comparison networks sorting sorting binary values sorting arbitrary numbers implementing symmetric...
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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)
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 Network
inputs outputs
n 1Sorting
Network...... n
...
Insertion Sort Network
inputs 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
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 Numbers
inputs 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
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001
Sorting Arbitrary Numbers
0
0
1
0
0
1
0
1
0
0
1
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001
Sorting Arbitrary Numbers
0
1
1
0
1
1
0
1
1
1
1
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001011 011
Sorting Arbitrary Numbers
1
0
0
0
0
1
0
1
0
0
1
0
inputs outputs
000 000
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
0
inputs outputs
000 000
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
1
inputs outputs
000 000
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
1
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001011 011100 001101 011110 101111 111
not sorted!
Sorting Arbitrary Numbers
Sorting Arbitrary Numbers
inputs 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
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001
Sorting Arbitrary Numbers
0
0
1
0
0
1
0
1
0
1
0
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001
Sorting Arbitrary Numbers
0
1
1
0
1
1
0
1
1
1
1
0
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001011 011
Sorting Arbitrary Numbers
1
0
0
0
0
1
0
1
0
1
0
0
inputs outputs
000 000
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
0
inputs outputs
000 000
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
0
inputs outputs
000 000
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
1
inputs outputs
000 000
Try all possible 0/1 sequences.
001 001010 001011 011100 001101 011110 011111 111
Sorting Arbitrary Numbers
inputs outputs
000 000
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 Principle
Suppose
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 if
1 otherwiseiax
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!
Batcher Sorting Network, n = 4
n = 4
n = 4
Batcher Sorting Network, n = 8
Lemma 1
Any subsequence of a sorted sequence is a sorted sequence.
00011111
00111
sorted
sorted
Lemma 2
For a sorted sequence, the number of 0’s in the even subsequence is either equal to, or one greater than, the number of 0’s in the odd subsequence.
00000111
0
0
0
1
0
0
1
1
sorted
even odd
denotes the the number of 0’s in y y
For two sorted sequences and : x x
11 OEOE xxxx
denotes the even subsequence of Ey y
denotes the odd subsequence of Oy y
Lemma 3
Lemma 3
00000111x
0
0
0
1
Ex
0
0
1
1
Ox
00011111
x¢
0
0
1
1
Ex¢
0
1
1
1
Ox¢
Lemma 3
1 OEO xxx
1 OEO xxx
For two sorted sequences and : x x
11 OEOE xxxx
(by Lemma 2)
(by Lemma 2)
Merge Network
0x
1x
2x
3x
0x
1x
2x
3x
sorted
sorted
Merge[4]
Merge[4]
sorted
0y
1y
2y
3y
4y
5y
6y
7y
Merge Network (pf.)
0x
1x
2x
3x
0x
1x
2x
3x
sorted
sorted
Merge[4]
Merge[4]
sorted
sorted
(by Lemma 1)
(by Lemma 1)
Merge Network (pf.)
Merge[4]
Merge[4]
sorted
sorted
Ex
Ox
Ex
Ox
OE xx OE xx and
differ by at most 1
By Lemma 3
Merge Network (pf.)
Merge[4]
Merge[4]
sorted
0y
1y
2y
3y
4y
5y
6y
7y
Ex
Ox
Ex
Ox
OE xx OE xx and
differ by at most 1
By Lemma 3
Merge Network (pf.)
Merge[4]
Merge[4]
Ex
Ox
Ex
Ox
OE xx OE xx and
differ by at most 1
By Lemma 3
0
0
1
1
0
0
0
1
0
0
0
0
0
1
1
1
Batcher Sorting Network
0x
1x
2x
3x
4x
Sort[4]
Sort[4]
0y
1y
2y
3y
4y
5y
6y
7y
5x
6x
7x
Merge[8] sorted
Merge[4]
Batcher Sorting Network, n = 4
Sort[2]
Sort[2]
Sort[4]
Sort[4]
Batcher Sorting Network, n = 8
Merge[8]
Sorting Networks
Batcher
n2log 1830depth
AKS (Chvátal)
)log1(log 2
1depth 22 nn
AKS (Ajtai, Komlós, Szemerédi) Network:),(logdepth nO based on expander graphs.
AKS better for 36592n
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
Balancer
Asynchronous token routing device
inputs outputs
1 bit of memory
balancedtoken counts
Balancer
Snapshot
inputs outputs
1 bit of memory
x
y
2
yx
2
yx
be
a a
a
ab b
b
c
c c
cd d
e
e
e
d
d
f
g f
g
f
f
g g
Data structure for multiprocessor coordination
step sequence
Counting Network
3
1
3
0
2
2
2
1
2
2
2
1
2
2
2
1
step sequence
Counting Network
Execution trace: token counts on all wires
Counting NetworkTokens are assigned value based on the output wirenumber.
1, 5, 9, 13, . . .
2, 6, 10, 14, . . .
3, 7, 11, 15, . . .
4, 8, 12, 16, . . .
CountingNetwork
• High throughput• Low contention
Advantages
Comparator
x
y
min(x, y)
max(x, y)
inputs outputs
Balancer
x
y
inputs outputs
2
yx
2
yx
Balancer
7
2
inputs outputs
4
5
Balancing Network
3
1
3
0
2
2
2
1
2
2
2
1
2
2
2
1
depth d
width n
Smooth Sequences
BalancingNetwork
......
0x
1x
1nx
for anysmooth property: 1 ji xx ji,
Smooth Sequences
BalancingNetwork
4
3
3
3
4
3
3
3
...
for anysmooth property: 1 ji xx ji,
Step Sequences
0x
1x
1nx
BalancingNetwork
......
for anystep property: 10 ji xx ji
Step Sequences
BalancingNetwork
3
3
3
3
4
4
4
4
...
for anystep property: 10 ji xx ji
Counting Network
CountingNetwork
...
Balancing network with step output sequences:
0x
1x
1nx
...for all inputs
for any
10 ji xx
ji
Sorting vs. Counting
Counts
BalancingNetwork
......
ComparisonNetwork
......
isomorphic
Sorts?
Sorting vs. Counting
BalancingNetwork
...
Counts
...Comparison
Network...
...
isomorphic
Sorts?
Sorting vs. Counting
BalancingNetwork
...
Counts
...Comparison
Network...
...
isomorphic
Sorts?
Sorting vs. Counting
If a balancing network counts, then its isomorphiccomparison network sorts, but not vice-versa.
Theorem
Sorting vs. Counting
• By 0/1 principle, we need only consider 0/1 inputs.
Counts Sorts
x
y
2
yx
2
yx
balancer
x
y
min(x, y)
max(x, y)
comparator
Sorting vs. Counting
• By 0/1 principle, we need only consider 0/1 inputs.• A step sequence of 0’s and 1’s is a sorted sequence of 0’s and 1’s.
Counts Sorts
comparator
1
0
0
1
1
0
0
1
balancer
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
1
2
0
3
4
Insertion Sort: a network which sorts but doesn’t count.
Sorts Counts
Sorting vs. Counting
16
0
16
0
0
0
0
32
0
0
8
8
16
0
0
12
4
12
4
0
8
8
12
2
2
10
5
10
5
2
7
8
10
5
2
9
8
8
5
2
Sorts Counts
Sorting vs. Counting
be
a a
a
ab b
b
c
c c
cd d
e
e
e
d
d
f
g f
g
f
f
g g
Data structure for multiprocessor coordination
step sequence
Counting Network
3
1
3
0
2
2
2
1
2
2
2
1
2
2
2
1
step sequence
Counting Network
Execution trace: token counts on all wires
Counting NetworkTokens are assigned value based on the output wirenumber.
1, 5, 9, 13, . . .
2, 6, 10, 14, . . .
3, 7, 11, 15, . . .
4, 8, 12, 16, . . .
CountingNetwork
• High throughput• Low contention
Advantages
Batcher Counting Network
n = 4inputs
Batcher sorting network also works as a counting network!
3
1
3
0
2
2
2
1
2
2
2
1
2
2
2
1
Lemma 1
Any subsequence of a step sequence is a step sequence.
33344444
33444
step
step
Lemma 2
33344444
3
3
4
4
even
3
4
4
4odd
step
For a step sequence, the sum of the even subsequence is either equal to, or one less than, the sum of the odd subsequence.
Sum of even subseq.: 3+3+4+4=14
Sum of odd subseq.: 3+4+4+4=15
Lemma 3
1 OEO xxx
1 OEO xxx
denotes the sum of y y
For two step sequences and : x x 11 OEOE xxxx
denotes the even subsequence of Ey y
denotes the odd subsequence of Oy y
Follows from:
(by Lemma 2)
(by Lemma 2)
33344444
x
3
3
4
4
Ex
3
4
4
4
Ox
99999999
x¢
9
9
9
9
Ex¢
9
9
9
9
Ox¢
14Ex
15Ox
36Ex
36Ox
Lemma 3
Merge Network
0x
1x
2x
3x
0x
1x
2x
3x
step
step
Merge[4]
Merge[4]
step
0y
1y
2y
3y
4y
5y
6y
7y
Merge Network (pf.)
0x
1x
2x
3x
0x
1x
2x
3x
Merge[4]
Merge[4]
step
step
step
step
Merge Network (pf.)
Merge[4]
Merge[4]
step
step
Ex
Ox
Ex
Ox
OE xx OE xx and
differ by at most 1
By Lemma 3
33344444
step
Merge Network (pf.)
Merging two step sequences whose sums differ by at most one:
3
3
4
4
OE xx
3
4
4
4
OE xx
33434444
not quite step
Merge Network (pf.)
Merging two step sequences whose sums differ by at most one:
3
4
4
4
OE xx
3
3
4
4
OE xx
Merge Network (pf.)
Merge[4]
Merge[4]
step
0y
1y
2y
3y
4y
5y
6y
7y
Ex
Ox
Ex
Ox
Batcher Counting Network
0x
1x
2x
3x
4x
Count[4]
Count[4]
0y
1y
2y
3y
4y
5y
6y
7y
5x
6x
7x
Merge[8] step
Verifying that a Network Counts
A balancing network with m balancers is a counting network if and only if it satisfies the step property for all input sequences with sum .
Theorem
m2
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