Indeterminate Strings, Prefix Arrays & Undirected Graphs

TL;DR

This paper explores the relationship between indeterminate strings, prefix arrays, and undirected graphs, establishing a graph model that characterizes feasible arrays and determines minimal alphabet sizes for indeterminate strings.

Contribution

It introduces a graph-based framework linking feasible arrays, indeterminate strings, and graph structures, providing bounds on alphabet size and a correspondence with simple graphs.

Findings

Every feasible array corresponds to some string, indeterminate or regular.

A one-to-one correspondence exists between labeled simple graphs and indeterminate strings.

The model provides a lower bound on the alphabet size for regular strings and determines minimal alphabet size for indeterminate strings.

Abstract

An integer array y = y[1..n] is said to be feasible if and only if y[1] = n and, for every i \in 2..n, i \le i+y[i] \le 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 Sigma; 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 |Sigma| 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 Sigma that have traditionally been studied.