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

Pointer-Machine Algorithms for Fully-Online Construction of Suffix Trees and DAWGs on Multiple Strings

“…Takagi et al [16] initiated the generalized problem of maintaining the suffix tree for a collection of strings in a fully-online manner, where a new character can be prepended to any string in the collection at any time. This fully-online scenario arises in real-time database systems e.g.…”

“…Takagi et al [16] also showed that Blumer et al's algorithm applied to a fully-online left-to-right DAWG construction requires at least Θ(N min(K, √ N )) work as well. They also showed how to maintain an implicit representation of the DAWG of O(N ) space which supports fully-online updates and simulates a DAWG edge traversal in O(log σ) time each.…”

“…When Weiner's algorithm is applied to fully-online right-to-left construction of STree(T ), the amortization in Analysis (c) does not work. Namely, it was shown by Takagi et al [16] that the number of redirected soft Weiner links is Θ(N min(K, √ N )) in the fully-online setting for multiple K strings. A simpler upper bound O(N K) immediately follows from an observation that the insertion of a new leaf for a string T i in T may also increase the depths of the leaves for all the other K − 1 strings T 1 , .…”

“…Takagi et al then obtained the aforementioned improved O(N min(K, √ N )) upper bound, and presented a lower bound instance that indeed requires Ω(N min(K, √ N )) work. It should also be noted that the original version of Weiner's algorithm that only maintains Boolean indicators for the existence of soft Weiner links, must also visit Θ(N min(K, √ N )) nodes [16].…”

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We deal with the problem of maintaining the suffix tree indexing structure for a fully-online collection of multiple strings, where a new character can be prepended to any string in the collection at any time. The only previously known algorithm for the problem, recently proposed by Takagi et al [Algorithmica 82(5): 1346-1377(2020], runs in O(N log σ) time and O(N ) space on the word RAM model, where N denotes the total length of the strings and σ denotes the alphabet size. Their algorithm makes heavy use of the nearest marked ancestor (NMA) data structure on semi-dynamic trees, that can answer queries and supports insertion of nodes in O(1) amortized time on the word RAM model.

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