String shuffle: Circuits and graphs

We show that shuffle, the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in AC0, but it is in AC1. The fact that shuffle is not in AC0 is shown by a reduction of parity to shuffle and invoking the seminal result of Furst et al., while the fact that it is in AC1 is implicit in the results of Mansfield. Together, the two results provide a lower and upper bound on the …