lecture 4 sorting networks. comparator comparator

Lecture 4Sorting Networks

Comparator

x ),min(' yxx

ycomparator

),max(' yxy

),min(' yxx

),max(' yxy

x

y

time)1(O

A Sorting Network9

5

2

6

5

9

2

6

2

5

6

9

2

5

6

9

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

Depth9

5

2

6

5

9

2

6

2

5

6

9

2

5

6

9

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

Depth = parallel time9

5

2

6

5

9

2

6

2

5

6

9

2

5

6

9

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

for.-end

;]1[

;1

and ][]1[ do

][ and 0 while

;1

];[begin do

][ to2for

keyiA

ii

iAiA

keyiAi

ji

jAkey

Alengthj

Insertion SortAarray input

3 1, ,6 5, 4, 2,

3 1, 6, 5, ,4 2,

3 1, 6, ,4 5, 2,

3 1, 6, 4, 5, ,2

3 1, 6, 4, ,2 5,

�

�

�

�

�

6 5, 4, ,3 2, 1,

6 5, ,3 4, 2, 1,

6 ,3 5, 4, 2, 1,

3 6, 5, 4, 2, 1,

�

�

�

�

3 6, 5, 4, 2, ,1

3 6, 5, 4, ,1 2,

3 6, 5, ,1 4, 2,

3 6, ,1 5, 4, 2,

3 ,1 6, 5, 4, 2,

�

�

�

�

�key

5a

1a

2a

6a

3a

4a

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

);,1,(Sort-Merge

begin

ProgramMain

]....[subarray in elements sortswhich

procedure a be ),,(Sort-MergeLet

nA

rpA

rpA

Procedure

then.-end

);,,,(Merge

);,1,(Sort-Merge

);,,(Sort-Merge

;2/)(begin then

if

),,(Sort-Merge

rqpA

rqA

qpA

rpq

rp

rpA

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

n

Lemma

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

ansformsnetwork tr the,function ingnondecreas

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

sequenceoutput theinto ),...,,( sequence

input theansformsnetwork tr comparison a If

2121

21

21

nn

n

n

bfbfbfafafaf

f

bbb

aaa

)),(min())(),(min( yxfyfxf

)(yf

)(xf

)),(max())(),(max( yxfyfxf

Proof of 0-1 Principal

ion.contradict a ,0)(

before 1)( putsnetwork Then the

. if 1

, if 0)(

Define . before putsnetwork the,

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

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

sortsnetwork heion that tcontradictfor Suppose

21

i

j

i

i

ijji

n

af

af

ax

axxf

aaaa

aaa

Bitonic Sequence

,8).(3,2,1,4,6 ,2),(1,4,6,8,3

.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)

1121

21

21

aaaaaa

aaa

aaa

nnR

n

n

n

The half-cleaner

bitonic

bitonicclean

bitonic

0

0

0

0

1

0

1

1

0

0

1

1

1

0

0

0

The half-cleaner

bitonic

bitonicclean

bitonic

0

0

1

0

1

1

1

1

0

0

1

1

1

1

1

0

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

0

0

10

0 1

0

0

Proof (case 2)

1

0

0

1

0

0

1

0

0

Proof (case 3)

1

0

0

1

0

0

1

0

0

1

1

Proof (case 4)

1

0

0

1

0

01

0

01

1

Proof (case 5)

1

0

0

1

0

1

1

1

1

1

1

Proof (case 6)

1

0

0

1

0

1

1

1

1

1

1

Proof (case 7)

1

0

0

1

0

1

1

1

1

0

0

Proof (case 8)

1

0

0

1

0

1

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

nO