A fast algorithm to generate Beckett-Gray codes: Extended Abstract

A Gray code is said to be cyclic if the last element and the first element in the sequence also differ by some pre-specified constant amount. The Gray code above demonstrates the cyclic property. It is well known that a cyclic Gray code of binary string corresponds to a Hamilton cycle in the n-dimensional hypercube. Figure 1 shows the 3-dimensional hypercube Q3 with the Hamilton cycle corresponding to the example Gray code above. In this paper, we are interested in finding a special type of Gray code named after Irish playright Samuel Beckett. One of his plays, “Quad”, had 4 actors and was divided into sixteen time periods. At the end of each time period, Beckett wished to have one of the four actors either entering or exiting the stage; he wished the play to begin and end with an empty stage and he wished each subset of actors to appear on stage exactly once. Observe that this problem is equivalent to finding a cyclic Gray code of length 4 binary strings. However, Beckett wanted an additional restriction on the scripting: the actor that leaves the stage must be the one who has currently been on stage for the longest time. If we apply this final restriction to cyclic Gray codes of binary strings we obtain what is known as a Beckett Gray code. The following is an example of a 5-bit Beckett-Gray code where the bit that has been a 1 for the longest time is underlined: