Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 433 — #441
11.3. Network Designs 433
À À
2 n
B n
2 n 2 nC1
B n
new inputs
B nC1
new outputs
Figure 11.6
Beneš Net B nC1 with 2 nC1 inputs from two B n ’s.
In the constructor step, we construct B nC1 out of two B n nets connected to a
new set of 2 nC1 input switches and also a new set of 2 nC1 output switches. This is
illustrated in Figure 11.6.
The ith and 2 n C ith new input switches are each connected to the same two
switches: the ith input switches of each of two B n components for i D 1; : : : ; 2 n ,
exactly as in the Butterfly net. In addition, the ith and 2 n Cith new output switches
are connected to the same two switches, namely, to the ith output switches of each
of two B n components.
Now, B nC1 is laid out in columns of height 2 nC1 by adding two more columns
of switches to the columns in B n . So, the B nC1 switches are arrayed in 2.n C 1/
columns. The total number of switches is the number of columns times the height
2.n C 1/2 nC1 of the columns.
All paths in B nC1 from an input switch to an output are length 2.n C 1/ 1, and
the diameter of the Beneš net with 2 nC1 inputs is this length plus two because of
the two edges connecting to the terminals.
So Beneš has doubled the number of switches and the diameter, but by doing so
he has completely eliminated congestion problems! The proof of this fact relies on
a clever induction argument that we’ll come to in a moment. Let’s first see how the