Computing minimum length representations of sets of words of uniform length

Motivated by text compression, the problem of representing sets of words of uniform length by partial words, i.e., sequences that may have some wildcard characters or holes, was recently considered and shown to be in PP. Polynomial-time algorithms that construct representations were described using graph theoretical approaches. As more holes are allowed, representations shrink, and if representation is given, the set can be reconstructed. We further study this problem by determining, for a binary alphabet, the largest possible value of the size of a set of partial words that is important in deciding the representability of a given set S   of words of uniform length. This largest value, surprisingly, is Σi=0|S|−12χ(i) where χ(i)χ(i) is the number of ones in the binary representation of i, a well-studied digital sum, and it is achieved when the cardinality of S   is a power of two. We show that circular representability is in PP and that unlike non-circular representability, it is easy to decide. We also consider the problem of computing minimum length representation (circular) total words, those without holes, and reduce it to a cost/flow network problem.