Bubble sort is an algorithm that sorts a collection of elements by repeatedly traversing the list from the first element to the last, comparing adjacent elements and swapping them if they are in the wrong order. This process is repeated until the list is fully sorted, requiring a number of passes equal to the size of the list minus one. The algorithm uses a flag to check if any swapping occurred on a given pass, and can stop early if the list is already sorted after an initial pass.

1. Bubble Sort
Merge Sort

2. Bubble Sort

3. Sorting
• Sorting takes an unordered collection and
makes it an ordered one.
5
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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
5
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35
42
77 101
1 2 3 4 5 6

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
5
12
35
42
77 101
1 2 3 4 5 6
Swap
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
5
12
35
77
42 101
1 2 3 4 5 6
Swap
35 77

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
5
12
77
35
42 101
1 2 3 4 5 6
Swap
12 77

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
5
77
12
35
42 101
1 2 3 4 5 6
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
5
77
12
35
42 101
1 2 3 4 5 6
Swap
5 101

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
77
12
35
42 5
1 2 3 4 5 6
101
Largest value correctly placed

11. The “Bubble Up” Algorithm
index <- 1
last_compare_at <- n – 1
loop
exitif(index > last_compare_at)
if(A[index] > A[index + 1]) then
Swap(A[index], A[index + 1])
endif
index <- index + 1
endloop

12. No, Swap isn’t built in.
Procedure Swap(a, b isoftype in/out Num)
t isoftype Num
t <- a
a <- b
b <- t
endprocedure // Swap
LB

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
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101
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 we’ll correctly
place all N elements.

15. “Bubbling” All the Elements
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101
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101
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1 2 3 4 5 6
101
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N
-
1

16. Reducing the Number of Comparisons
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1 2 3 4 5 6
101

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
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101

18. Putting It All Together

19. N is … // Size of Array
Arr_Type definesa Array[1..N] of Num
Procedure Swap(n1, n2 isoftype in/out Num)
temp isoftype Num
temp <- n1
n1 <- n2
n2 <- temp
endprocedure // Swap

20. procedure Bubblesort(A isoftype in/out Arr_Type)
to_do, index isoftype Num
to_do <- N – 1
loop
exitif(to_do = 0)
index <- 1
loop
exitif(index > to_do)
if(A[index] > A[index + 1]) then
Swap(A[index], A[index + 1])
endif
index <- index + 1
endloop
to_do <- to_do - 1
endloop
endprocedure // Bubblesort
Inner
loop
Outer
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”!
42
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5 77
1 2 3 4 5 6
101

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 <- true
loop
exitif ((to_do = 0) OR NOT(did_swap))
index <- 1
did_swap <- false
loop
exitif(index > to_do)
if(A[index] > A[index + 1]) then
Swap(A[index], A[index + 1])
did_swap <- true
endif
index <- index + 1
endloop
to_do <- to_do - 1
endloop

24. An Animated Example
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to_do
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N 8 did_swap true

25. An Animated Example
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to_do
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N 8 did_swap false

26. An Animated Example
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to_do
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N 8
Swap
did_swap false

27. An Animated Example
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to_do
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N 8
Swap
did_swap true

28. An Animated Example
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to_do
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N 8 did_swap true

29. An Animated Example
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to_do
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N 8
Swap
did_swap true

30. An Animated Example
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to_do
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N 8
Swap
did_swap true

31. An Animated Example
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to_do
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N 8 did_swap true

32. An Animated Example
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to_do
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N 8
Swap
did_swap true

33. An Animated Example
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to_do
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N 8
Swap
did_swap true

34. An Animated Example
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to_do
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N 8 did_swap true

35. An Animated Example
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to_do
index
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N 8
Swap
did_swap true

36. An Animated Example
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1 2 3 4 5 6 7 8
to_do
index
7
4
N 8
Swap
did_swap true

37. An Animated Example
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to_do
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N 8 did_swap true

38. An Animated Example
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1 2 3 4 5 6 7 8
to_do
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5
N 8
Swap
did_swap true

39. An Animated Example
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to_do
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N 8
Swap
did_swap true

40. An Animated Example
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to_do
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N 8 did_swap true

41. An Animated Example
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to_do
index
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N 8
Swap
did_swap true

42. An Animated Example
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to_do
index
7
6
N 8
Swap
did_swap true

43. An Animated Example
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23 42
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to_do
index
7
7
N 8 did_swap true

44. An Animated Example
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23 42
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to_do
index
7
7
N 8
Swap
did_swap true

45. An Animated Example
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23 98
1 2 3 4 5 6 7 8
to_do
index
7
7
N 8
Swap
did_swap true

46. After First Pass of Outer Loop
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to_do
index
7
8
N 8
Finished first “Bubble Up”
did_swap true

47. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
1
N 8 did_swap false

48. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
1
N 8 did_swap false
No Swap

49. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
2
N 8 did_swap false

50. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
2
N 8 did_swap false
Swap

51. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
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2
N 8 did_swap true
Swap

52. The Second “Bubble Up”
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to_do
index
6
3
N 8 did_swap true

53. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
3
N 8 did_swap true
Swap

54. The Second “Bubble Up”
33
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
3
N 8 did_swap true
Swap

55. The Second “Bubble Up”
33
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
4
N 8 did_swap true

56. The Second “Bubble Up”
33
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
4
N 8 did_swap true
No Swap

57. The Second “Bubble Up”
33
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
5
N 8 did_swap true

58. The Second “Bubble Up”
33
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
5
N 8 did_swap true
Swap

59. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
5
N 8 did_swap true
Swap

60. The Second “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
6
6
N 8 did_swap true

61. The Second “Bubble Up”
67
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
6
N 8 did_swap true
Swap

62. The Second “Bubble Up”
42
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23 98
1 2 3 4 5 6 7 8
to_do
index
6
6
N 8 did_swap true
Swap

63. After Second Pass of Outer Loop
42
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1 2 3 4 5 6 7 8
to_do
index
6
7
N 8 did_swap true
Finished second “Bubble Up”

64. The Third “Bubble Up”
42
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23 98
1 2 3 4 5 6 7 8
to_do
index
5
1
N 8 did_swap false

65. The Third “Bubble Up”
42
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1 2 3 4 5 6 7 8
to_do
index
5
1
N 8 did_swap false
Swap

66. The Third “Bubble Up”
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23 45 33 67
14 98
1 2 3 4 5 6 7 8
to_do
index
5
1
N 8 did_swap true
Swap

67. The Third “Bubble Up”
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23 45 33 67
14 98
1 2 3 4 5 6 7 8
to_do
index
5
2
N 8 did_swap true

68. The Third “Bubble Up”
42
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23 45 33 67
14 98
1 2 3 4 5 6 7 8
to_do
index
5
2
N 8 did_swap true
Swap

69. The Third “Bubble Up”
42
23
6 45 33 67
14 98
1 2 3 4 5 6 7 8
to_do
index
5
2
N 8 did_swap true
Swap

70. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
3
N 8 did_swap true

71. The Third “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
5
3
N 8 did_swap true
No Swap

72. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
4
N 8 did_swap true

73. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
4
N 8 did_swap true
Swap

74. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
4
N 8 did_swap true
Swap

75. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
5
N 8 did_swap true

76. The Third “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
5
5
N 8 did_swap true
Swap

77. The Third “Bubble Up”
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1 2 3 4 5 6 7 8
to_do
index
5
5
N 8 did_swap true
Swap

78. After Third Pass of Outer Loop
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to_do
index
5
6
N 8 did_swap true
Finished third “Bubble Up”

79. The Fourth “Bubble Up”
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23
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14 98
1 2 3 4 5 6 7 8
to_do
index
4
1
N 8 did_swap false

80. The Fourth “Bubble Up”
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14 98
1 2 3 4 5 6 7 8
to_do
index
4
1
N 8 did_swap false
Swap

81. The Fourth “Bubble Up”
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14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
4
1
N 8 did_swap true
Swap

82. The Fourth “Bubble Up”
45
23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
4
2
N 8 did_swap true

83. The Fourth “Bubble Up”
45
23
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6 98
1 2 3 4 5 6 7 8
to_do
index
4
2
N 8 did_swap true
No Swap

84. The Fourth “Bubble Up”
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23
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6 98
1 2 3 4 5 6 7 8
to_do
index
4
3
N 8 did_swap true

85. The Fourth “Bubble Up”
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23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
4
3
N 8 did_swap true
No Swap

86. The Fourth “Bubble Up”
45
23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
4
4
N 8 did_swap true

87. The Fourth “Bubble Up”
45
23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
4
4
N 8 did_swap true
No Swap

88. After Fourth Pass of Outer Loop
45
23
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6 98
1 2 3 4 5 6 7 8
to_do
index
4
5
N 8 did_swap true
Finished fourth “Bubble Up”

89. The Fifth “Bubble Up”
45
23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
3
1
N 8 did_swap false

90. The Fifth “Bubble Up”
45
23
14 33 42 67
6 98
1 2 3 4 5 6 7 8
to_do
index
3
1
N 8 did_swap false
No Swap

91. The Fifth “Bubble Up”
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
2
N 8 did_swap false

92. The Fifth “Bubble Up”
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23
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
2
N 8 did_swap false
No Swap

93. The Fifth “Bubble Up”
45
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
3
N 8 did_swap false

94. The Fifth “Bubble Up”
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
3
N 8 did_swap false
No Swap

95. After Fifth Pass of Outer Loop
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
4
N 8 did_swap false
Finished fifth “Bubble Up”

96. Finished “Early”
45
23
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6 98
1 2 3 4 5 6 7 8
to_do
index
3
4
N 8 did_swap false
We didn’t do any swapping,
so all of the other elements
must be correctly placed.
We can “skip” the last two
passes of the outer loop.

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

99. Questions?

100. Mergesort

101. Sorting
• Sorting takes an unordered collection and
makes it an ordered one.
5
12
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77 101
1 2 3 4 5 6
5 12 35 42 77 101
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-array’s 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
37 23 6 89 15 12 2 19

104. How to Remember Merge Sort?
That’s easy. Just
remember Mariah
Carey.
As a singing star, Ms.
Carey has perfected
the “wax-on” wave
motion--a clockwise
sweep of her hand
used to emphasize
lyrics.
The Maria
“Wax-on” Angle:
(q,t)
The Siren of Subquadratic Sorts

105. How To Remember Merge Sort?
Just as Mariah recursively moves
her hands into smaller circles, so
too does merge sort recursively
split an array into smaller
segments.
(q,t)
We need two such recursions,
one for each half of the split
array.

106. Algorithm
Mergesort(Passed an array)
if array size > 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. More TRUTH in CS
• We don’t 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.
s1 f1 s2 f2
LB

108. Algorithm
Mergesort(Passed an array)
if array size > 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)
LB

109. 67
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23 14 6 33
98 42

110. 67
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23 14 6 33
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67
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111. 67
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67
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23 14
98

112. 67
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67
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23 14
98
23
98

113. 67
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67
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23 14
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23
98
Merge

114. 67
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67
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45
23 14
98
23
98
23
Merge

115. 67
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67
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23 14 6 33
98 42
45
23 14
98
23
98
23 98
Merge

116. 67
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67
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98 42
45
23 14
98
23
98 45 14
23 98

117. 67
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67
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98 42
45
23 14
98
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98 45 14
Merge
23 98

118. 67
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98 42
67
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23 14 6 33
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45
23 14
98
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98 45 14
14
Merge
23 98

119. 67
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67
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45
23 14
98
23
98 45 14
45
Merge
23 98 14

120. 67
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23 14 6 33
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67
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45
23 14
98
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Merge
98 45
14
23

121. 67
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67
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23 14 6 33
98 42
45
23 14
98
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98 45 14
Merge
98 14
14
23 45

122. 67
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98 42
67
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23 14 6 33
98 42
45
23 14
98
23
98 45 14
Merge
23 14
14 23
98 45

123. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
Merge
23 98 45
14
14 23 45

124. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
Merge
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
23
98 45 14
67
6 33 42
23 98 45
14
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
23
98 45 14
67
6 33 42
67
6
23 98 45
14
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
23
98 45 14
67
6 33 42
67
6
Merge
23 98 45
14
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
23
98 45 14
67
6 33 42
67
6
6
Merge
23 98 45
14
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
23
98 45 14
67
6 33 42
67
6
67
Merge
23 98 45
14 6
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
23
98 45 14
67
6 33 42
67
6 33 42
23 98 45
14 67
6
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
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6
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
23
98 45 14
67
6 33 42
67
6 33 42
Merge
33
23 98 45
14 67
6
14 23 45 98

133. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
42
23 98 45
14 67
6 33
14 23 45 98

134. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 45 98

135. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 6 42
33
14 23 45 98 6
67

136. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 6 33
14 23 45 98 6 33
67 42

137. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 6 42
33
14 23 45 98 6 33 42
67

138. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 45 98 6 33 42 67

139. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
23 45 98 33 42 67
14 6

140. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
23 45 98 6 42 67
6
14 33

141. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 45 98 6 42 67
6 14
23 33

142. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 98 6 42 67
6 14 23
45 33

143. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 98 6 33 67
6 14 23 33
45 42

144. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 98 6 33 42
6 14 23 33 42
45 67

145. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 45 6 33 42
6 14 23 33 42 45
98 67

146. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 45 98 6 33 42 67
6 14 23 33 42 45 67

147. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
Merge
23 98 45
14 67
6 42
33
14 23 45 98 6 33 42 67
6 14 23 33 42 45 67 98

148. 67
45
23 14 6 33
98 42
67
45
23 14 6 33
98 42
45
23 14
98
23
98 45 14
67
6 33 42
67
6 33 42
23 98 45
14 67
6 42
33
14 23 45 98 6 33 42 67
6 14 23 33 42 45 67 98

149. 67
45
23 14 6 33
98 42
6 14 23 33 42 45 67 98

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

151. Questions?

152. Review?