Comparison Networks

Comparison Networks

Sorting• Sorting binary values• Sorting arbitrary numbers• Implementing symmetric functions

Sorting Algorithms

)log ( nnO comparisons to sort n elements

)(mO comparisons required to merge two arrays of size m/2

Order of comparisons not fixed in advance.

Mergesort(array[1,…,n] of Integers):begin Mergesort(array[1,…,n/2]); Mergesort(array[n/2+1,…,n]); Merge(array[1,…,n/2], array[n/2+1,…,n]);end

Not readily implementable in hardware.

Example

Sorting Networks

Order of comparisons fixed in advance.Readily implementable in hardware.

)log ( scomparison ofnumber ,)(logdepth 22 nnOnO

CDBA

ABCD

SortingNetwork

Sorting Networks (binary values)

SortingNetwork

10010011

00001111

inputs outputs

sorted

Comparator (2-sorter)

x

y

min(x, y)

max(x, y)

inputs outputs

C

Comparator (2-sorter)

inputs

x

y

min(x, y)

max(x, y)

AON Implementation

outputs

Comparator (2-sorter)

x

y

min(x, y)

max(x, y)

inputs outputs

Comparison Network

0

0

1

1

0

0

1

1

0

0

1

1

1

1

0

0

depth d

width n

Comparison Network

0

0

1

1

0

0

1

1

0

0

1

1

1

1

0

0

n / 2comparisonsper stage

d stages

Sorting NetworkAny ideas?

Sorting Networkinputs outputs

n 1Sorting

Network...... n

...

Insertion Sort Networkinputs outputs

depth 2n 3

Batcher Sorting NetworkNext Lecture

)(logdepth 2 nO

Sorting Arbitrary Numbers

x, y can be values from any linearly ordered set,e.g., integers, reals, etc.

inputs outputs

x

y

min(x, y)

max(x, y)

Comparison function:

C(X,Y) =1 if X > Y,0 otherwise.

Idea: use C(X,Y) to select the min and the max of X and Y.

X, Y: integers represented as m-bit binary strings.

Integer Comparator

Integer ComparatorX

C(X, Y)

YC(X, Y)

min(X, Y)

XC(X, Y)

YC(X, Y)

max(X, Y)

Sorting Arbitrary Numbers

9

6

2

2

6

9

2

9

6

sorted6

9

2

Sorting Arbitrary Numbers

1

4

5

1

4

5

1

5

4

sorted4

5

1

Sorting Arbitrary Numbers

3

0

7

3

0

7

3

7

0

0

7

3

not sorted

How can we verify if a network sorts all possible input sequences?

Sorting Arbitrary Numbersinputs outputs

Try all possible 0/1 sequences.

Sorting Arbitrary Numbers

0

0

0

0

0

0

0

0

0

0

0

0 000 000

inputs outputs

Try all possible 0/1 sequences.

Sorting Arbitrary Numbers

0

1

0

0

0

1

0

0

1

0

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001

Sorting Arbitrary Numbers

0

0

1

0

0

1

0

1

0

0

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001

Sorting Arbitrary Numbers

0

1

1

0

1

1

0

1

1

1

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011

Sorting Arbitrary Numbers

1

0

0

0

0

1

0

1

0

0

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001

Sorting Arbitrary Numbers

1

1

0

0

1

1

0

1

1

1

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011

Sorting Arbitrary Numbers

1

0

1

1

0

1

1

1

0

0

1

1inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011110 101

not sorted!

Sorting Arbitrary Numbers

1

1

1

1

1

1

1

1

1

1

1

1inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011110 101111 111

not sorted!

Sorting Arbitrary Numbers

Sorting Arbitrary Numbersinputs outputs

Try all possible 0/1 sequences.

Sorting Arbitrary Numbers

0

0

0

0

0

0

0

0

0

0

0

0 000 000

inputs outputs

Try all possible 0/1 sequences.

Sorting Arbitrary Numbers

0

1

0

0

0

1

0

0

1

0

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001

Sorting Arbitrary Numbers

0

0

1

0

0

1

0

1

0

1

0

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001

Sorting Arbitrary Numbers

0

1

1

0

1

1

0

1

1

1

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011

Sorting Arbitrary Numbers

1

0

0

0

0

1

0

1

0

1

0

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001

Sorting Arbitrary Numbers

1

1

0

0

1

1

0

1

1

1

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011

Sorting Arbitrary Numbers

1

0

1

0

1

1

1

1

0

1

1

0inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011110 011

Sorting Arbitrary Numbers

1

1

1

1

1

1

1

1

1

1

1

1inputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011110 011111 111

Sorting Arbitrary Numbersinputs outputs000 000

Try all possible 0/1 sequences.

001 001010 001011 011100 001101 011110 011111 111

all sorted!

Zero-One Principle

If a comparison network sorts all possible sequencesof 0’s and 1’s correctly, then it sorts all sequencesof arbitrary numbers correctly.

Lemma

0a

1a

Given

0b

1b

For a monotonically increasing function f,

Lemma

0a

1a

Given

0b

1b

For a monotonically increasing function f,

)( 0af

)( 1af

)( 0bf

)( 1bf

Proof: Lemma

),min( 10 aa

),max( 10 aa

0a

1a

Proof: Lemma

))(),(min( 10 afaf

))(),(max( 10 afaf

)( 0af

)( 1af

Proof: Lemma

f is monotonically increasing:yx )()( yfxf

)( 0af

)( 1af

))(),(min( 10 afaf

))(),(max( 10 afaf

Proof: Lemma

f is monotonically increasing:yx )()( yfxf

)),(min( 10 aaf

)),(max( 10 aaf

)( 0af

)( 1af

Proof: Lemma

f is monotonically increasing:

)( 0bf

)( 1bf

)( 0af

)( 1af

yx )()( yfxf

Generalization

0a

1a

2a

0b

1b

2b

0c

1c

2c

0d

1d

2d

Given

For a monotonically increasing function f,

)( 0af

)( 1af

)( 2af

)( 0bf

)( 1bf

)( 2bf

)( 0cf

)( 1cf

)( 2cf

)( 0df

)( 1df

)( 2df

Generalization

(by induction)

Proof: Zero-One PrincipleSuppose

b) there exists a sequence that it doesn’t sort, i.e., 110 ,...,, naaa

ji aa , such that ji aa but is placed before in the output.ja ia

a) the network sorts all sequences of 0’s and 1’s,

Define

f (x) =0 if1 otherwise

iax

Proof: Zero-One Principle

0a

1a

1na

0a

1a

1na

ja

ia

SortingNetwork

......

...

Proof: Zero-One Principle

SortingNetwork

)( 0af)( 1af

)( 1naf

)( 0af )( 1af

)( 1naf

)( jaf)( iaf

......

...

Proof: Zero-One Principle

SortingNetwork

)( 0af)( 1af

)( 1naf

)( 0af )( 1af

)( 1naf

......

...

10

contradiction!

Implementing XOR

SortingNetwork

00000001

00000111

00011111

X

01111111

XOR(X) =1 if odd0 otherwise

X

Implementing XOR

SortingNetworkX

XOR(X) =1 if odd0 otherwise

X

Symmetric Functions

SortingNetworkX

f (X) =1 if0 otherwise

Xfor some subset },...,0{ nof