Online recognition of dictionary with one gap

We formalize and examine the online Dictionary Recognition with One Gap problem (DROG) which is the following. Preprocess a dictionary D of d patterns each containing a special gap symbol that matches any string, so that given a text arriving online a character at a time, all patterns from D which are suffixes of the text that has arrived so far and have not been reported yet, are reported before the next character arrives. The gap symbols are associated with bounds determining possible lengths of matching strings. Online DROG captures the difficulty in a bottleneck procedure for cyber-security, as many digital signatures of viruses manifest themselves as patterns with a single gap. Following the work on the closely related online Dictionary Matching with One Gap problem (DMOG), we provide algorithms whose time cost depends linearly on δ(G_D), where G_D is a bipartite graph that captures the structure of D and δ(G_D) is the degeneracy of this graph. These algorithms are of practical interest since although δ(G_D) can be as large as d, and even larger if G_D is a multi-graph, it is typically a small constant in practice.