BANYAN NETWORKS JONATHAN CHAO CHEUK LAM

H. Jonathan Chao, Cheuk H. Lam, Eiji Oki Copyright 䊚 2001 John Wiley Sons, Inc. Ž . Ž . ISBNs: 0-471-00454-5 Hardback ; 0-471-22440-5 Electronic CHAPTER 5 BANYAN-BASED SWITCHES The very early theoretical work on multistage interconnection networks Ž . w x MINs was done in the context of circuit-switched telephone networks 7, 4 . The aim was to design a nonblocking multistage switch with number of crosspoints less than in a single-stage crossbar switch. After many such networks were studied and introduced for interconnecting multiple proces- sors and memories in parallel computer systems, several types of them, such w x as banyan and shuffle-exchange networks, were proposed 20, 12, 21 as switching fabrics because in them several cells can be routed in parallel and the switching function can be implemented regularly in hardware. In this chapter, we describe banyan-family switches, which have attracted many researchers, over more than two decades, for building interconnection networks. Section 5.1 classifies banyan-family switch architectures according to their nature and properties. Section 5.2 describes Batcher-sorting network switch architecture. Section 5.3 introduces output-contention resolution algo- rithms in banyan-family switches. Section 5.4 describes the Sunshine switch, which extends the Batcher᎐banyan switching architecture. Section 5.5 de- scribes some work on deflection routing over banyan-family networks. Sec- tion 5.6 introduces a self-routing copy network where the nonblocking property of banyan networks is generalized to support multicasting.

5.1 BANYAN NETWORKS

w x The banyan class of interconnection networks was originally defined in 9 . It has the property that there is exactly one path from any input to any output. 103 BANYAN-BASED SWITCHES 104 Ž . Ž . Fig. 5.1 Four different banyan-class networks: a shuffle-exchange omega net- Ž . Ž . Ž . work; b reverse shuffle-exchange network; c narrow-sense banyan network; d Ž . Ž . baseline network. We can see that a and b are isomorphic by interchanging the two shaded nodes. Figure 5.1 shows four networks belonging to this class: the shuffle-exchange Ž . network narrow-sense also called the omega network , the reverse shuffle- exchange network, the narrow-sense banyan network, and the baseline net- work. Ž . The principal common properties of these networks are: 1 they consist of 1 Ž . n s log N stages and Nr2 nodes per stage, 2 they have the self-routing 2 property that the unique n-bit destination address can be used to route a cell Ž . from any input to any output, each bit for one stage, and 3 their regularity and interconnection pattern are very attractive for VLSI implementation. Figure 5.2 shows a routing example in an 8 = 8 banyan network, where the bold lines indicate the routing paths. On the right hand side, the address of each output destination is labeled as a string of n bits, b ⭈⭈⭈ b . A cell’s 1 n destination address is encoded into the header of the cell. In the first stage, the most significant bit b is examined. If it is a 0, the cell will be forwarded 1 to the upper outgoing link; if it is a 1, the cell will be forwarded to the lower outgoing link. In the next stage, the next most significant bit b will be 2 1 A regular N = N network can also be constructed from identical b = b switching nodes in k stages, where N s b k . BANYAN NETWORKS 105 Fig. 5.2 An 8 = 8 banyan network. examined and the routing performed in the same manner. The internal blocking refers to the case where a cell is lost due to the contention on a link inside the network. Figure 5.3 shows an example of internal blocking in an 8 = 8 banyan network. However, the banyan network will be internally nonblocking if both conditions below are satisfied: 䢇 there is no idle input between any two active inputs; 䢇 the output addresses of the cells are in either ascending order or descending order. Fig. 5.3 Internal blocking in an 8 = 8 banyan network. BANYAN-BASED SWITCHES 106 Fig. 5.4 An example showing that the banyan network is nonblocking for sorted Ž . inputs. b Nonblocking sort᎐banyan network. See Figure 5.4. Suppose the banyan network is preceded by a network that concentrates the cells and sorts the cells according to their output destina- tions. The overall sort᎐banyan network will be internally nonblocking.

5.2 BATCHER-SORTING NETWORK