Indeterminate Strings, Prefix Arrays & Undirected Graphs
Manolis Christodoulakis, P. J. Ryan, W. F. Smyth, Shu Wang

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…
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Taxonomy
TopicsRNA and protein synthesis mechanisms · semigroups and automata theory · DNA and Biological Computing
