1. Comparison Networks Sorting Sorting binary values
Sorting arbitrary numbers
Implementing symmetric functions

2. Sorting Algorithms Example
Mergesort(array[1,…,n] of Integers):
begin
Mergesort(array[1,…,n/2]);
Mergesort(array[n/2+1,…,n]);
Merge(array[1,…,n/2], array[n/2+1,…,n]);
end
comparisons required to merge two arrays of size m/2
comparisons to sort n elements
Order of comparisons not fixed in advance.
Not readily implementable in hardware.

3. Sorting Networks Sorting Network C D B A A B C D
Order of comparisons fixed in advance.
Readily implementable in hardware.

4. Sorting Networks (binary values)
inputs
outputs 1
sorted
Sorting
Network 1 1 1 1 1 1 1

5. Comparator (2-sorter)
inputs
outputs x
min(x, y) C y
max(x, y)

6. Comparator (2-sorter) AON Implementation x min(x, y) max(x, y) y
inputs
outputs x
min(x, y)
max(x, y) y

7. Comparator (2-sorter)
inputs
outputs x
min(x, y) y
max(x, y)

8. Comparison Network 1 1 1 1
width n
depth d

9. Comparison Network 1 1 1 1
n / 2
comparisons
per stage
d stages

10. Sorting Network
Any ideas?

11. Sorting Network
inputs
outputs
Sorting
Network . n .
n 1

12. Insertion Sort Network
inputs
outputs
depth 2n 3

13. Batcher Sorting Network
Next Lecture

14. Sorting Arbitrary Numbers
inputs
outputs x
min(x, y) y
max(x, y)
x, y can be values from any linearly ordered set,
e.g., integers, reals, etc.

15. Integer Comparator X, Y: integers represented as m-bit binary strings.
Comparison function:
C(X,Y) =
1 if X > Y,
0 otherwise.
Idea: use C(X,Y) to select the min and the max
of X and Y.

16. Integer Comparator X C(X, Y) min(X, Y) Y C(X, Y) X C(X, Y) max(X, Y) Y

17. Sorting Arbitrary Numbers9 6 2 2 9 6 6 9 2 2 6 9
sorted

18. Sorting Arbitrary Numbers1 4 5 1 5 4 4 5 1 1 4 5
sorted

19. Sorting Arbitrary Numbers3 7 3 7 7 3 3 7
not sorted
How can we verify if a network sorts all possible input sequences?

20. Sorting Arbitrary Numbers
inputs
outputs
Try all possible 0/1 sequences.

21. Sorting Arbitrary Numbers
inputs
outputs
000
000
Try all possible 0/1 sequences.

22. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
Try all possible 0/1 sequences.

23. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
Try all possible 0/1 sequences.

24. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
Try all possible 0/1 sequences.

25. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
Try all possible 0/1 sequences.

26. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
Try all possible 0/1 sequences.

27. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
110
101
not sorted!
Try all possible 0/1 sequences.

28. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
110
101
not sorted!
111
111
Try all possible 0/1 sequences.

29. Sorting Arbitrary Numbers

30. Sorting Arbitrary Numbers
inputs
outputs
Try all possible 0/1 sequences.

31. Sorting Arbitrary Numbers
inputs
outputs
000
000
Try all possible 0/1 sequences.

32. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
Try all possible 0/1 sequences.

33. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
Try all possible 0/1 sequences.

34. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
Try all possible 0/1 sequences.

35. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
Try all possible 0/1 sequences.

36. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
Try all possible 0/1 sequences.

37. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
110
011
Try all possible 0/1 sequences.

38. Sorting Arbitrary Numbers
inputs
outputs 1 1 1 1
000
000
001
001
010
001
011
011
100
001
101
011
110
011
111
111
Try all possible 0/1 sequences.

39. Sorting Arbitrary Numbers
inputs
outputs
000
000
001
001
010
001
011
011
100
001
101
011
110
011
111
111
all sorted!
Try all possible 0/1 sequences.

40. Zero-One Principle
If a comparison network sorts all possible sequences
of 0’s and 1’s correctly, then it sorts all sequences
of arbitrary numbers correctly.

41. Lemma
Given
For a monotonically increasing function f,

42. Lemma
Given
For a monotonically increasing function f,

43. Proof: Lemma

44. Proof: Lemma

45. Proof: Lemma
f is monotonically increasing:

46. Proof: Lemma
f is monotonically increasing:

47. Proof: Lemma
f is monotonically increasing:

48. Generalization
Given

49. Generalization For a monotonically increasing function f,
(by induction)

50. Proof: Zero-One Principle
Suppose
b) there exists a sequence that it doesn’t sort, i.e.,
such that
but is placed before in the output.
a) the network sorts all sequences of 0’s and 1’s,
Define
f (x) =
0 if
1 otherwise

51. Proof: Zero-One Principle
Sorting
Network .

52. Proof: Zero-One Principle.
Sorting
Network . . . .

53. Proof: Zero-One Principle.
Sorting
Network . . 1
contradiction! . .

54. Implementing XOR
XOR(X) =
1 if odd
0 otherwise
Sorting
Network 1 1 1 1

55. Implementing XOR
XOR(X) =
1 if odd
0 otherwise
Sorting
Network

56. Symmetric Functions for some subset of f (X) = 1 if 0 otherwise
Sorting
Network