Bubble Sort Merge Sort

1
Bubble SortMerge Sort
Lecture 16
2
Bubble Sort
3
Sorting

• Sorting takes an unordered collection and makes
  it an ordered one.

1 2 3 4 5
6
101
12
5
35
42
77
1 2 3 4 5
6
4
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
12
101
5
35
42
77
5
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
Swap
12
101
5
35
42
77
6
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
Swap
12
101
5
35
77
42
7
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
Swap
12
101
5
77
35
42
8
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
77
101
5
12
35
42
No need to swap
9
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
Swap
77
101
5
12
35
42
10
"Bubbling Up" the Largest Element

• Traverse a collection of elements
• Move from the front to the end
• Bubble the largest value to the end using
  pair-wise comparisons and swapping

1 2 3 4 5
6
101
77
5
12
35
42
Largest value correctly placed
11
The Bubble Up Algorithm

• index lt- 1
• last_compare_at lt- n 1
• loop
• exitif(index gt last_compare_at)
• if(Aindex gt Aindex 1) then
• Swap(Aindex, Aindex 1)
• endif
• index lt- index 1
• endloop

12
No, Swap isnt built in.
LB

• Procedure Swap(a, b isoftype in/out Num)
• t isoftype Num
• t lt- a
• a lt- b
• b lt- t
• endprocedure // Swap

13
Items of Interest

• Notice that only the largest value is correctly
  placed
• All other values are still out of order
• So we need to repeat this process

1 2 3 4 5
6
101
77
5
12
35
42
Largest value correctly placed
14
Repeat Bubble Up How Many Times?

• If we have N elements
• And if each time we bubble an element, we place
  it in its correct location
• Then we repeat the bubble up process N 1
  times.
• This guarantees well correctly place all N
  elements.

15
Bubbling All the Elements
16
Reducing the Number of Comparisons
17
Reducing the Number of Comparisons

• On the Nth bubble up, we only need to do MAX-N
  comparisons.
• For example
• This is the 4th bubble up
• MAX is 6
• Thus we have 2 comparisons to do

18
Putting It All Together
19

• N is // Size of Array
• Arr_Type definesa Array1..N of Num
• Procedure Swap(n1, n2 isoftype in/out Num)
• temp isoftype Num
• temp lt- n1
• n1 lt- n2
• n2 lt- temp
• endprocedure // Swap

20

• procedure Bubblesort(A isoftype in/out Arr_Type)
• to_do, index isoftype Num
• to_do lt- N 1
• loop
• exitif(to_do 0)
• index lt- 1
• loop
• exitif(index gt to_do)
• if(Aindex gt Aindex 1) then
• Swap(Aindex, Aindex 1)
• endif
• index lt- index 1
• endloop
• to_do lt- to_do - 1
• endloop
• endprocedure // Bubblesort

Outer loop
Inner loop
21
Already Sorted Collections?

• What if the collection was already sorted?
• What if only a few elements were out of place and
  after a couple of bubble ups, the collection
  was sorted?
• We want to be able to detect this and stop
  early!

22
Using a Boolean Flag

• We can use a boolean variable to determine if any
  swapping occurred during the bubble up.
• If no swapping occurred, then we know that the
  collection is already sorted!
• This boolean flag needs to be reset after each
  bubble up.

23

• did_swap isoftype Boolean
• did_swap lt- true
• loop
• exitif ((to_do 0) OR NOT(did_swap))
• index lt- 1
• did_swap lt- false
• loop
• exitif(index gt to_do)
• if(Aindex gt Aindex 1) then
• Swap(Aindex, Aindex 1)
• did_swap lt- true
• endif
• index lt- index 1
• endloop
• to_do lt- to_do - 1
• endloop

24
An Animated Example
N
8
true
did_swap
to_do
7
index

67
45
23
14
6
33
98
42
1 2 3 4 5 6 7 8
25
An Animated Example
N
8
false
did_swap
to_do
7
index
1
67
45
23
14
6
33
98
42
1 2 3 4 5 6 7 8
26
An Animated Example
N
8
false
did_swap
to_do
7
index
1
Swap
67
45
23
14
6
33
98
42
1 2 3 4 5 6 7 8
27
An Animated Example
N
8
true
did_swap
to_do
7
index
1
Swap
67
45
98
14
6
33
23
42
1 2 3 4 5 6 7 8
28
An Animated Example
N
8
true
did_swap
to_do
7
index
2
67
45
98
14
6
33
23
42
1 2 3 4 5 6 7 8
29
An Animated Example
N
8
true
did_swap
to_do
7
index
2
Swap
67
45
98
14
6
33
23
42
1 2 3 4 5 6 7 8
30
An Animated Example
N
8
true
did_swap
to_do
7
index
2
Swap
67
98
45
14
6
33
23
42
1 2 3 4 5 6 7 8
31
An Animated Example
N
8
true
did_swap
to_do
7
index
3
67
98
45
14
6
33
23
42
1 2 3 4 5 6 7 8
32
An Animated Example
N
8
true
did_swap
to_do
7
index
3
Swap
67
98
45
14
6
33
23
42
1 2 3 4 5 6 7 8
33
An Animated Example
N
8
true
did_swap
to_do
7
index
3
Swap
67
14
45
98
6
33
23
42
1 2 3 4 5 6 7 8
34
An Animated Example
N
8
true
did_swap
to_do
7
index
4
67
14
45
98
6
33
23
42
1 2 3 4 5 6 7 8
35
An Animated Example
N
8
true
did_swap
to_do
7
index
4
Swap
67
14
45
98
6
33
23
42
1 2 3 4 5 6 7 8
36
An Animated Example
N
8
true
did_swap
to_do
7
index
4
Swap
67
14
45
6
98
33
23
42
1 2 3 4 5 6 7 8
37
An Animated Example
N
8
true
did_swap
to_do
7
index
5
67
14
45
6
98
33
23
42
1 2 3 4 5 6 7 8
38
An Animated Example
N
8
true
did_swap
to_do
7
index
5
Swap
67
14
45
6
98
33
23
42
1 2 3 4 5 6 7 8
39
An Animated Example
N
8
true
did_swap
to_do
7
index
5
Swap
98
14
45
6
67
33
23
42
1 2 3 4 5 6 7 8
40
An Animated Example
N
8
true
did_swap
to_do
7
index
6
98
14
45
6
67
33
23
42
1 2 3 4 5 6 7 8
41
An Animated Example
N
8
true
did_swap
to_do
7
index
6
Swap
98
14
45
6
67
33
23
42
1 2 3 4 5 6 7 8
42
An Animated Example
N
8
true
did_swap
to_do
7
index
6
Swap
33
14
45
6
67
98
23
42
1 2 3 4 5 6 7 8
43
An Animated Example
N
8
true
did_swap
to_do
7
index
7
33
14
45
6
67
98
23
42
1 2 3 4 5 6 7 8
44
An Animated Example
N
8
true
did_swap
to_do
7
index
7
Swap
33
14
45
6
67
98
23
42
1 2 3 4 5 6 7 8
45
An Animated Example
N
8
true
did_swap
to_do
7
index
7
Swap
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
46
After First Pass of Outer Loop
N
8
true
did_swap
to_do
7
Finished first Bubble Up
index
8
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
47
The Second Bubble Up
N
8
false
did_swap
to_do
6
index
1
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
48
The Second Bubble Up
N
8
false
did_swap
to_do
6
index
1
No Swap
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
49
The Second Bubble Up
N
8
false
did_swap
to_do
6
index
2
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
50
The Second Bubble Up
N
8
false
did_swap
to_do
6
index
2
Swap
33
14
45
6
67
42
23
98
1 2 3 4 5 6 7 8
51
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
2
Swap
33
45
14
6
67
42
23
98
1 2 3 4 5 6 7 8
52
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
3
33
45
14
6
67
42
23
98
1 2 3 4 5 6 7 8
53
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
3
Swap
33
45
14
6
67
42
23
98
1 2 3 4 5 6 7 8
54
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
3
Swap
33
6
14
45
67
42
23
98
1 2 3 4 5 6 7 8
55
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
4
33
6
14
45
67
42
23
98
1 2 3 4 5 6 7 8
56
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
4
No Swap
33
6
14
45
67
42
23
98
1 2 3 4 5 6 7 8
57
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
5
33
6
14
45
67
42
23
98
1 2 3 4 5 6 7 8
58
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
5
Swap
33
6
14
45
67
42
23
98
1 2 3 4 5 6 7 8
59
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
5
Swap
67
6
14
45
33
42
23
98
1 2 3 4 5 6 7 8
60
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
6
67
6
14
45
33
42
23
98
1 2 3 4 5 6 7 8
61
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
6
Swap
67
6
14
45
33
42
23
98
1 2 3 4 5 6 7 8
62
The Second Bubble Up
N
8
true
did_swap
to_do
6
index
6
Swap
42
6
14
45
33
67
23
98
1 2 3 4 5 6 7 8
63
After Second Pass of Outer Loop
N
8
true
did_swap
to_do
6
Finished second Bubble Up
index
7
42
6
14
45
33
67
23
98
1 2 3 4 5 6 7 8
64
The Third Bubble Up
N
8
false
did_swap
to_do
5
index
1
42
6
14
45
33
67
23
98
1 2 3 4 5 6 7 8
65
The Third Bubble Up
N
8
false
did_swap
to_do
5
index
1
Swap
42
6
14
45
33
67
23
98
1 2 3 4 5 6 7 8
66
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
1
Swap
42
6
23
45
33
67
14
98
1 2 3 4 5 6 7 8
67
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
2
42
6
23
45
33
67
14
98
1 2 3 4 5 6 7 8
68
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
2
Swap
42
6
23
45
33
67
14
98
1 2 3 4 5 6 7 8
69
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
2
Swap
42
23
6
45
33
67
14
98
1 2 3 4 5 6 7 8
70
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
3
42
23
6
45
33
67
14
98
1 2 3 4 5 6 7 8
71
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
3
No Swap
42
23
6
45
33
67
14
98
1 2 3 4 5 6 7 8
72
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
4
42
23
6
45
33
67
14
98
1 2 3 4 5 6 7 8
73
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
4
Swap
42
23
6
45
33
67
14
98
1 2 3 4 5 6 7 8
74
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
4
Swap
42
23
6
33
45
67
14
98
1 2 3 4 5 6 7 8
75
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
5
42
23
6
33
45
67
14
98
1 2 3 4 5 6 7 8
76
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
5
Swap
42
23
6
33
45
67
14
98
1 2 3 4 5 6 7 8
77
The Third Bubble Up
N
8
true
did_swap
to_do
5
index
5
Swap
45
23
6
33
42
67
14
98
1 2 3 4 5 6 7 8
78
After Third Pass of Outer Loop
N
8
true
did_swap
to_do
5
Finished third Bubble Up
index
6
45
23
6
33
42
67
14
98
1 2 3 4 5 6 7 8
79
The Fourth Bubble Up
N
8
false
did_swap
to_do
4
index
1
45
23
6
33
42
67
14
98
1 2 3 4 5 6 7 8
80
The Fourth Bubble Up
N
8
false
did_swap
to_do
4
index
1
Swap
45
23
6
33
42
67
14
98
1 2 3 4 5 6 7 8
81
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
1
Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
82
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
2
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
83
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
2
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
84
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
3
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
85
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
3
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
86
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
4
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
87
The Fourth Bubble Up
N
8
true
did_swap
to_do
4
index
4
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
88
After Fourth Pass of Outer Loop
N
8
true
did_swap
to_do
4
Finished fourth Bubble Up
index
5
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
89
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
1
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
90
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
1
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
91
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
2
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
92
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
2
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
93
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
3
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
94
The Fifth Bubble Up
N
8
false
did_swap
to_do
3
index
3
No Swap
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
95
After Fifth Pass of Outer Loop
N
8
false
did_swap
to_do
3
Finished fifth Bubble Up
index
4
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
96
Finished Early
N
8
false
did_swap
to_do
3
We didnt do any swapping,so all of the other
elementsmust be correctly placed. We can skip
the last twopasses of the outer loop.
index
4
45
23
14
33
42
67
6
98
1 2 3 4 5 6 7 8
97
Summary

• Bubble Up algorithm will move largest value to
  its correct location (to the right)
• Repeat Bubble Up until all elements are
  correctly placed
• Maximum of N-1 times
• Can finish early if no swapping occurs
• We reduce the number of elements we compare each
  time one is correctly placed

98
Truth in CS Act
LB

• NOBODY EVER USES BUBBLE SORT
• NOBODY
• NOT EVER
• BECAUSE IT IS EXTREMELY INEFFICIENT

99
Questions?
100
Mergesort
101
Sorting

• Sorting takes an unordered collection and makes
  it an ordered one.

1 2 3 4 5
6
101
12
5
35
42
77
1 2 3 4 5
6
102
Divide and Conquer

• Divide and Conquer cuts the problem in half each
  time, but uses the result of both halves
• cut the problem in half until the problem is
  trivial
• solve for both halves
• combine the solutions

103
Mergesort

• A divide-and-conquer algorithm
• Divide the unsorted array into 2 halves until the
  sub-arrays only contain one element
• Merge the sub-problem solutions together
• Compare the sub-arrays first elements
• Remove the smallest element and put it into the
  result array
• Continue the process until all elements have been
  put into the result array

104
Algorithm

• Mergesort(Passed an array)
• if array size gt 1
• Divide array in half
• Call Mergesort on first half.
• Call Mergesort on second half.
• Merge two halves.
• Merge(Passed two arrays)
• Compare leading element in each array
• Select lower and place in new array.
• (If one input array is empty then place
• remainder of other array in output array)

105
More TRUTH in CS
LB

• We dont really pass in two arrays!
• We pass in one array with indicator variables
  which tell us where one set of data starts and
  finishes and where the other set of data starts
  and finishes.
• Honest.

106
Algorithm
LB

• Mergesort(Passed an array)
• if array size gt 1
• Divide array in half
• Call Mergesort on first half.
• Call Mergesort on second half.
• Merge two halves.
• Merge(Passed two arrays)
• Compare leading element in each array
• Select lower and place in new array.
• (If one input array is empty then place
• remainder of other array in output array)

107
67
45
23
14
6
33
98
42
108
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
109
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
110
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
111
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
Merge
112
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
23
Merge
113
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
23
98
Merge
114
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
23
98
115
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
23
98
Merge
116
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
14
23
98
Merge
117
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
45
23
98
14
Merge
118
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
98
45
14
23
Merge
119
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
98
14
23
45
14
Merge
120
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
23
14
98
45
14
23
Merge
121
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
23
98
45
14
14
23
45
Merge
122
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
23
98
45
14
23
98
45
14
14
23
45
98
Merge
123
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
23
98
45
14
14
23
45
98
124
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
23
98
45
14
14
23
45
98
125
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
23
98
45
14
Merge
14
23
45
98
126
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
6
23
98
45
14
Merge
14
23
45
98
127
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
67
23
98
45
14
6
Merge
14
23
45
98
128
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
14
23
45
98
129
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
Merge
14
23
45
98
130
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
33
23
98
45
14
67
6
Merge
14
23
45
98
131
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
42
23
98
45
14
67
6
33
Merge
14
23
45
98
132
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
14
23
45
98
Merge
133
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
6
42
33
67
6
14
23
45
98
Merge
134
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
6
33
67
42
6
33
14
23
45
98
Merge
135
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
6
42
33
67
6
33
42
14
23
45
98
Merge
136
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
45
98
Merge
137
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
33
42
67
6
23
45
98
14
Merge
138
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
42
67
33
23
45
98
14
6
Merge
139
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
42
67
33
14
45
98
23
6
14
Merge
140
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
42
67
33
14
23
98
45
6
14
23
Merge
141
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
67
42
14
23
98
45
6
14
23
33
Merge
142
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
98
45
6
14
23
33
42
Merge
143
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
45
98
6
14
23
33
42
45
Merge
144
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
45
98
6
14
23
33
42
45
67
Merge
145
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
45
98
6
14
23
33
42
45
67
98
Merge
146
67
45
23
14
6
33
98
42
67
45
23
14
6
33
98
42
45
23
14
98
67
6
33
42
23
98
45
14
67
6
33
42
23
98
45
14
67
6
42
33
6
33
42
67
14
23
45
98
6
14
23
33
42
45
67
98
147
67
45
23
14
6
33
98
42
6
14
23
33
42
45
67
98
148
Summary

• Divide the unsorted collection into two
• Until the sub-arrays only contain one element
• Then merge the sub-problem solutions together

149
Thank you