New Bounds and Extended Relations Between Prefix Arrays, Border Arrays, Undirected Graphs, and Indeterminate Strings

We extend earlier works on the relation of prefix arrays of indeterminate strings to undirected graphs and border arrays. If integer array y is the prefix array for indeterminate string w, then we say w satisfies y. We use a graph theoretic approach to construct a string on a minimum alphabet size which satisfies a given prefix array. We relate the problem of finding a minimum alphabet size to finding edge clique covers of a particular graph, allowing us to bound the minimum alphabet size by n+n$n+\sqrt {n}$ for indeterminate strings, where n is the size of the prefix array. When we restrict ourselves to prefix arrays for partial words, we bound the minimum alphabet size by 2n$\left \lceil \sqrt {2n} \right \rceil $. Moreover, we show that this bound is tight up to a constant multiple by using Sidon sets. We also study the relationship between prefix arrays and border arrays. We give necessary and sufficient conditions for a border array and prefix array to be satisfied by the same indeterminate string. We show that the slowly-increasing property completely characterizes border arrays for indeterminate strings, whence there are exactly Cn distinct border arrays of size n for indeterminate strings (here Cn is the nth Catalan number). We give an algorithm to enumerate all prefix arrays for partial words of a given size, n. Our algorithm has a time complexity of n3 times the output size, that is, the number of valid prefix arrays for partial words of length n. We also bound the number of prefix arrays for partial words of a given size using Stirling numbers of the second kind.