Periodicity and Unbordered Segments of Words

In 1979 Ehrenfeucht and Silberger published an article [9] where the relationship between the length of a word and the maximum length of its unbordered factors (segments) was for the first time investigated. Periodicity and borderedness are two basic properties of words that play a rôle in many areas of computer science such as string searching algorithms [13, 3, 7], data compression [20, 6], and codes [2], which are classical examples, but also computational biology, e.g., sequence assembly [17] or superstrings [4], and serial data communications systems [5]. It is well known that these two word properties do not exist independently from each other. However, no clear relationship has been established so far, despite substantial recent progress in that area. We chose the title of Ehrenfeucht and Silberger’s paper for our essay to underline that this paper illustrates the way this line of research has evolved in the last 24 years. In Section 2 we give a historical overview on this line of research, its main results and conjectures so far. This will lead to the concept of Duval extensions which are introduced in Section 3. We conclude with Section 4. First, we shall introduce the main notations of this paper. We refer the reader to [14, 15] for more basic and general definitions. Consider a finite alphabet A of letters. Let A∗ denote the monoid of all finite words over A including the empty word, denoted by ε. Let w ∈ A∗. Then we can express w as a sequence of letters w(1)w(2) · · ·w(n) where w(i) ∈ A is a letter, for every 1 ≤ i ≤ n. Let 1 ≤ i ≤ j ≤ n, then w(i)w(i+1) · · ·w(j)is called a factor or segment of w. We denote the length n of w by |w|. Note, that |ε| = 0. A word w is called primitive if it cannot be factored such that w = u for some k ≥ 2. Let w = uv for some words u and v. Then vu is called conjugate of w. A nonempty word u is called a border of a word w, if w = uv = v′u for some words v and v′. We call w bordered, if it has a border that is shorter than w, otherwise w is called unbordered. Suppose w = uv, then u is called a prefix of w denoted by u ≤ w.