Lecture 4 Sorting Networks Comparator comparator A Sorting

Lecture 4 Sorting Networks

Comparator comparator

A Sorting Network 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 A sorting network is a comparison network which output monotone nondecreasing sequence for every input.

Depth 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 Depth is the maximum number of comparators on a path from an input wire to an output wire.

Depth = parallel time 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 Depth is the maximum number of comparators on a path from an input wire to an output wire.

Insertion Sort

key

Sorting network constructed from insertion sort.

How to construct a sorting network from merging sort?

Divide and Conquer • Divide the problem into subproblems. • Conquer the subproblems by solving them recursively. • Combine the solutions to subproblems into the solution for original problem.

Merge Sort

Procedure

Structure Sorting network Merging network Sorting network

Construction of Merging Network • 0 -1 principal. • Bitonic sorter. • Merging network.

0 -1 principal

Lemma

Proof of 0 -1 Principal

Bitonic Sequence

Bitonic 0 -1 Sequence

Some Properties

The half-cleaner bitonic 0 0 1 0 1 1 0 0 0 1 bitonic clean bitonic

The half-cleaner bitonic 0 0 1 1 1 0 1 bitonic clean

Lemma (a)One of two halfs is bitonic clean. (b) every number in the 1 st half ≤ any element in the 2 nd half.

Proof (case 1) 0 0 0 1 0 1 0

Proof (case 2) 0 0 1 0 0 0 1

Proof (case 3) 0 0 0 1 1 1 0 0 1

Proof (case 4) 0 0 0 1 1 0 0 0 1

Proof (case 5) 1 0 1 0 1 1 1

Proof (case 6) 1 1 0 1 0 1 1

Proof (case 7) 0 1 1 0 0 1 1

Proof (case 8) 0 1 1 0 1

bitonic sorted Half cleaners

sorted 0 0 0 1 0 0 0 sorted 0 1 1 1 0 1 1 Half cleaners Merging Network

Structure Sorting network Merging network Sorting network

Sorting Network Merging Networks

What we learnt in this lecture? • What is sorting network? • Depth = parallel time. • Sorting network from Merge sort.

Permutation Network • Switching network • Rearrangeability • Nework with 2 x 2 crossbars

Crossbar Switch A crossbar switch can realize any matching between Inputs and outputs.

3 -stage Clos Network 1 n n m

Rearrangeability Theorem

Network with 2 x 2 crossbars

Puzzle