 3398 42
674523 14 6 3398 42
2398
2398 2398 2398 2398 2398
45 14
45 1445 141445
14
14
4523
23
98 45
45
98
98
676 33 42
676 6766
6
67
67
33 4233 4233
33
42
42676 42336
6
67 33
33
4242
42
67
67
Merge
23 45 33 42 6714 6
Merge
23 6 42 67
6
14 3314 45 6 42 67
6 14
23 3323
23
45
33
42
42
67
45
67
67 98
17/07/2014 66

674523 14 6 3398 42
6 14 23 33 42 45 67 98
17/07/2014 67

 Quicksort is more widely used than any other sort.
 Quicksort is well-studied, not difficult to implement,
works well on a variety of data, and consumes fewer
resources that other sorts in nearly all situations.
 Quicksort is O(n*log n) time, and O(log n) additional
space due to recursion.
17/07/2014 68

Given a pivot, partition the elements of the array
such that the resulting array consists of:
1. One sub-array that contains elements >= pivot
2. Another sub-array that contains elements < pivot
The sub-arrays are stored in the original data array.
Partitioning loops through, swapping elements
below/above pivot.
17/07/2014 69

We are given array of n integers to sort:
17/07/2014 70
40 20 10 80 60 50 7 30 100

There are a number of ways to pick the pivot
element. In this example, we will use the first
element in the array:
17/07/2014 71
40 20 10 80 60 50 7 30 100

Given a pivot, partition the elements of the array
such that the resulting array consists of:
1. One sub-array that contains elements >= pivot
2. Another sub-array that contains elements < pivot
The sub-arrays are stored in the original data array.
Partitioning loops through, swapping elements
below/above pivot.
17/07/2014 72

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
17/07/2014 73

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
1. While data[too_big_index] <= data[pivot]
++too_big_index
17/07/2014 74

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
17/07/2014 75

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
17/07/2014 76

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
17/07/2014 77

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 78

40 20 10 80 60 50 7 30 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
17/07/2014 79

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 80

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
4. While too_small_index > too_big_index, go to 1.
17/07/2014 81

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 82

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 83

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 84

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 85

40 20 10 30 60 50 7 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
++too_big_index
--too_small_index
17/07/2014 86

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 87

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 88

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 89

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 90

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 91

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 92

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 93

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 94

++too_big_index
--too_small_index
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 95

++too_big_index
--too_small_index
5. Swap data[too_small_index] and data[pivot_index]
40 20 10 30 7 50 60 80 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 96

++too_big_index
--too_small_index
5. Swap data[too_small_index] and data[pivot_index]
7 20 10 30 40 50 60 80 100pivot_index = 4
[0] [1] [2] [3] [4] [5] [6] [7] [8]
17/07/2014 97

17/07/2014 98
7 20 10 30 40 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
<= data[pivot] > data[pivot]

17/07/2014 99
7 20 10 30 40 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
<= data[pivot] > data[pivot]

 Quicksort illustrates the operation of the basic
algorithm. When the array is partitioned, one element is
in place on the diagonal, the left subarray has its upper
corner at that element, and the right subarray has its
lower corner at that element. The original file is
divided into two smaller parts that are sorted
independently. The left subarray is always sorted first,
so the sorted result emerges as a line of black dots
moving right and up the diagonal.
17/07/2014 100

 In this ,we can understand that every sorting will
depends upon one another.
 So, we should understand sorting is the best
starting for every representation.
 From this, Quick sort will easily perform sorting.
17/07/2014 101

DESIGN AND ANALYSIS OF ALGORITHM (DAA)

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