Shuffle-Exchange Network with Deflection Routing

DEFLECTION ROUTING 117 Fig. 5.13 An 8 = 8 shuffle-exchange network.

5.5.2 Shuffle-Exchange Network with Deflection Routing

Ž . w x Consider an N = N shuffle᎐exchange network SN 15 with n s log N 2 stages, each consisting of Nr2 2 = 2 switch elements. Figure 5.13 shows an Ž . 8 = 8 SN. Switch nodes in each stage are labeled by an n y 1 -bit binary Ž . number from top to bottom. The upper input output of a node is labeled 0, Ž . and the lower input output is labeled 1. A cell will be forwarded to output 0 Ž . Ž . 1 at stage i if the ith most significant bit of its destination address is 0 1 . The interconnection between two consecutive stages is called shuffle ex- Ž . change. The output a of node X s a a . . . a is connected to the input n 1 2 ny1 Ž . a of node Y s a a . . . a of the subsequent stage. The link between node 1 2 3 n ² : X and node Y is labeled a , a . The path of a cell from input to output is n 1 completely determined by the source address S s s . . . s and the destina- 1 n tion address D s d ⭈⭈⭈ d . It can be expressed symbolically as follows: 1 n S s s ⭈⭈⭈ s 1 n ² : ² : y , s d , s 1 1 2 6 6 s . . . s s . . . s d Ž . Ž . 2 n 3 n 1 ² : ² : d , s d , s 2 3 iy1 i 6 6 ⭈⭈⭈ s . . . s d . . . d Ž . iq1 n 1 iy1 ² : ² : d , s d , s i iq1 ny1 n 6 6 ⭈⭈⭈ d ⭈⭈⭈ d Ž . 1 ny1 ² : d , 0 n 6 d . . . d s D. 1 n BANYAN-BASED SWITCHES 118 The node sequence along the path is embedded in the binary string Ž . s . . . s d . . . d , represented by an n y 1 -bit window moving one bit per 2 n 1 ny1 stage from left to right. Ž . The state of a cell traveling in the SN can be represented by a pair R, X , where R is its current routing tag and X is the label of the node that the cell Ž . resides. At the first stage, the cell is in state d . . . d , s . . . s . The state n 1 2 n transition is determined by the self-routing algorithm as follows: exchange 6 r . . . r , x x . . . x r . . . r , x x . . . x Ž . Ž . 1 k 1 2 ny1 1 ky1 1 2 ny1 input label x output label r n k shuffle 6 r . . . r , x . . . x r . Ž . 1 ky1 2 ny1 k ² : r , x k 1 Notice that the routing bit used in the switching node is removed from the routing tag after each stage, before the node label is shuffled to the next Ž . stage. Finally, the cell will reach the state d d . . . d , from which the n 1 ny1 following 2 = 2 element will switch the cell to the destination. When a contention occurs at a switch node, one of the cells will be successfully routed while the other one will be deflected to the wrong output. As a result, only the nondeflected cells can ever reach their desired outputs. Ž . The deflected cells can restart routing with routing tag reset to d . . . d n 1 again at the deflection point, and if the SN is extended to consist of more than n stages, those deflected cells can reach the destination at later stages. As some cells will reach their destinations after fewer stages than others, a multiplexer is needed to collect cells that reach physical links of the same logical address at different stages. A cell will eventually reach its destination address with good probability provided that the number of stages L is sufficiently large. If it cannot reach the destination after L stages, it is considered lost.

5.5.3 Dual Shuffle-Exchange Network with Error-Correcting Routing