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Mapping of one model into other model

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Mapping of one model into other model

1. 1. MAPPING OF ONE MODEL INTO OTHER MODEL<br />Submitted by:-<br />RATIKA AGARWAL<br />M.Tech.-CSE<br />
2. 2. Embedding:<br />Relationship between two networks<br /> G(V,E)  G’(V’,E’)<br />Graph G, G’<br />Vertices V, V’<br />Edges E, E’<br />MAP<br />E E’<br />V  V’<br />
3. 3. Process Mapping (Eg.- 2-d Mesh)<br />Underlying Architecture<br />Processes and their interactions<br />Mapping of Processes to nodes<br />
4. 4. Need of Mapping:<br />G(V,E)  G’(V’,E’)<br /><ul><li>Communication structure of an algorithm differs from the interconnection architecture of the intended parallel machine.
5. 5. Number of processes required by the algorithm exceeds the number of processors available.</li></li></ul><li>Types of network topologies: (Continued..)<br />Linear Arrays<br />2-d Mesh<br />Binary Tree<br />
6. 6. Types of network topologies: (Hypercube)<br />2-d Hypercube<br />0-d Hypercube<br />1-d Hypercube <br />3-d Hypercube<br />4-d Hypercube<br />
7. 7. Mapping techniques for graphs:<br />Topology embedding:<br />Embed a communication pattern into a given interconnection topology.<br /><ul><li>Linear Array in a Hypercube .
8. 8. 2-D Mesh in a Hypercube.
9. 9. Linear Array in a 2-D Mesh.
10. 10. 2-D Mesh in a Linear Array.</li></ul> And more…<br />
11. 11. Metrics used forEmbedding:<br />DILATION<br />CONGESTION<br />EXPANSION<br />
12. 12. <ul><li>Expansion</li></ul>Expansion = No.of Nodes in V’ = |V’| / |V|<br /> No.of Nodes in V<br />Where ( |V|<|V’| )<br />
13. 13. <ul><li>Congestion</li></ul>Congestion of k: ‘k’Es to one E’<br />k>1<br />
14. 14. <ul><li>Dilation</li></ul>Dilation of k: One E to ‘k’ E’s.<br />
15. 15. MAPPING<br />
16. 16. Linear Array/Ring into Hypercube<br />MAP<br />Linear Array 2d nodes<br />d-dimesional Hypercube<br />Node ‘i’<br />Node G(i,d)<br />G(i, d) : Gray code of ith node of Array.<br />RGC : Binary Reflected Gray Code.<br />G(0,1) = 0<br />G(1,1) = 1<br />G(i,x) for i<2x<br />G(i,x+1)<br />2x + G[ (2x+1 –1)-i, x] for i>= 2x<br />
17. 17. Example: Say d = 3<br />2d = 23= 8, i.e., 8 nodes in ring and hypercube.<br />
18. 18. Gray Coding<br />ALTERNATIVE METHOD<br />3 bit RGC<br />2 bit RGC<br />1 bit RGC<br />0<br />1<br />2<br />3<br />4<br />5<br />6<br />7<br />0 0 0 0 00 <br />1 0 1 0 01 <br /> 1 1 0 11<br /> 1 0 0 10 <br /> 1 10 <br /> 1 11 <br /> 1 01<br /> 1 00<br />G(0,1) = 0<br />G(1,1) = 1<br />Congestion=1<br />Dilation=1<br /><Mapped Model><br />
19. 19. Gray Coding<br />ALTERNATIVE METHOD<br />3 bit RGC<br />2 bit RGC<br />1 bit RGC<br />0<br />1<br />2<br />3<br />4<br />5<br />6<br />7<br />0 0 0 0 00 <br />1 0 1 0 01 <br /> 1 1 0 11<br /> 1 0 0 10 <br /> 1 10 <br /> 1 11 <br /> 1 01<br /> 1 00<br />G(0,1) = 0<br />G(1,1) = 1<br />Congestion=1<br />Dilation=1<br /><Mapped Model><br />
20. 20. 2-D Mesh into Hypercube<br /><ul><li>Wraparound Mesh</li></ul>MAP<br />2r*2s nodes of Mesh (r+s) dimension Hypercube of 2r+s nodes<br />Node (i, j) Node G(i, r) || G(j, s)<br />MAP<br /><ul><li> ‘||’ is concatenation of two Gray codes</li></li></ul><li>Example for 2*4 mesh…<br />Dilation=Congestion =1<br />
21. 21. <ul><li>Same row of Mesh  ‘r’ identical MSBs in mapped node
22. 22. Same column of Mesh  ‘s’ identical LSBs in mapped node</li></ul>00 00<br />01 00<br />11 00<br />10 00<br />10 11<br />10 10<br />10 01<br />
23. 23. Linear Array into 2-D Mesh<br />MAP<br />16 node linear array 2-d Mesh <br />Solid lines: links in array<br />Normal lines: links in the mesh<br />Congestion=1<br />
24. 24. 2-D Mesh into Linear Array<br />Congestion =5<br />Congestion =5<br />Congestion =5<br />Solid lines: links in array<br />Normal lines: links in the mesh<br /><ul><li> Inversion of the previous mapping.</li></li></ul><li>Summary<br /><ul><li>Applications of Embedding:
25. 25. To minimize the communication overhead of a parallel algorithm.
26. 26. Laying out circuits on chips.
27. 27. Reasons for embedding:
28. 28. An algorithm can run on many architecture.
29. 29. Algorithm mapped on interconnection network to achieve maximum parallelism.</li></ul>THANK YOU<br />