Inferring an indeterminate string from a prefix graph

An indeterminate string (or, more simply, just a string) x = x[1..n] on an alphabet Σ is a sequence of nonempty subsets of Σ. We say that x[i1] and x[i2] match (written x[i1] ≈ x[i2]) if and only if x[i1]∩x[i2] 6= ∅. A feasible array is an array y = y[1..n] of integers such that y[1] = n and for every i ∈ 2..n, y[i] ∈ 0..n−i+1. A prefix table of a string x is an array π = π[1..n] of integers such that, for every i ∈ 1..n, π[i] = j if and only if x[i..i+j−1] is the longest substring at position i of x that matches a prefix of x. It is known from [CRSW14] that every feasible array is a prefix table of some indetermintate string. A prefix graph P = Py is a labelled simple graph whose structure is determined by a feasible array y. In this paper we show, given a feasible array y, how to use Py to construct a lexicographically least indeterminate string on a minimum alphabet whose prefix table π = y.