The all-pairs suffix-prefix (APSP) problem is a classical problem in string processing which has important applications in bioinformatics. Given a set \(\mathcal {S} = \{S_1, \ldots , S_k\}\) of k strings, the APSP problem asks one to compute the longest suffix of \(S_i\) that is a prefix of \(S_j\) for all \(k^2\) ordered pairs \(\langle S_i, S_j \rangle \) of strings in \(\mathcal {S}\) . In this paper, we consider the dynamic version of the APSP problem that allows for insertions of new strings to the set of strings. Our objective is, each time a new string \(S_i\) arrives to the current set \(\mathcal {S}_{i-1} = \{S_1, \ldots , S_{i-1}\}\) of \(i-1\) strings, to compute (1) the longest suffix of \(S_i\) that is a prefix of \(S_j\) and (2) the longest prefix of \(S_i\) that is a suffix of \(S_j\) for all \(1 \le j \le i\) . We propose an O(n)-space data structure which computes (1) and (2) in \(O(|S_i| \log \sigma + i)\) time for each new given string \(S_i\) , where n is the total length of the strings.