(a) radix-8 torus.
(b) folded implementation.
Figure 7.7: Decreasing the longest cable length in a torus (a) by “folding” it (b).
very regular cabling diagram is less difficult to correctly cable. Items (d) and (e) point out that torus
and mesh networks have both shorter links, and just a few different cable lengths to interconnect
nodes in different cabinets as shown in Figure 7.7 b. Keeping the cables short 3 eliminates the need for
repeaters or expensive optical links, and allows for high-speed serial point-to-point communication,
with signal rates in excess of 10 Gb/s commonplace over a few meters.
The flattened butterfly is another example of a topology with a lot of packaging locality. A
k -ary n -flat will have k switches co-located with each cabinet, where each switch has a minimum of
p ports (Equation 7.2 ).
p ≥ (n − 1 )(k − 1 ) + k
Each switch will use k electrical links to connect to its hosts, and another 2 k
1 links to interconnect
the other switches 4 (call this dimension 1) as shown in Figure 7.8 . The total number of electrical
links used (Equation 7.3 )
1 ) (7.3)
Assume that all inter-cabinet links for the remaining n − 2 dimensions will require optics since it is
likely they will be in excess of a few meters. Each switch connects k −
e = k + (k −
1 ports to cabinets in the same
row (dimension 2), and another k − 1 ports to cabinets in the same column (dimension 3) as shown
in Figure 7.8 . More generally, the number of optical links in a k -ary n -flat is given by Equation 7.4 .
o = (n −
2 )(k −
The fraction of electrical links in the network is given by Equation 7.5 . For example, an 8-ary 5-flat
with 32k nodes, will use about 42% low cost, electrical links.
k + (k −
f e =
3 We consider “short” distances as cables shorter than 5m.
4 These could be arranged as a stack of 1U switches co-located with the hosts, for instance.