1. CSE 291 Interconnection Networks Winter 2007 Lecture 6 February 5 2007 Prof. Chung-Kuan Cheng University of California San Diego

2. Circulant Networks G(n; S) Example: G(16; 1,4)

3. Circulant Network Properties 1. k-regular 2. Strongly connected iff it is connected 3. Strongly connected iff 4. Connectivity K(G)=k if G is connected and n is prime 5. Connectivity K(G)= if G is connected and n is not prime

4. Circulant Networks (cont.) k-ary n-fly  Butterfly k-ary n-cube  1. n-regular 2. Connectivity k = n 3. Diameter n(d-1)

5. Pyramid Networks PN(n) is adjacent to 4 vertices at level i+1 Level i is a mesh Level 0 vertex (1,1,0) is the root

6. Pyramid Network Properties 1. 2. 3. Min degree = 3, max degree = 9 4. Diameter 2n … …… … … …

7. Butterfly Networks BN(n) iff x=y or x differs from y in precisely the (i+1)th bit level

8. Ω Networks iff  (1) y is a left cyclic shift of x; or  (2) y is a left cyclic shift of x and then change the last bit Remark: The routing is identical for all

9. Ω Network is isomorphic to Butterfly Network Ω(n) BF(n) Left shift & change the last bit Change the (i+1)th bit

10. Shuffle-Exchange Networks SE(n) and are adjacent iff  (1) x & y differ in precisely the last digit; or  (2) x is a left or right cyclic shift of y Properties:  (1)  (2) 3-regular  (3) Diameter=2n-1

11. Circuit Switching Rearrangeable: connect all inputs & outputs when reroute is allowed Nonblocking in the wide sense: connect all new inputs & outputs if the routing is suitably performed Nonblocking in the strict sense: connect all new inputs & outputs with no assumptions on the routing