Efficient Realization of Hypercube Algorithms on Optical Arrays*
Hong Shen
Department of Computing & MathsManchester Metropolitan University, UK
( Joint work with Yawen Chen done at JAIST)
Outline
Introduction Our Schemes Conclusions Open Problems
Introduction
000 001
011
100 101
110111
010
D2
D3
D3D2
D3
D3
D2
D2
D1
D1D1
D1
000
001
011
100
101
110
111
010
1 2 3
Characteristic: in each time unit i=1,2,…,n only the ith dimensional edges can
be used.
a wide class of hypercube algorithms (FFT algorithm, uniaxial algorithm,etc)
Introduction
11
1 2 3 4 5 6 70
1 1
2 2 2 2
3 3 3 31 2 3 4 5 6 7
Example:8-node hypercube embedded on 8-node linear arrayStandard embedding (optimal for traditional measure of congestion, Congestion= 5 link3)Step1: 4 edges on link 4Step2: 2 edges on link 2, 6Step3: 1 edge on link 1,3,5,7
Embedding
000
001
011
100
101
110
111
010
1 2 3
Given a physical network structure and a set of required connections Select a suitable path for each connection and assign a wavelength to the
path, such that the following two constraints are satisfied:
Introduction
1.Wavelength continuity constraint
---- a lightpath must use the same wavelength on all the links along its path from source to destination node.
2. Distinct wavelength constraint
---- all lightpaths using the same link (fiber) must be assigned distinct wavelengths.
Parallel transmission characteristic of WDM optical
…2
w
2
w
…1, 2, …, w
Optical fiber
1 1
Goal: Minimize the number of wavelengths
Parallel FFT Communication Pattern (N=2n)
n steps: performed step by step in sequence The communications during the ith step: performed in parallel
The number of wavelengths required to realize parallel FFT communications on optical networks is the maximum among the n steps.
Our goal is try to minimize the number of wavelengths.
Introduction
What is the minimum number of wavelengths to realize parallel FFT communication on some regular WDM optical networks?
Number of wavelengths for realizing FFT on optical networks on G>=Dimensional Congestion of hypercube on G
Conventional embedding
Standard embedding is optimal for the traditional measure of Congestion
Embed the ith node of FFT communication on the ith node of array
11
1 2 3 4 5 6 70
1 1
2 2 2 2
3 3 3 31 2 3 4 5 6 7
wavelength requirement: N/2
Shift-reversal embedding
wavelength requirement: 3N/8
11
1 23 456 70
112
222
3 3 331 2 3 4 5 6 7
1
1
1 23 456 7 0
11
2
222
3
3 3371 2 3 4 5 6
reverse order
Shift operation for 2n-3 timesreverse embedding
Cross Embedding
wavelength requirement: N/4+111
1 2 34 5 6 70
1 1
2 2 2 2
3 3 3 31 2 6 73 54
cross operation
Cross(NL, NR)
cross order
NL NR
* Xn is the increasing order of indices in binary representations of 2n FFT nodes.
* NL and NR: node arrangement with 2n-1 nodes numbered from left to right in ascending order starting from 0.
* Cross operation: Put node i of NR between node 2n-2+i and node 2n-2+i+1 of NL for i=0, 1, 2, …, 2n-2-2
Lattice Embedding(1)
k=0
kth layer
k=n
k+1
Nodes
connections
dimensional i connections
For n=4
12 connections
3 dimensional i connections
2
2
00010010
0011
0100
01010110
0111
0000
2
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33
3
4
44
44
4
1100 1010
1011
1001
11011110
1111
1000
2
2
3
3
3
2
42
4
1
11
11
1
1
3 21
Our solution is based on the lattice form of hypercube.
Lattice Embedding(2)
3
2
0001100101
0011
01001
01010110
0111
00001
2
3
33
3
4
44
44
4
1100 1010
1011
1001
11011110
11111
10001
2
2
3
3
3
2
42
4
1
11
11
1
1
3 21
3
2
0001000100
00110
01000
0101001100
01110
00000
2
3
33
3
4
44
44
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11000 10100
10110
10010
1101011100
11110
10000
2
2
3
3
3
2
42
4
1
11
11
1
1
3 21
5
5 5 5 5
5 5 5 5 5 5
5
5
5 5 5
Lattice form (n=5)
For n=5
30 connections
6 dimensional i connections
Lattice Embedding(3)
layer 0layer 1
layer 2
layer 4
layer 3
2
2
00010010
0011
0100
01010110
0111
0000
2
3
33
3
4
44
44
4
1100 1010
1011
1001
11011110
1111
1000
2
2
3
3
3
2
42
4
1
11
11
1
1
3 21
Lattice Embedding:
Embed the node layer by layer
22
00010010 00110100 0101 0110 01110000
2
3 3 3 3
4 44 44
1100 1010 10111001 11011110 11111000
22
3 3 3
2
4
2
4
1
111 1
1
32
11
4
Lattice Embedding(4)
layer k-1 layer k layer k+1
Proof: Number of wavelengths>=dimensional edges passing the inter-layer edges
* inner-layer edges: the edges on optical array connecting the nodes embedded within the same layer
inter-layer edge
… … … …
W>=
dimensional i connections
W>=
Lattice Embedding(5)
layer k-1 layer k layer k+1
Proof: Number of wavelengths<=dimensional edges passing the inner-layer edges
* inner-layer edges: the edges on optical array connecting the nodes embedded within the same layer
inner-layer edge
… … … ……
W<=
W<=
Lattice Embedding(6)
Stirling’s formula:
=<W<=
Wavelength requirement:
Lattice Embedding(7)minimum number:
layer k-1 layer k layer k+1inner-layer edge
… … … ……
wwp p
n1 n2
2i
1i
1i 2
i
u0 uj
1
number of nodes between n0 and nj, whose ith bit is 0:
Lattice Embedding(8)
2
2
00010010
0011
0100
01010110
0111
0000
2
3
33
3
4
44
44
4
1100 1010
1011
1001
11011110
1111
1000
2
2
3
3
3
2
42
4
1
11
11
1
1
3 21
for n is even, each node has n/2 0s on the n/2th row :
2
1
For n is even W
Minimum can be achieved when
Lattice Embedding(9)
0011 0101 01101100 1010 1001
11 1 1
1 1
n/2-1 l ayer n/2+1 l ayer
n/2 l ayer
22 22 2
2
3
33 3
3
3
4
4
444
4
n/2 layer Nodes indices 0011 1100 0101 1010 1001 0110
Nodes Indices of array 5 6 7 8 9 10
Example: FFT4 16-node optical array(4 wavelengths)
the number of nodes, whose ith bit is 0, between u0 and uj , is equal to at most n1/2+1.
… …n1 n2
u0 uj
…
Lattice Embedding(10)
FFT5 32-node linear array(7 wavelengths)
(n-1)/2 layer Nodes indices 00011 01100 10001 00110 11000
Nodes Indices of array 6 7 8 9 10
(n-1)/2 layer Nodes indices 00101 01010 10100 01001 10010
Nodes Indices of array 11 12 13 14 15
for n is odd, each node has (n+1)/2 0s on the (n-1)/2th row :
For n is odd, W
Minimum can be achieved when
ConclusionsConclusions
We provided a new measure, dimensional congestion, for embedding hypercube on other graphs.
This new measure has great significance in practice. Wavelength requirement analysis of parallel FFT communication on optical networks is an interesting example.
We have proposed several schemes for embedding parallel FFT on optical networks. The results outperforms the traditional embedding schemes for embedding hypercube on other graphs, such as standard embedding, xor embedding.
Open ProblemsOpen Problems
What is the optimal value of dimensional congestion on array or other topologies?
How can we find the embedding schemes which can achieve the theoretical lower bound?
One obvious lower bound for dimensional congestion on linear array is dimensional bisection Ω(NloglogN/logN).
("Introduction to parallel algorithms and architectures: array, trees,
hypercubes” Problem 3.8 Show that any bisection of an N-node hypercube requires the
removal of at least Ω(NloglogN/logN) dimension d edges for some d<=logN.)
Thank you!Thank you!