Two-tone sequencing network based on Batcher comparator

1. Batcher Comparator

The Batcher comparator refers to the maximum value of max{x,y} and the minimum value min{x,y} if the input x, Y is given at the two inputs and the output is at two outputs. As shown in 1, we specify a minimum output value for the upper output of the Batcher comparator and a maximum output value at the output end.

2. Double-tone sequence

A so-called double-tone sequence (Bitonic Sequence) is a sequence, such as a sequence (23,10,8,3,5,7,11,78), consisting of a non-strict sequence x and a non-strict minus sequence Y (where the smallest element of x is exactly the largest element of Y). 23,10,8 is the subtraction sequence, the 8,3,5,7,11,78 is the increment sequence, and 8 is the intermediate element. The specific mathematical definition is Jiangzi de:

Definition: A sequence a1,a2,..., an is a double-tone sequence (Bitonic Sequence), if

(1) There is an AK (1≤k≤n) that makes a1≥ ... ≥ak≤ ... ≤an is established, or

(2) sequence capable of cyclic displacement satisfying conditions (1)

In general, the two-tone sequence has an intermediate element, the left side of which is an increment sequence (or subtraction sequence), and the right side is a minus sequence (or an increment sequence). 2 (the downward and upward slashes represent the subtraction sequence and the increment sequence, respectively):

3. Batcher theorem

　　The Batcher theorem is to say that any one of the double-tone series of 2n is cut in half from the middle, divided into equal length two sequences x and Y, and then x and y in the same position of the element Xi and Yi, small to the min sequence, large put to the max sequence. The resulting max sequence and min sequence are also two-tone sequences. And each element of the min sequence is less than or equal to each element of the max sequence. 3 is shown below:

4. Double-tuned merging network based on Batcher theorem

Using the Batcher theorem, we can divide any two-tone sequence A, which is 2n long, into a min sequence and a max sequence, then divide the min sequence and the max sequence, and so on, until N=1. Finally, all sub-sequences containing two elements are then merged into a fully ordered sequence. 4 is shown below:

5. Two-Tone sorting network

The input of a double-tuned merge network must be a double-tone sequence. If you want to turn it into a sort network for any input sequence, you first need to convert the input sequence to a double-tone sequence. The main idea is: the sequence of length 2n is regarded as an ordered sequence of n 2 elements, and any two ordered sequence can be composed of a 4-element double-modulation sequence, then a double-modulation sequence of N/2 4 elements can be obtained, and the double-modulation sequence of 4 element is obtained by merging the network with a 4-input double-tune. The ordered sequence of two 4 elements can be composed of a 8-element two-tone sequence, and then a sequential sequence of 8 elements is obtained by a 8-input double-tuned network; and so on, until the end of an ordered sequence of 2n elements. 5 is shown below:

Reference documents:

1. Batcher Merge Network

2. Double-tone sequencing

3. Selection and sequencing of comparator networks

4. Parallel computing-structure, algorithm, and Programming (3rd edition) Guoliang Chen

Two-tone sort network based on the Batcher comparator