MSc/MRes courses

1
Efficient Realization of Hypercube Algorithms on
Optical Arrays
Hong Shen Department of Computing
Maths Manchester Metropolitan University, UK (
Joint work with Yawen Chen done at JAIST)
2
Outline

• Introduction
• Our Schemes
• Conclusions
• Open Problems

3
Introduction

• a wide class of hypercube algorithms (FFT
  algorithm, uniaxial algorithm,etc)

• Characteristic
• in each time unit i1,2,,n only the ith
  dimensional edges can be used.

4
Introduction
Embedding
Example 8-node hypercube embedded on 8-node
linear array Standard embedding (optimal for
traditional measure of congestion, Congestion 5
link3) Step1 4 edges on link 4 Step2 2 edges on
link 2, 6 Step3 1 edge on link 1,3,5,7
5
Introduction

• Parallel transmission characteristic of WDM
  optical

?1
?1
?2
?2
?1, ?2, , ?w


?w
?w
Optical fiber

• 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

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.

• Goal Minimize the number of wavelengths

6
Introduction

• Parallel FFT Communication Pattern (N2n)
• 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.

• 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 GgtDimensional Congestion of
  hypercube on G

7
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

wavelength requirement N/2
8
Shift-reversal embedding
reverse order
Shift operation for 2n-3 times
reverse embedding
wavelength requirement 3N/8
9
Cross Embedding
cross operation Cross(NL, NR)
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-2i and node 2n-2i1 of NL
for i0, 1, 2, , 2n-2-2
Xn is the increasing order of indices in
binary representations of 2n FFT nodes.
cross order
wavelength requirement N/41
10
Lattice Embedding(1)

• Our solution is based on the lattice form of
  hypercube.

k0
kth layer
Nodes
connections
dimensional i connections
k1
For n4 12 connections 3 dimensional i
connections
kn
11
Lattice Embedding(2)
Lattice form (n5)
For n5 30 connections 6 dimensional i
connections
12
Lattice Embedding(3)
Lattice Embedding Embed the node layer by layer
layer 2
layer 3
layer 1
layer 0
layer 4
13
Lattice Embedding(4)
Wgt
Proof Number of wavelengthsgtdimensional edges
passing the inter-layer edges inner-layer
edges the edges on optical array connecting the
nodes embedded within the same layer
Wgt
dimensional i connections
inter-layer edge
layer k1
layer k
layer k-1
14
Lattice Embedding(5)
Wlt
Proof Number of wavelengthsltdimensional edges
passing the inner-layer edges inner-layer
edges the edges on optical array connecting the
nodes embedded within the same layer
Wlt
inner-layer edge
layer k1
layer k
layer k-1
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Lattice Embedding(6)
ltWlt
Stirlings formula
Wavelength requirement
16
Lattice Embedding(7)
minimum number
1
number of nodes between n0 and nj, whose ith bit
is 0
inner-layer edge
layer k1
layer k
layer k-1
17
Lattice Embedding(8)
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
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Lattice Embedding(9)
the number of nodes, whose ith bit is 0, between
u0 and uj , is equal to at most n1/21.

Example FFT4 ?16-node optical array(4
wavelengths)
n/2 layer Nodes indices 0011 1100 0101 1010 1001 0110
Nodes Indices of array 5 6 7 8 9 10
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Lattice Embedding(10)
for n is odd, each node has (n1)/2 0s on the
(n-1)/2th row
For n is odd, W
Minimum can be achieved when

• 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
20
Conclusions

• 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.

21
Open 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 O(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 O(NloglogN/logN) dimension d
edges for some dltlogN.)
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Thank you!