Fully-Online Suffix Tree and Directed Acyclic Word Graph Construction for Multiple Texts

Takuya Takagi, Shunsuke Inenaga, Hiroki Arimura, Dany Breslauer, Diptarama Hendrian

Abstract: We consider the construction of the suffix tree and the directed acyclic word graph (DAWG) indexing data structures for a collection \(\mathcal {T}\) of texts, where a new symbol may be appended to any text in \(\mathcal {T} = \{T_1, \ldots , T_K\}\), at any time. This fully-online scenario, which arises in dynamically indexing multi-sensor data, is a natural generalization of the long solved semi-online text indexing problem, where texts \(T_1, \ldots , T_{k}\) are permanently fixed before the next text \(T_{k+1}\) is processed for each k (\(1 \le k < K\)). We first show that a direct application of Weiner’s right-to-left online construction for the suffix tree of a single text to fully-online multiple texts requires superlinear time. This also means that Blumer et al.’s left-to-right online construction for the DAWG of a single text requires superlinear time in the fully-online setting. We then present our fully-online versions of these algorithms that run in \(O(N \log \sigma )\) time and O(N) space, where N is the total length of the texts in \(\mathcal {T}\) and \(\sigma \) is their alphabet size. We then show how to extend Ukkonen’s left-to-right online suffix tree construction to fully-online multiple strings, with the aid of Weiner’s suffix tree for the reversed texts.