Efficient Realization of Hypercube Algorithms on Optical Arrays*

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 i=1,2,…,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 1 1 • Parallel transmission characteristic of WDM optical 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 (N=2n) • n steps: performed step by step in sequence • The communications during the ith step: performed in parallel • 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 • 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.

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-2+i and node 2n-2+i+1 of NL for i=0, 1, 2, …, 2n-2-2 * Xn is the increasing order of indices in binary representations of 2n FFT nodes. cross order wavelength requirement: N/4+1

10. Lattice Embedding(1) • Our solution is based on the lattice form of hypercube. k=0 kth layer Nodes connections dimensional i connections k+1 For n=4 12 connections 3 dimensional i connections k=n

11. Lattice Embedding(2) Lattice form (n=5) For n=5 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) W>= 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 W>= dimensional i connections inter-layer edge layer k+1 layer k layer k-1

14. Lattice Embedding(5) W<= 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 W<= inner-layer edge layer k+1 layer k layer k-1

15. Lattice Embedding(6) =<W<= Stirling’s 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 k+1 layer k layer k-1

17. Lattice Embedding(8) for nis even,each node has n/2 0s on the n/2th row : 2 1 For n is even W Minimum can be achieved when

18. Lattice Embedding(9) the number of nodes, whose ith bit is 0, between u0 and uj , is equal to at most n1/2+1. … Example: FFT4 16-node optical array(4 wavelengths)

19. Lattice Embedding(10) for nis odd,each node has (n+1)/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)

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 bisectionΩ(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.)

22. Thank you!