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

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?

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 ?

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?

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

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 

The Zero-One Principle - 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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] ?

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] ?