1. Sorting Networks Characteristics The basic unit: a comparator
Implemented in hardware
Parallel operations are possible
The basic unit: a comparator
Comparison occurs when both inputs are available
Comparison time is O(1)
A comparator sorts two inputs

2. A Comparison Network
Characteristics: four inputs, four outputs, five comparators, depth of three
This comparison network is a sorter; how can this be explained in common sense terms?

3. Some Thought Problems
Why must any sorting network with n inputs have at least a depth of lg n ?
Why must a sorting network with n inputs perform at least n lg n comparisons ?

4. A Sorting Network Based on Insertion Sort
Explain why this is based on insertion sort
General characteristics for an n input network
What will be the depth of the network?
How many comparators are there in the network?

5. The Zero-One Principle - 1
The basic idea is that is you can prove an n input comparison network sorts the 2n inputs with 0 or 1 correctly then it will sort any data correctly
The proof of the lemma has two basic steps
Induction is then used to prove the lemma

6. The Zero-One Principle - 2
Induction is based on the depth of the network
The result for depth zero is trivial
For depth d, it is proved for depth d+1 using the result on the previous page
The outputs are at a specific depth, so it is true at the outputs too
An example with f =  x/2 

7. The Zero-One Principle - 3

8. A Bitonic Sorting Network
What is a bitonic sequence?
A sequence that increases then decreases, such as { 3, 5, 7, 9, 12, 14, 11, 8, 7, 4, 2 } or such a sequence that has been circularly shifted
For inputs of 0 and 1, a bitonic sequence has one of two forms: 0i1j0k or 1i0j1k
Steps in the process
Build a “half-cleanser” network
Build a bitonic sorter recursively by halving the size from n to n/2 to n/4 until a base case of 2 is reached
Show the computation time is lg n

9. A Half-Cleaner
A half-cleaner with bitonic input produces one of two possible outputs

10. Proof of Lemma - 1
Assume inputs of the form 0i1j0k ; there are four cases, two are shown here

11. Proof of Lemma - 2 The other two cases
There is a symmetric proof for inputs of the form 1i0j1k

12. Building the Bitonic Sorter
Steps
A half-cleanser of size n is introduced
Two parallel bitonic sorters of size n/2 are attached
A recursive unraveling continues until the base case is reached
The final circuit is a bitonic sorter of size n

13. Depth of the Sorter
This result has been shown for a zero-one bitonic sorter
In a manner analogous to the zero-one theorem, it can be shown this same network can sort bitonic sequences of any arbitrary numbers

14. The First Stage of a Merger
An intuitive approach
Given two sorted sequences X and Y; reverse the Y to give YR, the concatenated sequence X YR is bitonic
Perform a bitonic sort on the sequence X YR
The first stage of the merger compares input i with input n-i+1
Comparing Merger[n] and Half-Cleanser[n] shows that the outputs of the bottom of the first stage are reversed

15. A Merging Network
The merging network is completed by attaching bitonic sorters to the outputs of the first stage
An 8 input merger is shown below where two sorted sequences of length four are merged into a sorted sequence of length eight

16. A Sorting Network
The sorting network is produced by unwinding the mergers from right to left

17. The Depth of Sorter[n] Calculating the depth recursively
The depth of sorter[n] is the depth of sorter[n/2] plus the depth of the merger[n], which is lg n
Note the the two sorter[n/2] run in parallel so it only appears once in the recurrence relation

18. An Odd-Even Transposition Sorter
An odd-even transposition sorter is based on a bubble sort algorithm
Here is the sorter for eight inputs

19. Name(s)______________________________
Use the zero-one principle to prove the this comparison network is a sorting network

20. Name(s)______________________________
How many bitonic sequences containing 0 or 1 are there?
Prove a bitonic sorter of size n, a power of 2, contains exactly ( n lg n) comparators

21. Name(s)______________________________
Design a network that merges 1 item with n-1 sorted items to produce n sorted items
Show that to produce a sequence of length n, the merger must have a depth of at least lg n

22. Name(s)______________________________
How many comparators are needed to build sorter[n] ? (assume n is a power of 2)
Prove the depth of sorter[n] is exactly (lg n) (lg n + 1)/2

23. Name(s)______________________________
How many comparators are needed to build odd-even[n] ? (assume n is a power of 2)
What is the depth of odd-even[n] ?

24. Name(s)______________________________
How many comparators are needed to build insertion_sorter[n] ? (assume n is a power of 2)
What is the depth of insertion_sorter[n] ?