1

Sorting Networks

2

10

94

3

21

87

109

4321

87

Sorting

3

Sorting Network

(Bitonic Sorting Network)

4

Sorting Network

(Bitonic Sorting Network)

5

Comparator

x

y

max(x,y)

min(x,y)

6

Comparatormax

min

1

9

9

1

7

Comparatormax

min

100

4

100

4

8

9

Levels1 2 3

Depth = 3

Width = 4

10

1825

11

18

25

Level 1

12

1

8

2

5

Level 2

13

1

8

25

Level 3

14

1

8

25

1825

Input Output

15

10

94

3

21

87

109

4321

87

Input Output

16

10

94

3

21

87

109

4321

87

1 2 3 4 5 6levels

17

Sort(n)

Sort(n/2)

Sort(n/2)

Merge(n)In Out

)(log2 nOdepth

Recursive construction of Bitonic Sorting Network

18

Merge(n)

Merge(n/2)

Merge(n/2)

)(lognOdepth

Recursive construction of merging network

19

Induction BasisSort(2)

Merge(2)

20

Width

Sorting network depth:

)(log2log...4log2loglog 2nOnnn

n

)(log2 nO

Mergerwidth n

Totaldepth

2n

4n 2

21

Counting Networks

22

The Counting Problem

0

Shared variable

23

0

Token = Increment request

Shared variable

24

1

Increment request

Shared variable

0

25

1

Shared variable

26

1

Increment request

Shared variable

27

2

Shared variable

1Increment request

28

2

Shared variable

29

2

Increment request

Shared variable

30

3

Increment request

Shared variable

2

31

6

Sequential Bottleneck

Shared variable

The requests have to be serialized

01 2 3

45

32

Counting Network

Inputs Outputs

33

Counting Network

34

Balancer

Inputs Outputs

Token

35

Balancer

Inputs Outputs

36

Balancer

37

Balancer

38

Balancer

39

Balancer

40

Balancer

41

Balancer

42

Balancer

All tokens together

43

Balancer

Step property

44

Balancer

Step property

Another example

45

Balancer

Step property

Another example

46

Balancer

Step property

Another example

47

Counting Network

(Bitonic Counting Network)

48

Counting Network

49

Counting Network

50

Counting Network

51

Counting Network

52

Counting Network

53

Counting Network

54

Counting Network

55

Counting Network

56

Counting Network

57

Counting Network

58

Counting Network

59

Counting Network

60

Counting Network

61

Counting Network

62

Counting Network

63

Counting Network

64

Counting Network

65

Counting Network

All tokens

66

Counting Network

All tokens Step property

67

Counting Network

Another example Step property

68

Counting Network

Another example Step property

69

Counting Network

Another example Step property

70

Counting Network

Another example Step property

71

Counting Network

ParallelismMany increment requests are processed simultaneously

72

Counting

0123

Token = Increment request

Output Shared variables

73

Counting

0123

Output Shared variables

74

Counting

4123

0Returned value

Shared variable value is increased by 4 (output width)

75

Counting

4123

0

Increment request

76

Counting

4123

0

77

Counting

4523

0

1

Returned value

78

Counting

4523

0

1

Increment request

79

Counting

4563

0

1

2

Returned value

80

Counting

4563

0

1

2

81

Counting

4567

0

1

2

3

Returned value

82

Counting

4567

0

1

2

3

83

Counting

8567

0

1

2

3

4Returned value

84

Counting

8567

0

1

2

3

4

85

Counting

8967

0

1

2

3

4

5

Returned value

86

Counting

8967

0

1

2

3

4

5

All tokens

87

Bitonic Counting Network

Isomorphic to Bitonic Sorting Network

88

Count(n)

Count(n/2)

Count(n/2)

Merge(n)In Out

)(log2 nOdepth

X

Y

Z

89

Merge(n)

Merge1(n/2)

Merge2(n/2)

)(lognOdepth

X

Y

Z

StepProp.

StepProp.

StepProp.

90

X

0x1x2x3x4x5x6x7x

oddX

0x1x2x3x4x5x6x7x

evenX

0x1x2x3x4x5x6x7x

Even subsequence

Odd subsequence

91

Merge(n)

Merge1(n/2)

Merge2(n/2)

X

Y

Z

evenX

oddY

evenY

oddX

A

B

92

Theorem:

Proof:The proof is by induction on n

For the merger is a balancer andthe claim holds trivially.

2nInduction Basis:

The merger(n) produces output Z with step property if both inputs X and Y have step property

93

Induction Hypothesis:

Merge(n/2)

If step sequence

If step sequence

Then stepsequence

Assume that every merger of size n/2 and smallerperforms merging properly

94

Merge(n)

Induction step:

If step

If stepThen step

We want to show that:

X

YZ

Suppose and have the step property.Then we show that has the step property

X YZ

95

0x

2x

4x

6x

1x

3x

5x

7x

evenX

oddX

0x1x2x3x4x5x6x7x

X

If has the step propertythen and have the step property

XevenX oddX

Tokens

96

Merge1(n/2)evenX

oddYA

Merge2(n/2)evenY

oddX

B

step

step

step

step

step

step

Therefore, from induction hypothesis:

Output sequence

Output sequence

97

First, we show: 1||||1 BA

Where denotes the total number oftokens in sequence

||AA

98

We have: |||||| oddeven YXA

|||||| oddeven XYB

Merger 1

Merger 2

99

|||| oddeven XX 1|||| oddeven XXor0x1x2x3x4x5x6x7x

0x1x2x3x4x5x6x7x

Since has the step property:X

Therefore: 1||||0 oddeven XX

Similarly: 1||||0 oddeven YY

100

|)||(||)||(||||| evenoddoddeven YYXXBA

1||||0 oddeven XX

1||||0 oddeven YY

1||||1 BA

101

Now, we show that has the step propertyZ

1||||1 BA

A Bc

i i

ccc

1c 1c1c1c

ccccc

1c1c1c

There is at most one wire where the two sequences differ

#tokens

i

#tokensstep step

102

A

B

c

i

i

ccc1c1c1c1c

ccccc

1c1c1c

cccc

c1c

1c1c

1c1c

cccc

c1c

1c1c

1c1c

Z

Stepproperty

Mer

ge1(

n/2)

Mer

ge(n

/2)

Merge(n)