Lecture 4Sorting Networks

Comparator

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A Sorting Network9

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A sorting network is a comparison network which output monotone nondecreasing sequence for every input.

Depth9

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Depth is the maximum number of comparators on apath from an input wire to an output wire.

Depth = parallel time9

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Depth is the maximum number of comparators on apath from an input wire to an output wire.

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3 1, 6, 4, ,2 5,

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Sorting network constructed from insertion sort.

3-2n depth

How to construct a sortingnetwork 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

end

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begin

ProgramMain

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Procedure

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Structure

1a 1a

1a

Sortingnetwork

Sortingnetwork

Mergingnetwork

1a

2/na

2/1 na

na

Construction of Merging Network

• 0-1 principal.

• Bitonic sorter.

• Merging network.

0-1 principal

correctly.

numbersarbitrary of sequences all sortsit then

correctly, s1' and s0' of sequence possible 2

allsort input th network wi comparison a Ifn

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Lemma

)).(),...,(),(( into ))(),...,(),((

ansformsnetwork tr the,function ingnondecreas

lly monotonicaany for then ,),...,,(

sequenceoutput theinto ),...,,( sequence

input theansformsnetwork tr comparison a If

2121

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Proof of 0-1 Principal

ion.contradict a ,0)(

before 1)( putsnetwork Then the

. if 1

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Define . before putsnetwork the,

somefor is,That correctly.not ),...,,(

seqnce a is but there correctly, sequences 1-0 all

sortsnetwork heion that tcontradictfor Suppose

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Bitonic Sequence

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.decreasing

monotone then and increasing monotone

first become toshifted circularly becan

or decreasing monotone then and increasing

monotonefirst isit if bitonic is ),...,,( 21 naaa

Bitonic 0-1 Sequence

kji 010

kji 101

Some Properties

).,...,,(),...,,(

is so bitonic, is ),...,,( If (2)

bitonic.

isit then ,increasing monotone then and

decreasing monotonefirst is ),...,,( If (1)

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The half-cleaner

bitonic

bitonicclean

bitonic

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The half-cleaner

bitonic

bitonicclean

bitonic

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Lemma

(a)One of two halfs is bitonic clean.

(b) every number in the 1st half ≤ any element in the 2nd half.

Proof (case 1)

1

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0 1

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0

Proof (case 2)

1

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Proof (case 3)

1

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1

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1

Proof (case 4)

1

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01

0

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1

Proof (case 5)

1

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1

1

1

1

1

1

Proof (case 6)

1

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Proof (case 7)

1

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Proof (case 8)

1

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1

11

00

Half cleaners

bitonic sorted

Half cleaners

sorted

sorted

sorted

0

0

11

0001

0

0000

11

1

Merging Network )(lg depth nO

Structure

1a 1a

1a

Sortingnetwork

Sortingnetwork

Mergingnetwork

1a

2/na

2/1 na

na

Merging Networks

Sorting Network

))((lg depth 2nO

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 betweenInputs and outputs.

3-stage Clos Network

1

m

n

n

n

n

Rearrangeability

.

ifonly and if blerearrangea isnetwork Clos stage-3

outputs. and inputsbetween matchingany

realizecan it if blerearrangea isnetwork switchingA

nm

Theorem

Network with 2x2 crossbars

y.recursivel dconstructe becan crossbars

22only th network wi )lg3(-depth blerearrangeaA n

Puzzle

d?constructejust that we

network )(lg-depth for the thisdo can we ,Especially

?comparator ah switch witcrossbar 22

each replaceby network sorting a into switchscrossbar

22only th network win permutatioevery transformCan we

switch.crossbar 22 a with comparatorevery

replacingby network) switching lerearrageab a (i.e.,network

n permutatio a into ed transformbecan network sortingAny

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