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arXiv · 1406.3289

Indeterminate Strings, Prefix Arrays & Undirected Graphs

Manolis Christodoulakis·P. J. Ryan·W. F. Smyth·Shu Wang
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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.

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Manolis Christodoulakis, P. J. Ryan, W. F. Smyth, Shu Wang. 2014-06-12. Indeterminate Strings, Prefix Arrays & Undirected Graphs. https://arxiv.org/abs/1406.3289

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