(a) Consider a 3-stage Clos network like the one we saw in class, with
N = 4, and k = n = m = 2, i.e. there is no expansion between the first
and second stage. Because k < 2n−1, the switch is not guaranteed to
be strictly non-blocking. Prove that it is in fact strictly non-blocking,
or find a counter-example to show that it isn’t.
(b) Joe Engineer believes that if we add connections according to the
following algorithm, a Clos network is strictly non-blocking when k =
Algorithm: When adding new connections, if possible, use the center
stage block that already has the most connections through it. If not
possible, try the second most congested center stage block; and so on,
until a block is found. Ties are broken at random; and once added,
connections are never rearranged.
(i) With this algorithm, is the switch in part (a) strictly non-blocking?
(ii) Is the switch strictly non-blocking for N > 4? (Hint: Try N = 9,
k = n = m = 3.)