efficient realization of hypercube algorithms on optical arrays* hong shen department of computing...

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

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

Our solution is based on the lattice form of hypercube.


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

4

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


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


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>=


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<=


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:


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


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

…


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!

efficient realization of hypercube algorithms on optical arrays* hong shen department of computing...

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