Incremental computation of the set of period sets

Abstract

Overlaps between words are crucial in many areas of computer science, such as code design, stringology, and bioinformatics. A self overlapping word is characterized by its periods and borders. A period of a word $u$ is the starting position of a suffix of $u$ that is also a prefix $u$, and such a suffix is called a border. Each word of length, say $n>0$, has a set of periods, but not all combinations of integers are sets of periods. Computing the period set of a word $u$ takes linear time in the length of $u$. We address the question of computing, the set, denoted $Γ_n$, of all period sets of words of length $n$. Although period sets have been characterized, there is no formula to compute the cardinality of $Γ_n$ (which is exponential in $n$), and the known dynamic programming algorithm to enumerate $Γ_n$ suffers from its space complexity. We present an incremental approach to compute $Γ_n$ from $Γ_{n-1}$, which reduces the space complexity, and then a constructive certification algorithm useful for verification purposes. The incremental approach defines a parental relation between sets in $Γ_{n-1}$ and $Γ_n$, enabling one to investigate the dynamics of period sets, and their intriguing statistical properties. Moreover, the period set of a word $u$ is the key for computing the absence probability of $u$ in random texts. Thus, knowing $Γ_n$ is useful to assess the significance of word statistics, such as the number of missing words in a random text.