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 G x . 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.

Inferring an indeterminate string from a prefix graph

“…Similarly, the requirement to achieve a lexicographically least solution leads to a recurring dependence on alphabet size σ that expresses itself in the time complexity. Even though it may be true that σ ≤ n, nevertheless it seems clear that σ can be much larger than in the regular case, where it has been shown [7,6] that σ ≤ log 2 n .…”

“…Following [6], for a given feasible array y = y[1..n], we define a corresponding graph P = P y , on which our algorithm will be based: Definition 4. Let P = (V, E) be a labelled graph with vertex set V = {1, 2, .…”

“…The time requirement is Θ(|E + |), thus O(n 2 ) in the worst case, since E + can contain Θ(n 2 ) edges [6].…”

“…In this paper we study strings in their full generality, hence the following definitions: The prefix table is an important data structure for strings: it identifies all the borders, hence all the periods, of every prefix of x [6]. It was originally introduced to facilitate the computation of repetitions in regular strings [16], see also [20]; and for regular strings, prefix table and border array are equivalent, since each can be computed from the other in linear time [5].…”

Constructing an indeterminate string from its associated graph

“…In this thesis, we seek to extend the connections between graph theory and stringology explored in [5]. We consider a strin g x = x[1.…”