Interconnection Networks Lecture 3: Topologies

1. Interconnection NetworksLecture 3: Topologies Jan. 17th, 2007 Prof. Chung-Kuan Cheng Transcribed by: Mohammad Al-Fares

2. Interconnect Issues • One pair of terminals: • Latency -> Credit-based, Buffer size • Multiple terminals: conflict on resources • Interconnect -> Topology • Router -> Switches, arbitration, buffers

3. Topology Types

4. Graph Construction • Line Graphical Method • Cayley Method • Cartesian Product Method

5. Line Graph (undirected) • Given G = (V, E): • L(G) has vertex set V(L(G)) = E(G) • Two vertices are linked by an edge iff they are adjacent as edges of G

6. a 1 (a,b) w 3 (a,c) 3 d u x 4 z 1 c 2 2 (b,c) y 4 (c,d) b G L(G) Line Graph (undirected) • Example:

7. w (a,b,c) 1 (a,b) w 3 (a,c) u (a,b,c) x (a,b,c) u x z 2 (b,c) y 4 (c,d) z (a,c,d) y (b,c,d) L(G) Line Graph (undirected) • Can be recursive (e.g. Ln(G) ) L2(G)

8. Line Graph of Digraph • Given G = (V, E): • L(G) has vertex set V(L(G)) = E(G) iff s.t. (i.e. any edge corresponds to a trace in original graph)

9. Line Graph (undirected) • Example: L(G) G(V,E)

10. Cayley Method • Given Γ a non-trivial finite group, and S a non-empty subset of Γ, without the identity element e of Γ. iff

11. Cayley Method • Example: • Γ = { 0, 1, …, n-1 } • Operation: i – j mod n • S = { 1, 3 } • V = {v0 , …, vn-1 }

12. 1 2 3 0 5 4 = Cayley Method • Example (cont.) Note:

13. Cartesian Product • Constructed by:

14. 1 01 11 00 0 10 Cartesian Product • Example:

15. Hypercube Qn • Definition: • Cartesian Product:

16. Hypercube Qn