Succinct online dictionary matching with improved worst-case guarantees

In the online dictionary matching problem the goal is to preprocess a set of patterns D = {P₁, . . . , P_d} over alphabet Σ, so that given an online text (one character at a time) we report all of the occurrences of patterns that are a suffix of the current text before the following character arrives. We introduce a succinct Aho-Corasick like data structure for the online dictionary matching problem. Our solution uses a new succinct representation for multi-labeled trees, in which each node has a set of labels from a universe of size λ. We consider lowest labeled ancestor (LLA) queries on multi-labeled trees, where given a node and a label we return the lowest proper ancestor of the node that has the queried label. In this paper we introduce a succinct representation of multi-labeled trees for λ = ω(1) that support LLA queries in O(log log λ) time. Using this representation of multi-labeled trees, we introduce a succinct data structure for the online dictionary matching problem when σ = ω(1). In this solution the worst case cost per character is O(log log σ + occ) time, where occ is the size of the current output. Moreover, the amortized cost per character is O(1 + occ) time.