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

2. Interconnect Issues 1. One pair of terminals: Latency -> Credit-based, Buffer size 2. Multiple terminals: conflict on resources Interconnect -> Topology Router -> Switches, arbitration, buffers

3. Topology Types Type# nodes# edgesdegreesdiameterconnectivity Hypercube2n2n n 2 n-1 nnn De Bruijndndn d n+1 dnd-1 Kautzd n + d n-1 d n+1 + d n dnd Circulantdndn n d n nn (d-1)n Butterfly(n+1) d n n 2 n+1 4n- Pyramid⅓ (4 n+1 -1)4 (4 n - 2 n )(3, 9)2n-

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

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

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 = {v 0, …, v n-1 }

12. Cayley Method Example (cont.) 0 12 3 45 = Note:

13. Cartesian Product Constructed by:

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

15. Hypercube Q n Definition: (1) Cartesian Product: (2)

16. Hypercube Q n