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Properties and Applications of Circulant and Pyramid Networks in Interconnection Systems

This lecture discusses the principles of circulant networks, with a focus on G(n; S) structures and their key properties, such as k-regularity, connectivity, and diameter. The presentation illustrates examples like G(16; 1, 4) and further explores pyramid and butterfly networks. It covers their topology, defining characteristics such as adjacency patterns, routing mechanisms, and the concept of nonblocking circuit switching. These properties are crucial for understanding how these networks enable efficient data routing and interconnectivity in computational systems.

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Properties and Applications of Circulant and Pyramid Networks in Interconnection Systems

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  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

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