1. Lecture 16
Bubble Sort Merge Sort

2. Bubble Sort

3. Sorting
Sorting takes an unordered collection and makes it an ordered one. 77 42 35 12
101 5 5 12 35 42 77
101

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 77 42 35 12
101 5

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
Swap 42 77 77 42 35 12
101 5

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
Swap 35 77 42 77 35 12
101 5

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
Swap 12 77 42 35 77 12
101 5

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 42 35 12 77
101 5
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
Swap 5
101 42 35 12 77
101 5

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
101 42 35 12 77 5
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. LB Procedure Swap(a, b isoftype in/out Num)
t isoftype Num
t <- a
a <- b
b <- 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
101 42 35 12 77 5
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 Elements77 12 35 42 5
101
N - 1 5 42 12 35 77
101 42 5 35 12 77
101 42 35 5 12 77
101 42 35 12 5 77
101

16. Reducing the Number of Comparisons12 35 42 77
101 5 77 12 35 42 5
101 5 42 12 35 77
101 42 5 35 12 77
101 42 35 5 12 77
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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 42 5 35 12 77
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. Outer loop Inner loop procedure Bubblesort(A isoftype in/out Arr_Type)
to_do, index isoftype Num
to_do <- N – 1
loop
exitif(to_do = 0)
index <- 1
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
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 35 12 5 77
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
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

24. An Animated Example N 8 did_swap true to_do 7 index 98 23 45 14 6 6733 42

25. An Animated Example N 8 did_swap false to_do 7 index 1 98 23 45 14 667 33 42

26. An Animated Example N 8 did_swap false to_do 7 index 1 Swap 98 23 4514 6 67 33 42

27. An Animated Example N 8 did_swap true to_do 7 index 1 Swap 23 98 45 146 67 33 42

28. An Animated Example N 8 did_swap true to_do 7 index 2 23 98 45 14 6 6733 42

29. An Animated Example N 8 did_swap true to_do 7 index 2 Swap 23 98 45 146 67 33 42

30. An Animated Example N 8 did_swap true to_do 7 index 2 Swap 23 45 98 146 67 33 42

31. An Animated Example N 8 did_swap true to_do 7 index 3 23 45 98 14 6 6733 42

32. An Animated Example N 8 did_swap true to_do 7 index 3 Swap 23 45 98 146 67 33 42

33. An Animated Example N 8 did_swap true to_do 7 index 3 Swap 23 45 14 986 67 33 42

34. An Animated Example N 8 did_swap true to_do 7 index 4 23 45 14 98 6 6733 42

35. An Animated Example N 8 did_swap true to_do 7 index 4 Swap 23 45 14 986 67 33 42

36. An Animated Example N 8 did_swap true to_do 7 index 4 Swap 23 45 14 698 67 33 42

37. An Animated Example N 8 did_swap true to_do 7 index 5 23 45 14 6 98 6733 42

38. An Animated Example N 8 did_swap true to_do 7 index 5 Swap 23 45 14 698 67 33 42

39. An Animated Example N 8 did_swap true to_do 7 index 5 Swap 23 45 14 667 98 33 42

40. An Animated Example N 8 did_swap true to_do 7 index 6 23 45 14 6 67 9833 42

41. An Animated Example N 8 did_swap true to_do 7 index 6 Swap 23 45 14 667 98 33 42

42. An Animated Example N 8 did_swap true to_do 7 index 6 Swap 23 45 14 667 33 98 42

43. An Animated Example N 8 did_swap true to_do 7 index 7 23 45 14 6 67 3398 42

44. An Animated Example N 8 did_swap true to_do 7 index 7 Swap 23 45 14 667 33 98 42

45. An Animated Example N 8 did_swap true to_do 7 index 7 Swap 23 45 14 667 33 42 98

46. After First Pass of Outer LoopN 8
did_swap
true
to_do 7
Finished first “Bubble Up”
index 8 23 45 14 6 67 33 42 98

47. The Second “Bubble Up” N 8 did_swap false to_do 6 index 1 23 45 14 667 33 42 98

48. The Second “Bubble Up” N 8 did_swap false to_do 6 index 1 No Swap 2345 14 6 67 33 42 98

49. The Second “Bubble Up” N 8 did_swap false to_do 6 index 2 23 45 14 667 33 42 98

50. The Second “Bubble Up” N 8 did_swap false to_do 6 index 2 Swap 23 4514 6 67 33 42 98

51. The Second “Bubble Up” N 8 did_swap true to_do 6 index 2 Swap 23 14 4567 33 42 98

52. The Second “Bubble Up” N 8 did_swap true to_do 6 index 3 23 14 45 6 6733 42 98

53. The Second “Bubble Up” N 8 did_swap true to_do 6 index 3 Swap 23 14 4567 33 42 98

54. The Second “Bubble Up” N 8 did_swap true to_do 6 index 3 Swap 23 14 645 67 33 42 98

55. The Second “Bubble Up” N 8 did_swap true to_do 6 index 4 23 14 6 45 6733 42 98

56. The Second “Bubble Up” N 8 did_swap true to_do 6 index 4 No Swap 23 1445 67 33 42 98

57. The Second “Bubble Up” N 8 did_swap true to_do 6 index 5 23 14 6 45 6733 42 98

58. The Second “Bubble Up” N 8 did_swap true to_do 6 index 5 Swap 23 14 645 67 33 42 98

59. The Second “Bubble Up” N 8 did_swap true to_do 6 index 5 Swap 23 14 645 33 67 42 98

60. The Second “Bubble Up” N 8 did_swap true to_do 6 index 6 23 14 6 45 3367 42 98

61. The Second “Bubble Up” N 8 did_swap true to_do 6 index 6 Swap 23 14 645 33 67 42 98

62. The Second “Bubble Up” N 8 did_swap true to_do 6 index 6 Swap 23 14 645 33 42 67 98

63. After Second Pass of Outer Loop8
did_swap
true
to_do 6
Finished second “Bubble Up”
index 7 23 14 6 45 33 42 67 98

64. The Third “Bubble Up” N 8 did_swap false to_do 5 index 1 23 14 6 45 3342 67 98

65. The Third “Bubble Up” N 8 did_swap false to_do 5 index 1 Swap 23 14 645 33 42 67 98

66. The Third “Bubble Up” N 8 did_swap true to_do 5 index 1 Swap 14 23 645 33 42 67 98

67. The Third “Bubble Up” N 8 did_swap true to_do 5 index 2 14 23 6 45 3342 67 98

68. The Third “Bubble Up” N 8 did_swap true to_do 5 index 2 Swap 14 23 645 33 42 67 98

69. The Third “Bubble Up” N 8 did_swap true to_do 5 index 2 Swap 14 6 2345 33 42 67 98

70. The Third “Bubble Up” N 8 did_swap true to_do 5 index 3 14 6 23 45 3342 67 98

71. The Third “Bubble Up” N 8 did_swap true to_do 5 index 3 No Swap 14 623 45 33 42 67 98

72. The Third “Bubble Up” N 8 did_swap true to_do 5 index 4 14 6 23 45 3342 67 98

73. The Third “Bubble Up” N 8 did_swap true to_do 5 index 4 Swap 14 6 2345 33 42 67 98

74. The Third “Bubble Up” N 8 did_swap true to_do 5 index 4 Swap 14 6 2333 45 42 67 98

75. The Third “Bubble Up” N 8 did_swap true to_do 5 index 5 14 6 23 33 4542 67 98

76. The Third “Bubble Up” N 8 did_swap true to_do 5 index 5 Swap 14 6 2333 45 42 67 98

77. The Third “Bubble Up” N 8 did_swap true to_do 5 index 5 Swap 14 6 2333 42 45 67 98

78. After Third Pass of Outer LoopN 8
did_swap
true
to_do 5
Finished third “Bubble Up”
index 6 14 6 23 33 42 45 67 98

79. The Fourth “Bubble Up” N 8 did_swap false to_do 4 index 1 14 6 23 3342 45 67 98

80. The Fourth “Bubble Up” N 8 did_swap false to_do 4 index 1 Swap 14 6 2333 42 45 67 98

81. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 1 Swap 6 14 2333 42 45 67 98

82. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 2 6 14 23 33 4245 67 98

83. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 2 No Swap 6 1423 33 42 45 67 98

84. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 3 6 14 23 33 4245 67 98

85. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 3 No Swap 6 1423 33 42 45 67 98

86. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 4 6 14 23 33 4245 67 98

87. The Fourth “Bubble Up” N 8 did_swap true to_do 4 index 4 No Swap 6 1423 33 42 45 67 98

88. After Fourth Pass of Outer LoopN 8
did_swap
true
to_do 4
Finished fourth “Bubble Up”
index 5 6 14 23 33 42 45 67 98

89. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 1 6 14 23 33 4245 67 98

90. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 1 No Swap 6 1423 33 42 45 67 98

91. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 2 6 14 23 33 4245 67 98

92. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 2 No Swap 6 1423 33 42 45 67 98

93. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 3 6 14 23 33 4245 67 98

94. The Fifth “Bubble Up” N 8 did_swap false to_do 3 index 3 No Swap 6 1423 33 42 45 67 98

95. After Fifth Pass of Outer LoopN 8
did_swap
false
to_do 3
Finished fifth “Bubble Up”
index 4 6 14 23 33 42 45 67 98

96. Finished “Early” N 8 did_swap false to_do 3
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.
index 4 6 14 23 33 42 45 67 98

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 EVERLB
Truth in CS Act
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. 77 42 35 12
101 5 5 12 35 42 77
101

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?
¦(q,t)
Just as Mariah recursively moves her hands into smaller circles, so too does merge sort recursively split an array into smaller segments.
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 LB 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

108. Algorithm LB 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)

109. 98 23 45 14 6 67 33 42

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111. 98 23 45 14 6 67 33 42 98 23 45 14 6 67 33 42 98 23 45 14

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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Merge

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

149. 98 23 45 14 6 67 33 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