Editing Nonblocking minimal spanning switch (section)

===Completely connected 3-layer switches===

The next approach was to break apart the crossbar switch into three layers of smaller crossbar switches. There would be an "input layer", a "middle layer" and an "output layer."  The smaller switches are less massive, more reliable, and generally easier to build, and therefore less expensive.

A telephone system only has to make a one-to-one connection.  Intuitively this seems to mean that the number of inputs and the number of outputs can always be equal in each subswitch, but intuition does not prove this can be done nor does it tell us how to do so.  Suppose we want to synthesize a 16 by 16 crossbar switch.  The design could have 4 subswitches on the input side, each with 4 inputs, for 16 total inputs.  Further, on the output side, we could also have 4 output subswitches, each with 4 outputs, for a total of 16 outputs.  It is desirable that the design use as few wires as possible, because wires cost real money.  The least possible number of wires that can connect two subswitches is a single wire.  So, each input subswitch will have a single wire to each middle subswitch.  Also, each middle subswitch will have a single wire to each output subswitch.

The question is how many middle subswitches are needed, and therefore how many total wires should connect the input layer to the middle layer.  Since telephone switches are symmetric (callers and callees are interchangeable), the same logic will apply to the output layer, and the middle subswitches will be "square", having the same number of inputs as outputs.

The number of middle subswitches depends on the algorithm used to allocate connection to them.  The basic algorithm for managing a three-layer switch is to search the middle subswitches for a middle subswitch that has unused wires to the needed input and output switches.  Once a connectible middle subswitch is found, connecting to the correct inputs and outputs in the input and output switches is trivial.

Theoretically, in the example, only four central switches are needed, each with exactly one connection to each input switch and one connection to each output switch.  This is called a "minimal spanning switch,"  and managing it was the holy grail of the Bell Labs' investigations.

However, a bit of work with a pencil and paper will show that it is easy to get such a minimal switch into conditions in which no single middle switch has a connection to both the needed input switch and the needed output switch.   It only takes four calls to partially block the switch.  If an input switch is half-full, it has connections via two middle switches.  If an output switch is also half full with connections from the other two middle switches, then there is no remaining middle switch which can provide a path between that input and output.

For this reason, a "simply connected nonblocking switch" 16x16 switch with four input subswitches and four output switches was thought to require 7  middle switches; in the worst case an almost-full input subswitch would use three middle switches, an almost-full output subswitch would use three different ones, and the seventh would be guaranteed to be free to make the last connection.  For this reason, sometimes this switch arrangement is called a "2''n''&minus;1 switch", where ''n'' is the number of input ports of the input subswitches.

The example is intentionally small, and in such a small example, the reorganization does not save many switches.  A 16×16 crossbar has 256 contacts, while a 16×16 minimal spanning switch has 4×4×4×3 = 192 contacts.

As the numbers get larger, the savings increase.  For example, a 10,000 line exchange would need 100 million contacts to implement a full crossbar.  But three layers of 100 100×100 subswitches would use only 300 10,000 contact subswitches, or 3 million contacts.

Those subswitches could in turn each be made of 3×10 10×10 crossbars, a total of 3000 contacts, making 900,000 for the whole exchange; that is a far smaller number than 100 million.