Lecture 4 Sorting Networks

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Presentation transcript:

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 clean 1 1 bitonic 1 1 bitonic 1 1

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

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

Proof (case 1) 1 1 1

Proof (case 2) 1 1 1

Proof (case 3) 1 1 1 1 1

Proof (case 4) 1 1 1 1 1

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

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

Proof (case 7) 1 1 1 1 1 1

Proof (case 8) 1 1 1 1 1 1

bitonic sorted Half cleaners

sorted 1 sorted 1 sorted 1 1 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 2x2 crossbars

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

3-stage Clos Network 1 n n n n m

Rearrangeability Theorem

Network with 2x2 crossbars

Puzzle