Indeterminate strings, prefix arrays & undirected graphs

Abstract

An integer array y= y[1. .n] is said to be feasible if and only if y[1] = n and, for every i∈. 2. .n, i≤. i+ y[i]. ≤. n+. 1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet σ otherwise it is said to be regular. A feasible array y is said to be regular if and only if it is the prefix array of some regular string. We show using a graph model that every feasible array of integers is a prefix array of some (indeterminate or regular) string, and for regular strings corresponding to y, we use the model to provide a lower bound on the alphabet size. We show further that there is a 1-1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size σ of an indeterminate string x based on its associated graph Gx. Thus, in this sense, indeterminate strings are a more natural object of combinatorial interest than the strings on elements of σ that have traditionally been studied.