  This paper investigates the (quasi-)periodicity of a string when the string
is edited. A string $C$ is called a cover (as known as a quasi-period) of a
string $T$ if each character of $T$ lies within some occurrence of $C$. By
definition, a cover of $T$ must be a border of $T$; that is, it occurs both as
a prefix and as a suffix of $T$. In this paper, we focus on the changes in the
longest border and the shortest cover of a string when the string is edited
only once. We propose a data structure of size $O(n)$ that computes the longest
border and the shortest cover of the string in $O(\ell \log n)$ time after an
edit operation (either insertion, deletion, or substitution of some string) is
applied to the input string $T$ of length $n$, where $\ell$ is the length of
the string being inserted or substituted. The data structure can be constructed
in $O(n)$ time given string $T$.
