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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)

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Outline n Introduction n Our Schemes n Conclusions n Open Problems

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Introduction n Characteristic: in each time unit i=1,2,…,n only the ith dimensional edges can be used. n a wide class of hypercube algorithms (FFT algorithm, uniaxial algorithm,etc)

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Introduction 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 Embedding

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n Given a physical network structure and a set of required connections n 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. n Parallel transmission characteristic of WDM optical … 2 w 2 w … 1, 2, …, w Optical fiber 1 1 n Goal: Minimize the number of wavelengths

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n Parallel FFT Communication Pattern (N=2 n) u n steps: performed step by step in sequence u The communications during the ith step: performed in parallel n The number of wavelengths required to realize parallel FFT communications on optical networks is the maximum among the n steps. n Our goal is try to minimize the number of wavelengths. Introduction n 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

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Conventional embedding n Standard embedding is optimal for the traditional measure of Congestion n Embed the ith node of FFT communication on the ith node of array wavelength requirement: N/2

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Shift-reversal embedding wavelength requirement: 3N/8 reverse order Shift operation for 2 n-3 times reverse embedding

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Cross Embedding wavelength requirement: N/4+1 cross operation Cross(N L, N R ) cross order * X n is the increasing order of indices in binary representations of 2 n FFT nodes. * N L and N R : node arrangement with 2 n -1 nodes numbered from left to right in ascending order starting from 0. * Cross operation: Put node i of N R between node 2 n-2 +i and node 2 n-2 +i+1 of N L for i=0, 1, 2, …, 2 n-2 -2

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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 Our solution is based on the lattice form of hypercube.

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Lattice Embedding(2) Lattice form (n=5) For n=5 30 connections 6 dimensional i connections

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Lattice Embedding(3) layer 0 layer 1 layer 2 layer 4 layer 3 Lattice Embedding: Embed the node layer by layer

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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>=

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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<=

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Lattice Embedding(6) Stirling’s formula: =

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Lattice Embedding(7) minimum number: layer k-1 layer k layer k+1 inner-layer edge 1 number of nodes between n 0 and n j, whose ith bit is 0:

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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) n/2 layer Nodes indices Nodes Indices of array Example: FFT4 16-node optical array(4 wavelengths) the number of nodes, whose ith bit is 0, between u 0 and u j, is equal to at most n 1 /2+1. …

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Lattice Embedding(10) FFT5 32-node linear array(7 wavelengths) (n-1)/2 layer Nodes indices Nodes Indices of array (n-1)/2 layer Nodes indices Nodes Indices of array 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

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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.

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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 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.)

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Thank you!

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