CSE 291 a Interconnection Networks Lecture 7 February

CSE 291 -a Interconnection Networks Lecture 7: February 7, 2007 Prof. Chung-Kuan Cheng CSE Dept, UC San Diego Winter 2007 Transcribed by Thomas Weng

Topics • Circuit Switching - Definitions: Nonblocking, rearrangeable, strict. • • Crossbar Clos Networks

Crossbar • • • n inputs, n outputs, n 2 switches Rearrangeable, strict and wide nonblocking If n is small, this is usually the best choice. 1 x x x 2 x x x n x x x 1 2 n

Engineering Issues 1. 2. Physical layout (what to do with many nodes? ) Control (packet switching) Paper - Black. Widow: High-Radix Clos Networks, S. Scott, D. Abts, J. Kim, W. J. Dally 1 2 n 1 2 3 n

Physical Layout (example) 8 x 8 crossbar . . . . 1 8 . . . . Goes to row 1, row 2, … , row 8 8 wires per row 64 horizontal wires 64 wires in between each signal 8 x 64 vertical wires in all 64 8 x 8 switches

Clos Network: Three Stage Clos(m, n, r) n n x 1 m 1 1 2 2 2 . . r m r n n n

Clos Network (continued) 1. 2. rn inputs, rn outputs 2 rnm + mr 2 switches (this is less than r 2 n 2) Clos(m, n, r) is rearrangeable iff m >= n Let m = n 1. rn inputs 2 rn 2 + nr 2 switches = (2 n + r)rn (a crossbar is rn 2 switches) Optimal choice of n and r?

Clos Network - Proof: By induction Clos(1, 1, r) – you have r boxes, each box is 1 x 1 n n n 1 2 1 r r 2 . . r r This is a crossbar, which we know is rearrangeable.

Clos Network – Proof (cont) Assume that for the case Clos(n-1, r), n>=2, the statement is true. For the case Clos(n, n, r), we use the first switch in the middle to reduce the requirement to Clos(n-1, r). n n n x m 1 1 1 . . r n n

Clos Network – Proof (cont) Permutation 1 2. n Output side (i, j) for p(j) = i p 1 p 2 Each box is a node with degree=n n n 1 . pn n . 2 n rx n Each box will have n input edges 1 2 3 4 5 1 Bipartite graph 2 n (r-1)n+1. . n 2 n+1. Each box will have n outputs n n n 1 2 3 4 5 Perfect matching Because n inputs, n outputs, we can always find a perfect matching. If we take out a middle box, and now have (n-1) inputs, (n-1) outputs.

Clos Network – Strictly nonblocking Clos Network is non-blocking in strict sense when m >= 2 n-1. n n x (2 n– 1) 1 1 1 2 2 2 . . r Each box has 2 n-1 output pins 2 n-1 r n n n Each box has 2 n-1 input pins

Clos Network – Proof by contradiction From i to j, we cannot make connection, e. g. from 1 to 2, we cannot make connection. Only time we can’t make a connection is if all paths are taken. Input i has taken n-1 signals, output j has taken n-1 signals. Thus, at most 2 n-2 paths are taken. However, we have 2 n-1 boxes for 2 n-1 distinct paths between i and j. So we will always have at least one path to go through.

Clos Network – Proof (cont) At most n-1 boxes taken from 1, and n-1 boxes taken from 2, so 2 n-2 boxes are taken. n n x (2 n– 1) 1 1 1 n-1 2 2 2 . . r 2 n-1 r n n n

Clos Network – More than three stages Clos Networks: Adding wires to reduce switches. Can we do better? Add even more wires to reduce number of switches? Yes! By increasing number of stages. Change middle stage box into another 3 -stage Clos Network, this gives us 5 -stage Clos Network. Can repeat this process! Replace with 3 -stage Clos Network n n 1 1 1 . . . r m r n n

Clos Network – More than three stages (cont) C(1) = N 2 switches (crossbar) C(3) = 6 N 3/2 – 3 N C(5) = 16 N 4/3 – 14 N + 3 N 2/3 C(7) = 36 N 5/4 – 46 N + 20 N 3/4 – 3 N 1/2 C(9) = 76 N 6/5 – 130 N + 86 N 4/5 – 26 N 3/5 + 3 N 2/5 (if N is huge, we want more levels)

Benes Network Start with Butterfly Network. What if we flip this, and repeat this network to the other side? This is Benes Network. . . 0 1 . . n-1 n 2 n 2 x 2 n inputs: (2 n+1)2 nx 4 switches N inputs: 2 N(2 log 2 N-1) switches